Dumitru Băleanu

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New fractional derivatives with nonlocal and non-singular kernel: Theory and application to heat transfer model

2016-01-01

In this manuscript we proposed a new fractional derivative with non-local and no-singular kernel. We presented some useful properties of the new derivative and applied it to solve the fractional … In this manuscript we proposed a new fractional derivative with non-local and no-singular kernel. We presented some useful properties of the new derivative and applied it to solve the fractional heat transfer model.

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A new collection of real world applications of fractional calculus in science and engineering

2018-04-24

HongGuang Sun , Yong Zhang , Dumitru Băleanu +2 more

Fractional Calculus

2011-11-20

Dumitru Băleanu , Kai Diethelm , Enrico Scalas +1 more

New Trends in Nanotechnology and Fractional Calculus Applications

2009-12-10

Dumitru Băleanu , Zi̇ya B. Güvenç , J. A. Tenreiro Machado

A new study on the mathematical modelling of human liver with Caputo–Fabrizio fractional derivative

2020-02-25

Dumitru Băleanu , Amin Jajarmi , Hakimeh Mohammadi +1 more

New properties of conformable derivative

2015-12-03

Abdon Atangana , Dumitru Băleanu , Ahmed Alsaedi

Abstract Recently, the conformable derivative and its properties have been introduced. In this work we have investigated in more detail some new properties of this derivative and we have proved … Abstract Recently, the conformable derivative and its properties have been introduced. In this work we have investigated in more detail some new properties of this derivative and we have proved some useful related theorems. Also, some new definitions have been introduced.

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Discrete fractional logistic map and its chaos

2013-09-19

Guo–Cheng Wu , Dumitru Băleanu

Anomalous diffusion expressed through fractional order differential operators in the Bloch–Torrey equation

2007-12-05

Richard L. Magin , Osama Abdullah , Dumitru Băleanu +1 more

On a new class of fractional operators

2017-08-22

Fahd Jarad , Ekin Uğurlu , Thabet Abdeljawad +1 more

This manuscript is based on the standard fractional calculus iteration procedure on conformable derivatives. We introduce new fractional integration and differentiation operators. We define spaces and present some theorems related … This manuscript is based on the standard fractional calculus iteration procedure on conformable derivatives. We introduce new fractional integration and differentiation operators. We define spaces and present some theorems related to these operators.

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Caputo-type modification of the Hadamard fractional derivatives

2012-08-10

Fahd Jarad , Thabet Abdeljawad , Dumitru Băleanu

Abstract Generalization of fractional differential operators was subjected to an intense debate in the last few years in order to contribute to a deep understanding of the behavior of complex … Abstract Generalization of fractional differential operators was subjected to an intense debate in the last few years in order to contribute to a deep understanding of the behavior of complex systems with memory effect. In this article, a Caputo-type modification of Hadamard fractional derivatives is introduced. The properties of the modified derivatives are studied.

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Fractional Calculus

2012-01-01

Dumitru Băleanu , Kai Diethelm , Enrico Scalas +1 more

Stability analysis of Caputo fractional-order nonlinear systems revisited

2011-08-12

Hadi Delavari , Dumitru Băleanu , Jalil Sadati

A new fractional model and optimal control of a tumor-immune surveillance with non-singular derivative operator

2019-08-01

Dumitru Băleanu , Amin Jajarmi , Samaneh Sadat Sajjadi +1 more

In this paper, we present a new fractional-order mathematical model for a tumor-immune surveillance mechanism. We analyze the interactions between various tumor cell populations and immune system via a system … In this paper, we present a new fractional-order mathematical model for a tumor-immune surveillance mechanism. We analyze the interactions between various tumor cell populations and immune system via a system of fractional differential equations (FDEs). An efficient numerical procedure is suggested to solve these FDEs by considering singular and nonsingular derivative operators. An optimal control strategy for investigating the effect of chemotherapy treatment on the proposed fractional model is also provided. Simulation results show that the new presented model based on the fractional operator with Mittag–Leffler kernel represents various asymptomatic behaviors that tracks the real data more accurately than the other fractional- and integer-order models. Numerical simulations also verify the efficiency of the proposed optimal control strategy and show that the growth of the naive tumor cell population is successfully declined.

On some new properties of fractional derivatives with Mittag-Leffler kernel

2017-12-06

Dumitru Băleanu , Arran Fernandez

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A Hamiltonian Formulation and a Direct Numerical Scheme for Fractional Optimal Control Problems

2007-09-01

Om P. Agrawal , Dumitru Băleanu

This paper deals with a direct numerical technique for Fractional Optimal Control Problems (FOCPs). In this paper, we formulate the FOCPs in terms of Riemann—Liouville Fractional Derivatives (RLFDs). It is … This paper deals with a direct numerical technique for Fractional Optimal Control Problems (FOCPs). In this paper, we formulate the FOCPs in terms of Riemann—Liouville Fractional Derivatives (RLFDs). It is demonstrated that right RLFDs automatically arise in the formulation even when the dynamics of the system is described using left RLFDs only. For numerical computation, the FDs are approximated using the Grunwald—Letnikov definition. This leads to a set of algebraic equations that can be solved using numerical techniques. Two examples, one time-invariant and the other time-variant, are considered to demonstrate the effectiveness of the formulation. Results show that as the order of the derivative approaches an integer value, these formulations lead to solutions for integer order system. The approach requires dividing of the entire time domain into several sub-domains. Further, as the sizes of the sub-domains are reduced, the solutions converge to unique solutions. However, the convergence is slow. A scheme that improves the convergence rate will be considered in a future paper. Other issues to be considered in the future include formulations using other types of derivatives, nonlinear and stochastic fractional optimal controls, existence and uniqueness of the solutions, and the error analysis.

Fractal heat conduction problem solved by local fractional variation iteration method

2012-11-29

Xiao‐Jun Yang , Dumitru Băleanu

This paper points out a novel local fractional variational iteration method for processing the local fractional heat conduction equation arising in fractal heat transfer. This paper points out a novel local fractional variational iteration method for processing the local fractional heat conduction equation arising in fractal heat transfer.

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Integration by parts and its applications of a new nonlocal fractional derivative with Mittag-Leffler nonsingular kernel

2017-03-13

Thabet Abdeljawad , Dumitru Băleanu

In this manuscript we define the right fractional derivative and its corresponding right fractional integral for the newly suggested nonlocal fractional derivative with Mittag-Leffler kernel.Then, we obtain the related integration … In this manuscript we define the right fractional derivative and its corresponding right fractional integral for the newly suggested nonlocal fractional derivative with Mittag-Leffler kernel.Then, we obtain the related integration by parts formula.We use the Q-operator to confirm our results.The related Euler-Lagrange equations are reported and one illustrative example is discussed.

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Caputo-Fabrizio Derivative Applied to Groundwater Flow within Confined Aquifer

2016-03-16

Abdon Atangana , Dumitru Băleanu

The model of the movement of subsurface water via the geological formation called aquifer was extended using a newly proposed derivative with fractional order. An alternative derivative to that of … The model of the movement of subsurface water via the geological formation called aquifer was extended using a newly proposed derivative with fractional order. An alternative derivative to that of Caputo-Fabrizio with fractional order was presented. The relationship between both derivatives was presented. The new equation was solved analytically using some integral transforms. The exact solution is therefore compared to experimental data obtained from the settlement of the University of the Free State in South Africa. The numerical simulation shows the agreement of the experimental data with an analytical solution for some values of fractional order.

On the generalized fractional derivatives and their Caputo modification

2017-05-12

Fahd Jarad , Thabet Abdeljawad , Dumitru Băleanu

In this manuscript, we define the generalized fractional derivative onWe present some of the properties of generalized fractional derivatives of these functions and then we define their Caputo version. In this manuscript, we define the generalized fractional derivative onWe present some of the properties of generalized fractional derivatives of these functions and then we define their Caputo version.

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Analysis of time-fractional hunter-saxton equation: a model of neumatic liquid crystal

2016-01-01

Abdon Atangana , Dumitru Băleanu , Ahmed Alsaedi

Abstract In this work, a theoretical study of diffusion of neumatic liquid crystals was done using the concept of fractional order derivative. This version of fractional derivative is very easy … Abstract In this work, a theoretical study of diffusion of neumatic liquid crystals was done using the concept of fractional order derivative. This version of fractional derivative is very easy to handle and obey to almost all the properties satisfied by the conventional Newtonian concept of derivative. The mathematical equation underpinning this physical phenomenon was solved analytically via the so-called homotopy decomposition method. In order to show the accuracy of this iteration method, we constructed a Hilbert space in which we proved its stability for the time-fractional Hunder-Saxton equation.

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On a Fractional Operator Combining Proportional and Classical Differintegrals

2020-03-06

Dumitru Băleanu , Arran Fernandez , Ali Akgül

The Caputo fractional derivative has been one of the most useful operators for modelling non-local behaviours by fractional differential equations. It is defined, for a differentiable function f ( t … The Caputo fractional derivative has been one of the most useful operators for modelling non-local behaviours by fractional differential equations. It is defined, for a differentiable function f ( t ) , by a fractional integral operator applied to the derivative f ′ ( t ) . We define a new fractional operator by substituting for this f ′ ( t ) a more general proportional derivative. This new operator can also be written as a Riemann–Liouville integral of a proportional derivative, or in some important special cases as a linear combination of a Riemann–Liouville integral and a Caputo derivative. We then conduct some analysis of the new definition: constructing its inverse operator and Laplace transform, solving some fractional differential equations using it, and linking it with a recently described bivariate Mittag-Leffler function.

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A hybrid Caputo fractional modeling for thermostat with hybrid boundary value conditions

2020-03-24

Dumitru Băleanu , Sina Etemad , Shahram Rezapour

Abstract We provide an extension for the second-order differential equation of a thermostat model to the fractional hybrid equation and inclusion versions. We consider boundary value conditions of this problem … Abstract We provide an extension for the second-order differential equation of a thermostat model to the fractional hybrid equation and inclusion versions. We consider boundary value conditions of this problem in the form of the hybrid conditions. To prove the existence of solutions for our hybrid fractional thermostat equation and inclusion versions, we apply the well-known Dhage fixed point theorems for single-valued and set-valued maps. Finally, we give two examples to illustrate our main results.

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Lagrangian Formulation of Classical Fields within Riemann-Liouville Fractional Derivatives

2005-01-01

Dumitru Băleanu , Sami I. Muslih

The classical fields with fractional derivatives are investigated by using the fractional Lagrangian formulation. The fractional Euler-Lagrange equations were obtained and two examples were studied. The classical fields with fractional derivatives are investigated by using the fractional Lagrangian formulation. The fractional Euler-Lagrange equations were obtained and two examples were studied.

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Analysis of the model of HIV-1 infection of $CD4^{+}$ T-cell with a new approach of fractional derivative

2020-02-13

Dumitru Băleanu , Hakimeh Mohammadi , Shahram Rezapour

Abstract By using the fractional Caputo–Fabrizio derivative, we investigate a new version for the mathematical model of HIV. In this way, we review the existence and uniqueness of the solution … Abstract By using the fractional Caputo–Fabrizio derivative, we investigate a new version for the mathematical model of HIV. In this way, we review the existence and uniqueness of the solution for the model by using fixed point theory. We solve the equation by a combination of the Laplace transform and homotopy analysis method. Finally, we provide some numerical analytics and comparisons of the results.

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On Fractional Derivatives with Exponential Kernel and their Discrete Versions

2017-08-01

Thabet Abdeljawad , Dumitru Băleanu

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On exact traveling-wave solutions for local fractional Korteweg-de Vries equation

2016-08-01

Xiao‐Jun Yang , J. A. Tenreiro Machado , Dumitru Băleanu +1 more

This paper investigates the Korteweg-de Vries equation within the scope of the local fractional derivative formulation. The exact traveling wave solutions of non-differentiable type with the generalized functions defined on … This paper investigates the Korteweg-de Vries equation within the scope of the local fractional derivative formulation. The exact traveling wave solutions of non-differentiable type with the generalized functions defined on Cantor sets are analyzed. The results for the non-differentiable solutions when fractal dimension is 1 are also discussed. It is shown that the exact solutions for the local fractional Korteweg-de Vries equation characterize the fractal wave on shallow water surfaces.

New exact solutions of Burgers’ type equations with conformable derivative

2016-07-07

Yücel Çenesiz , Dumitru Băleanu , Ali Kurt +1 more

In this paper, the new exact solutions for some nonlinear partial differential equations are obtained within the newly established conformable derivative. We use the first integral method to establish the … In this paper, the new exact solutions for some nonlinear partial differential equations are obtained within the newly established conformable derivative. We use the first integral method to establish the exact solutions for time-fractional Burgers' equation, modified Burgers' equation, and Burgers–Korteweg–de Vries equation. We report that this method is efficient and it can be successfully used to obtain new analytical solutions of nonlinear FDEs.

Chaos analysis and asymptotic stability of generalized Caputo fractional differential equations

2017-03-18

Dumitru Băleanu , Guo–Cheng Wu , Shengda Zeng

Fractional calculus: A survey of useful formulas

2013-09-01

Duarte Valério , Juan J. Trujillo , M. Rivero +2 more

Discrete fractional differences with nonsingular discrete Mittag-Leffler kernels

2016-09-05

Thabet Abdeljawad , Dumitru Băleanu

In this manuscript we propose the discrete versions for the recently introduced fractional derivatives with nonsingular Mittag-Leffler function. The properties of such fractional differences are studied and the discrete integration … In this manuscript we propose the discrete versions for the recently introduced fractional derivatives with nonsingular Mittag-Leffler function. The properties of such fractional differences are studied and the discrete integration by parts formulas are proved. Then a discrete variational problem is considered with an illustrative example. Finally, some more tools for these derivatives and their discrete versions have been obtained.

On Caputo modification of the Hadamard fractional derivatives

2014-01-07

Yusuf Y. Gambo , Fahd Jarad , Dumitru Băleanu +1 more

Abstract This paper is devoted to the study of Caputo modification of the Hadamard fractional derivatives. From here and after, by Caputo-Hadamard derivative, we refer to this modified fractional derivative … Abstract This paper is devoted to the study of Caputo modification of the Hadamard fractional derivatives. From here and after, by Caputo-Hadamard derivative, we refer to this modified fractional derivative (Jarad et al. in Adv. Differ. Equ. 2012:142, 2012, p.7). We present the generalization of the fundamental theorem of fractional calculus (FTFC) in the Caputo-Hadamard setting. Also, several new related results are presented.

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Controllability of fractional evolution nonlocal impulsive quasilinear delay integro-differential systems

2011-04-20

Amar Debbouche , Dumitru Băleanu

New Derivatives on the Fractal Subset of Real-Line

2016-01-29

Alireza Khalili Golmankhaneh , Dumitru Băleanu

In this manuscript we introduced the generalized fractional Riemann-Liouville and Caputo like derivative for functions defined on fractal sets. The Gamma, Mittag-Leffler and Beta functions were defined on the fractal … In this manuscript we introduced the generalized fractional Riemann-Liouville and Caputo like derivative for functions defined on fractal sets. The Gamma, Mittag-Leffler and Beta functions were defined on the fractal sets. The non-local Laplace transformation is given and applied for solving linear and non-linear fractal equations. The advantage of using these new nonlocal derivatives on fractals subset of real-line lies in the fact that they are used for better modelling of processes with memory effect.

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Some existence results on nonlinear fractional differential equations

2013-04-02

Dumitru Băleanu , Shahram Rezapour , Hakimeh Mohammadi

In this paper, by using fixed-point methods, we study the existence and uniqueness of a solution for the nonlinear fractional differential equation boundary-value problem D(α)u(t)=f(t,u(t)) with a Riemann-Liouville fractional derivative … In this paper, by using fixed-point methods, we study the existence and uniqueness of a solution for the nonlinear fractional differential equation boundary-value problem D(α)u(t)=f(t,u(t)) with a Riemann-Liouville fractional derivative via the different boundary-value problems u(0)=u(T), and the three-point boundary condition u(0)=β(1)u(η) and u(T)=β(2)u(η), where T>0, t∈I=[0,T], 0<α<1, 0<η<T, 0<β(1)<β(2)<1.

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A new fractional exothermic reactions model having constant heat source in porous media with power, exponential and Mittag-Leffler laws

2019-05-03

Devendra Kumar , Jagdev Singh , Kumud Tanwar +1 more

On Fractional Operators and Their Classifications

2019-09-08

Dumitru Băleanu , Arran Fernandez

Fractional calculus dates its inception to a correspondence between Leibniz and L’Hopital in 1695, when Leibniz described “paradoxes” and predicted that “one day useful consequences will be drawn” from them. … Fractional calculus dates its inception to a correspondence between Leibniz and L’Hopital in 1695, when Leibniz described “paradoxes” and predicted that “one day useful consequences will be drawn” from them. In today’s world, the study of non-integer orders of differentiation has become a thriving field of research, not only in mathematics but also in other parts of science such as physics, biology, and engineering: many of the “useful consequences” predicted by Leibniz have been discovered. However, the field has grown so far that researchers cannot yet agree on what a “fractional derivative” can be. In this manuscript, we suggest and justify the idea of classification of fractional calculus into distinct classes of operators.

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A spectral tau algorithm based on Jacobi operational matrix for numerical solution of time fractional diffusion-wave equations

2014-04-03

A. H. Bhrawy , E. H. Doha , Dumitru Băleanu +1 more

Two analytical methods for time-fractional nonlinear coupled Boussinesq–Burger’s equations arise in propagation of shallow water waves

2016-03-11

Sunil Kumar , Amit Kumar , Dumitru Băleanu

On the analysis of vibration equation involving a fractional derivative with Mittag‐Leffler law

2019-09-13

Devendra Kumar , Jagdev Singh , Dumitru Băleanu

The present article deals with a fractional extension of the vibration equation for very large membranes with distinct special cases. The fractional derivative is considered in Atangana‐Baleanu sense. A numerical … The present article deals with a fractional extension of the vibration equation for very large membranes with distinct special cases. The fractional derivative is considered in Atangana‐Baleanu sense. A numerical algorithm based on homotopic technique is employed to examine the fractional vibration equation. The stability analysis is conducted for the suggested scheme. The maple software package is utilized for numerical simulation. In order to illustrate the effects of space, time, and order of Atangana‐Baleanu derivative on the displacement, the outcomes of this study are demonstrated graphically. The results revel that the Atangana‐Baleanu fractional derivative is very efficient in describing vibrations in large membranes.

Analysis of regularized long-wave equation associated with a new fractional operator with Mittag-Leffler type kernel

2017-10-19

Devendra Kumar , Jagdev Singh , Dumitru Băleanu +1 more

EXACT TRAVELING-WAVE SOLUTION FOR LOCAL FRACTIONAL BOUSSINESQ EQUATION IN FRACTAL DOMAIN

2017-06-02

Xiao‐Jun Yang , J. A. Tenreiro Machado , Dumitru Băleanu

The new Boussinesq-type model in a fractal domain is derived based on the formulation of the local fractional derivative. The novel traveling wave transform of the non-differentiable type is adopted … The new Boussinesq-type model in a fractal domain is derived based on the formulation of the local fractional derivative. The novel traveling wave transform of the non-differentiable type is adopted to convert the local fractional Boussinesq equation into a nonlinear local fractional ODE. The exact traveling wave solution is also obtained with aid of the non-differentiable graph. The proposed method, involving the fractal special functions, is efficient for finding the exact solutions of the nonlinear PDEs in fractal domains.

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A fractional differential equation model for the COVID-19 transmission by using the Caputo–Fabrizio derivative

2020-06-18

Dumitru Băleanu , Hakimeh Mohammadi , Shahram Rezapour

We present a fractional-order model for the COVID-19 transmission with Caputo-Fabrizio derivative. Using the homotopy analysis transform method (HATM), which combines the method of homotopy analysis and Laplace transform, we … We present a fractional-order model for the COVID-19 transmission with Caputo-Fabrizio derivative. Using the homotopy analysis transform method (HATM), which combines the method of homotopy analysis and Laplace transform, we solve the problem and give approximate solution in convergent series. We prove the existence of a unique solution and the stability of the iteration approach by using fixed point theory. We also present numerical results to simulate virus transmission and compare the results with those of the Caputo derivative.

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A new fractional analysis on the interaction of HIV with<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"><mml:msup><mml:mtext>CD4</mml:mtext><mml:mo>+</mml:mo></mml:msup></mml:math>T-cells

2018-06-23

Amin Jajarmi , Dumitru Băleanu

A Central Difference Numerical Scheme for Fractional Optimal Control Problems

2009-02-04

Dumitru Băleanu , Özlem Defterli , Om P. Agrawal

This paper presents a modified numerical scheme for a class of fractional optimal control problems where a fractional derivative (FD) is defined in the Riemann—Liouville sense. In this scheme, the … This paper presents a modified numerical scheme for a class of fractional optimal control problems where a fractional derivative (FD) is defined in the Riemann—Liouville sense. In this scheme, the entire time domain is divided into several sub-domains, and a FD at a time node point is approximated using a modified Grünwald—Letnikov approach. For the first-order derivative, the proposed modified Grünwald— Letnikov definition leads to a central difference scheme. When the approximations are substituted into the fractional optimal control equations, it leads to a set of algebraic equations which are solved using a direct numerical technique. Two examples, one time-invariant and the other time-variant, are considered to study the performance of the numerical scheme. Results show that 1) as the order of the derivative approaches an integer value, these formulations lead to solutions for the integer-order system, and 2) as the sizes of the sub-domains are reduced, the solutions converge. It is hoped that the present scheme would lead to stable numerical methods for fractional differential equations and optimal control problems.

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Fractional modeling of blood ethanol concentration system with real data application

2019-01-01

Sania Qureshi , Abdullahi Yusuf , Asif Ali Shaikh +2 more

In this study, a physical system called the blood ethanol concentration model has been investigated in its fractional (non-integer) order version. The three most commonly used fractional operators with singular … In this study, a physical system called the blood ethanol concentration model has been investigated in its fractional (non-integer) order version. The three most commonly used fractional operators with singular (Caputo) and non-singular (Atangana-Baleanu fractional derivative in the Caputo sense—ABC and the Caputo-Fabrizio—CF) kernels have been used to fractionalize the model, whereas during the process of fractionalization, the dimensional consistency for each of the equations in the model has been maintained. The Laplace transform technique is used to determine the exact solution of the model in all three cases, whereas its parameters are fitted through the least-squares error minimization technique. It is shown that the fractional versions of the model based upon the Caputo and ABC operators estimate the real data comparatively better than the original integer order model, whereas the CF yields the results equivalent to the results obtained from the integer-order model. The computation of the sum of squared residuals is carried out to show the performance of the models along with some graphical illustrations.

The Hamilton formalism with fractional derivatives

2006-06-07

Eqab M. Rabei , Khaled I. Nawafleh , R. S. Hijjawi +2 more

A new fractional modelling and control strategy for the outbreak of dengue fever

2019-08-27

Amin Jajarmi , Sadia Arshad , Dumitru Băleanu

An Efficient Numerical Method for Fractional SIR Epidemic Model of Infectious Disease by Using Bernstein Wavelets

2020-04-10

Sunil Kumar , Ali Ahmadian , Ranbir Kumar +4 more

In this paper, the operational matrix based on Bernstein wavelets is presented for solving fractional SIR model with unknown parameters. The SIR model is a system of differential equations that … In this paper, the operational matrix based on Bernstein wavelets is presented for solving fractional SIR model with unknown parameters. The SIR model is a system of differential equations that arises in medical science to study epidemiology and medical care for the injured. Operational matrices merged with the collocation method are used to convert fractional-order problems into algebraic equations. The Adams–Bashforth–Moulton predictor correcter scheme is also discussed for solving the same. We have compared the solutions with the Adams–Bashforth predictor correcter scheme for the accuracy and applicability of the Bernstein wavelet method. The convergence analysis of the Bernstein wavelet has been also discussed for the validity of the method.

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On the global existence of solutions to a class of fractional differential equations

2009-08-29

Dumitru Băleanu , Octavian G. Mustafa

A new and efficient numerical method for the fractional modeling and optimal control of diabetes and tuberculosis co-existence

2019-09-01

Amin Jajarmi , Behzad Ghanbari , Dumitru Băleanu

The main objective of this research is to investigate a new fractional mathematical model involving a nonsingular derivative operator to discuss the clinical implications of diabetes and tuberculosis coexistence. The … The main objective of this research is to investigate a new fractional mathematical model involving a nonsingular derivative operator to discuss the clinical implications of diabetes and tuberculosis coexistence. The new model involves two distinct populations, diabetics and nondiabetics, while each of them consists of seven tuberculosis states: susceptible, fast and slow latent, actively tuberculosis infection, recovered, fast latent after reinfection, and drug-resistant. The fractional operator is also considered a recently introduced one with Mittag-Leffler nonsingular kernel. The basic properties of the new model including non-negative and bounded solution, invariant region, and equilibrium points are discussed thoroughly. To solve and simulate the proposed model, a new and efficient numerical method is established based on the product-integration rule. Numerical simulations are presented, and some discussions are given from the mathematical and biological viewpoints. Next, an optimal control problem is defined for the new model by introducing four control variables reducing the number of infected individuals. For the control problem, the necessary and sufficient conditions are derived and numerical simulations are given to verify the theoretical analysis.

A New Analysis of the Ab-Fractal-Fractional Operators Using the Generalized Gamma Function in Geometric Patterns

2025-06-03

Rabha W. Ibrahim , Dumitru Băleanu , Yeliz Karaca +1 more

Fractional Systems with Multi-Parameters Fractional Derivatives

2025-05-28

Sami I. Muslih , Om P. Agrawal , Dumitru Băleanu

Generalized Hat Functions for Fractional Delay Optimal Control Problems With ψ$$ \psi $$‐Caputo Fractional Derivative

2025-05-21

Shahram Karami , Mohammad Heydari , Dumitru Băleanu +1 more

ABSTRACT The primary focus of this study is to examine delay optimization problems, subject to a dynamical system that involves ‐Caputo fractional derivative. The generalized hat functions are applied to … ABSTRACT The primary focus of this study is to examine delay optimization problems, subject to a dynamical system that involves ‐Caputo fractional derivative. The generalized hat functions are applied to develop a computational technique for addressing such problems. To achieve this, a new matrix for the ‐Riemann–Liouville fractional integral of the generalized hat functions is derived. The proposed approach utilizes the generalized hat functions to approximate the control and state variables, successfully converting the solution of the primary problem into a set of algebraic equations. The established algorithm is a straightforward and effective mathematical technique for numerically solving this family of problems. Finally, several examples are examined to validate the accuracy and applicability of the proposed approach.

Nonlinear Dynamical Control, Computer Simulation and Optimization Systems

2025-05-19

Dumitru Băleanu , Necati Özdemir , Yeliz Karaca +3 more

Flip Bifurcation and Numerical Study of Crimean-Congo Hemorrhagic Fever with Sustainable Fractional Approach

2025-05-12

Aqeel Ahmad , Muhammad Rafiq , Muhammad Farman +5 more

Computational analysis of fractional model of convective radial fins with temperature dependent thermal conductivity

2025-05-08

Jagdev Singh , Jitendra Kumar , Devendra Kumar +1 more

Quantization of Dissipative Anisotropic Harmonic Oscillators

2025-05-01

Bashar M. Al-Khamiseh , Yazen M. Alawaideh , Dumitru Băleanu +4 more

In this study, we developed a Hamiltonian framework for the anisotropic harmonic oscillator and applied the Hamilton-Jacobi equations, which are particularly well-suited for this approach. We were able to find … In this study, we developed a Hamiltonian framework for the anisotropic harmonic oscillator and applied the Hamilton-Jacobi equations, which are particularly well-suited for this approach. We were able to find the action function by including boundary conditions in the study of a dissipative anisotropic harmonic oscillator. To quantize the system, we employed three distinct methods: the WKB approximation, canonical quantization, and the creation and annihilation operator technique. The quantization of the Heisenberg equations was achieved by reformulating them using Poisson bracket quantum variables. Through the application of creation and annihilation operators, we derived the energy levels and quantum states of the system. The results demonstrated complete consistency across the different methods. This model exhibits strong agreement among the solutions derived from various approaches, highlighting its importance in advancing the understanding of dissipative quantum systems.

Comparative Results on Stability Analysis for Three Dimensional Functional Equations

2025-05-01

Jagjeet Jakhar , Dumitru Băleanu , S.M. Sharma +4 more

This study investigates the stability of a three-dimensional cubic functional equation within several mathematical frameworks, including $(n, \beta)$-normed spaces, non-Archimedean $(n, \beta)$-normed spaces, and random normed spaces. The theoretical stability … This study investigates the stability of a three-dimensional cubic functional equation within several mathematical frameworks, including $(n, \beta)$-normed spaces, non-Archimedean $(n, \beta)$-normed spaces, and random normed spaces. The theoretical stability results are validated through experimental approaches, offering practical insight into the behavior of these functional equations. A comparative analysis is provided, highlighting differences in stability dynamics across the various spaces. Notably, the introduction of $(n, \beta)$-normed spaces and their non-Archimedean counterparts presents a novel framework for analyzing stability, while the inclusion of random normed spaces adds a stochastic dimension to the analysis. The experimental validation further strengthens the practical application of the stability results, distinguishing this study from traditional approaches.

Retraction notice to “The investigation of Fe3O4 atomic aggregation in a nanochannel in the presence of magnetic field: Effects of nanoparticles distance center of mass, temperature and total energy via molecular dynamics approach” [J. Mol. Liq. 348 (2022) 118400]

2025-05-01

Xinglong Liu , Moram A. Fagiry , S. Mohammad Sajadi +5 more

Retraction notice to “The molecular dynamics study of atomic management and thermal behavior of Al-water nanofluid: A two phase unsteady simulation” [J. Mol. Liquids 340 (2021) 117286]

2025-05-01

Yunhong Shi , Seyedmahmoodreza Allahyari , S. Mohammad Sajadi +6 more

Retraction notice to “The investigation of energy management and atomic interaction between coronavirus structure in the vicinity of aqueous environment of H2O molecules via molecular dynamics approach” [J. Mol. Liq. 341 (2021) 117430]

2025-05-01

Huihui Guo , Mohd Yazid Bajuri , Hussam Alrabaiah +5 more

Dynamical and computational analysis of fractional Whitham-Broer-Kaup equations arising in shallow-water waves

2025-05-01

Amit Kumar , Devendra Kumar , Dumitru Băleanu

Solving Time-Fractional Fisher Models by Non-Polynomial Splines in Terms of Logarithmic Derivatives

2025-04-23

H. M. Srivastava , Majeed A. Yousif , Pshtiwan Othman Mohammed +3 more

Finite difference <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si0125.svg"><mml:mi>β</mml:mi></mml:math>-fractional approach for solving the time-fractional FitzHugh–Nagumo equation

2025-04-16

Majeed A. Yousif , Dumitru Băleanu , Mohamed Abdelwahed +2 more

Exploring soliton dynamics and wave interactions in an extended Kadomtsev-Petviashvili-Boussinesq equation

2025-04-15

Nauman Raza , Adil Jhangeer , Zeeshan Amjad +2 more

Solutions comparative for a novel model of fractional delta difference type

2025-04-14

Pshtiwan Othman Mohammed , H. M. Srivastava , Nejmeddine Chorfi +3 more

Unveiling the dynamics of fractional optical solitons in cubic–quintic–septimal nonlinear Schrödinger equations

2025-04-12

Asma Rashid Butt , Tasnim Fatima , Younès Chahlaoui +2 more

Dynamical behaviors of fractional neutral stochastic integro-differential delay systems with impulses

2025-03-31

Dhanalakshmi Kasinathan , Ramkumar Kasinathan , Ravikumar Kasinathan +2 more

On a hybrid class of p -Laplacian boundary value problems with modified Mittag-Leffler kernel: application to nonlinear and mixed pharmacokinetic models

2025-03-06

Suresh Ramasamy , K. Velusamy , Dumitru Băleanu +1 more

On the solutions of generalized Cauchy differential equations and diffusion equations with k-Hilfer-Prabhakar derivative

2025-02-01

Ved Prakash Dubey , Jagdev Singh , Sarvesh Dubey +2 more

Modified hat functions for constrained fractional optimal control problems with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" id="d1e3357" altimg="si5.svg"><mml:mi>ψ</mml:mi></mml:math>-Caputo derivative

2025-02-01

Sh. Karami , Mohammad Heydari , Dumitru Băleanu +1 more

Discrete Legendre polynomials method to solve the coupled nonlinear Caputo–Hadamard fractional Ginzburg–Landau equations

2025-02-01

Mohammad Heydari , Dumitru Băleanu , Mustafa Bayram

A hybrid method based on the classical/piecewise Chebyshev cardinal functions for multi-dimensional fractional Rayleigh–Stokes equations

2025-01-30

M. Hosseininia , Mohammad Heydari , Dumitru Băleanu +1 more

A deep neural network analysis of fractional omicron mathematical model with vaccination and booster dose

2025-01-24

Mati ur Rahman , Salah Boulaaras , Saira Tabassum +1 more

Conformable Schrödinger equation in D-dimensional space

2025-01-24

Eqab M. Rabei , Mohamed Ghaleb Al-Masaeed , Sami I. Muslih +1 more

In this paper, we investigate a time-dependent conformable Schr&ouml;dinger equation of order 0 < &beta; &le; 1, in fractional space domains of space dimension, 0 < Ds &le; 3. We … In this paper, we investigate a time-dependent conformable Schr&ouml;dinger equation of order 0 < &beta; &le; 1, in fractional space domains of space dimension, 0 < Ds &le; 3. We examine a specific example within the realm of free particle conformable Schr&ouml;dinger wave mechanics, focusing on both N-Polar and NCartesian coordinates systems. We find that the conformable quantities align with the regular counterparts when &beta; = 1.

An efficient numerical method for fractional order nonlinear two-point boundary value problem occurring in chemical reactor theory

2025-01-21

Devendra Kumar , Hunney Nama , Dumitru Băleanu

Erratum — ON NEW SOLUTIONS OF THE NORMALIZED FRACTIONAL DIFFERENTIAL EQUATIONS

2025-01-20

Mohammed Al-Refai , Dumitru Băleanu

A Predictor-Corrector Algorithm for IVPs in Frame of Generalized Fractional Operator with Mittag-Leffler Kernels

2025-01-17

Ibrahim Slimane , Zaid Odibat , Dumitru Băleanu

Ulam-hyres stability analysis and fractional operator implications on the Covid-19 virus dynamics with long-term vaccination effects

2025-01-08

Muhammad Farman , Saba Jamil , Evren Hınçal +3 more

Abstract The COVID-19 pandemic has necessitated the development of highly efficient mathematical models to manage its spread, particularly concerning vaccination strategies. Traditional models, however, often fail to account for memory … Abstract The COVID-19 pandemic has necessitated the development of highly efficient mathematical models to manage its spread, particularly concerning vaccination strategies. Traditional models, however, often fail to account for memory effects observed in real-world scenarios, which can be effectively captured using fractional derivatives. This paper introduces a novel COVID-19 model incorporating fractional-order derivatives to reflect better the dependence of the pandemic’s growth on historical events. By approximating the non-locality of fractional derivatives through a generalized Mittag-Leffler kernel, the model effectively captures long-term vaccination effects. To enhance the accuracy of numerical results, the model also integrates the concept of a two-step Lagrange polynomial. The local asymptotic stability of the disease-free equilibrium point is examined through sensitivity and qualitative analyses, particularly using the basic reproduction number, $$ R_0 $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>R</mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:math> . The proposed scheme is rigorously validated through fixed point theory, ensuring the model is evidence-based. A case study conducted in Saudi Arabia demonstrates the effectiveness of the proposed model, yielding successful verification through real-world numerical analysis. The results indicate that incorporating fractional derivatives leads to a more robust model, particularly in assessing the long-term impact of vaccination on the COVID-19 epidemic. The findings underscore the significant role of vaccination in reducing $$ R_0 $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>R</mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:math> and achieving a disease-free state. This work highlights the potential of fractional operators in epidemiological modeling, providing crucial insights into strategies for preventing the transmission of COVID-19 and similar diseases.

Optimal solution and sensitivity analysis of fractional-order mathematical modeling of the omicron variant and malaria co-infection

2025-01-02

Attiq ul Rehman , Ram Singh , Naveen Sharma +2 more

Analytic solutions of a generalized complex multi-dimensional system with fractional order

2025-01-01

Dumitru Băleanu , Rabha W. Ibrahim

Abstract The Duhamel principle is a mathematical principle that allows us to solve linear partial differential equations. This system is generalized by the concept of the <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>k</m:mi> </m:math> … Abstract The Duhamel principle is a mathematical principle that allows us to solve linear partial differential equations. This system is generalized by the concept of the <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>k</m:mi> </m:math> k -symbol fractional calculus. We demonstrate that in specific functional domains, the suggested system produces a global solution. Convergence toward the stable point is investigated. Some special cases are illustrated including the rotated Koebe function, which is used for creating the optimal solutions, where these functions are convex univalent in the open unit disk.

A New Operator Approach for Solving Time-Fractional Nonlinear Burgess Equation

2025-01-01

Hassen Arfaoui , Mohamed Kharrat , Hacene Mecheri +1 more

Existence and Controllability for Second-Order Neutral Functional Integro-differential Equations with Infinite Delay and Random Effects

2025-01-01

Gunasekar Tharmalingam , Madhavi Srinivasan , Shyam Sundar Santra +1 more

A New Mathematical Integral Transformation for Advancing Solutions to Differential Equations with Scientific Applications

2025-01-01

Gunasekar Tharmalingam , Prabakaran Raghavendran , Shyam Sundar Santra +1 more

Introduction to the Special Issue on Mathematical Aspects of Computational Biology and Bioinformatics-II

2025-01-01

Dumitru Băleanu , Carla M. A. Pinto , Sunil Kumar

Treatment of Cancer Disease with Modified ABC Derivative: Mathematical Analysis and Modeling

2025-01-01

Muhammad Farman , Khadija Jamil , Dumitru Băleanu +1 more

On generalized asymmetric harmonic oscillator with quadratic nonlinearity within fractional variational principles

2024-12-17

Dumitru Băleanu , Amin Jajarmi , Özlem Defterli +2 more

This work studies the nonlinear fractional dynamics of asymmetric harmonic oscillators. The classical description of the physical system is generalized using the principles of fractional variational analysis. As a system … This work studies the nonlinear fractional dynamics of asymmetric harmonic oscillators. The classical description of the physical system is generalized using the principles of fractional variational analysis. As a system of two-coupled fractional differential equations with a quadratic nonlinear component, the fractional Euler–Lagrange equations of the motion of the corresponding system are obtained. The Adams–Bashforth predictor-corrector numerical approach is used to approximate the system’s outcomes, which are then simulated comparatively with respect to various model parameter values, including mass, linear and quadratic nonlinear stiffness, and the order of the fractional derivative. The simulations provided the possibility of investigating various dynamical behaviours within the same physical model that is generalized by the use of fractional operators.

Inverse problems for discrete Hermite nabla difference equation

2024-12-16

Babak Shiri , Guang Yang , Dumitru Băleanu

Analytical study of existence, uniqueness, and stability in impulsive neutral fractional Volterra-Fredholm equations

2024-12-16

Prabakaran Raghavendran , Th. Gunasekar , Shyam Sundar Santra +2 more

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Bifurcation Analysis, Sensitivity Analysis, and Jacobi Elliptic Function Structures to a Generalized Nonlinear Schrödinger Equation

2024-12-09

K. Hosseini , Evren Hınçal , Farzaneh Alizadeh +2 more

Positivity and uniqueness of solutions for Riemann–Liouville fractional problem of delta types

2024-11-29

H. M. Srivastava , Pshtiwan Othman Mohammed , Dumitru Băleanu +3 more

Fractional Calculus Approach for Variational Problems: Characterization of Sufficient Optimality Conditions and Duality

2024-11-17

Ved Prakash Dubey , Devendra Kumar , Jagdev Singh +1 more

Bessel statistical convergence: New concepts and applications in sequence theory

2024-11-14

Ibrahim S. Ibrahim , Majeed A. Yousif , Pshtiwan Othman Mohammed +3 more

This research introduces novel concepts in sequence theory, including Bessel convergence, Bessel boundedness, Bessel statistical convergence, and Bessel statistical Cauchy sequences. These concepts establish new inclusion relations and related results … This research introduces novel concepts in sequence theory, including Bessel convergence, Bessel boundedness, Bessel statistical convergence, and Bessel statistical Cauchy sequences. These concepts establish new inclusion relations and related results within mathematical analysis. Additionally, we extend the first and second Korovkin-type approximation theorems by incorporating Bessel statistical convergence, providing a more robust and comprehensive framework than existing results. The practical implications of these theorems are demonstrated through examples involving the classical Bernstein operator and Fejér convolution operators. This work contributes to the foundational understanding of sequence behavior, with potential applications across various scientific disciplines.

An $$\varrho $$-uniformly convergent technique for singularly perturbed problems, with an interior turning point occurring in chemical processes

2024-11-11

Parvin Kumari , Devendra Kumar , Dumitru Băleanu

Piecewise second kind Chebyshev functions for a class of piecewise fractional nonlinear reaction–diffusion equations with variable coefficients

2024-11-07

Mohammad Heydari , Dumitru Băleanu , Mustafa Bayram

Hyers - Ulam Stability of Ebola and Nipah Virus Co-infection Fractional Dynamics

2024-11-06

DHIVYA SUNDAR , Vediyappan Govindan , Pandiarajan Vignesh +3 more

In this research, the goal is to formulate a mathematical model that predicts the transference mechanisms of Ebola virus (EBOV) and Nipah virus (NIV) infections. They utilize fractional-order derivatives to … In this research, the goal is to formulate a mathematical model that predicts the transference mechanisms of Ebola virus (EBOV) and Nipah virus (NIV) infections. They utilize fractional-order derivatives to describe the behavior of the viruses, particularly focusing on the difference in disease manifestation between humans and fruit bats, the putative natural reservoirs. Additionally, they employ fixed-point theory to analyze the qualitative aspects of their model. Also, we investigate the stability of the model using Ulam-Hyers-type results. Furthermore, we utilize the fractional Atangana–Baleanu integral and the Adams–Milton numerical method using MATLAB to provide graphical representations and insights into the behavior of the viruses under various conditions of transference.

On a Hybrid Class of $p$-Laplacian Initial Value Problems with Modified Mittag-Leffler Kernel

2024-10-31

Suresh Ramasamy , V. Kavitha , Dumitru Băleanu +1 more

In this work, we establish key results on the existence theory for a category of initial value problems (IVPs) involving hybrid fractional integro-differential equations (HFIDEs) with a $p$-Laplacian operator, utilizing … In this work, we establish key results on the existence theory for a category of initial value problems (IVPs) involving hybrid fractional integro-differential equations (HFIDEs) with a $p$-Laplacian operator, utilizing the modified Mittag-Leffler kernel. By employing Krasnoselskii and Banach fixed point theorems (FPTs), we determine the conditions required for the existence of solutions. Additionally, we examine the Hyers-Ulam (H-U) stability of the problem. Lastly, we present an example to confirm our theoretical results.

Computational analysis for fractional model of coupled Whitham-Broer-Kaup equation

2024-10-31

Jagdev Singh , Arpita Gupta , Dumitru Băleanu

Hyers-Ulam stability of systems of incommensurate fractional differential equations.

2024-10-18

Babak Shiri , Yong Shi , Dumitru Băleanu

Hyers-Ulam (HU) stability for systems of incommensurate Fractional Differential Equations (FDEs) is studied. The existence and uniqueness of the mild solutions are obtained in the space of discontinuous functions. The … Hyers-Ulam (HU) stability for systems of incommensurate Fractional Differential Equations (FDEs) is studied. The existence and uniqueness of the mild solutions are obtained in the space of discontinuous functions. The related analysis for Laplace transforms is stated and it is used for HU stability. It is shown that the HU stability is equivalent to the convergence of the perturbed system toward the original system when the error of the perturbed source vanishes. Numerical methods based on HU stability are proposed. The analysis is investigated within some examples.

Uniqueness Results Based on Delta Fractional Operators for Certain Boundary Value Problems

2024-10-15

Pshtiwan Othman Mohammed , Ravi P. Agarwal , Dumitru Băleanu +3 more

This paper primarily lies in presenting existence and uniqueness analysis of boundary fractional difference equations of a special Riemann–Liouville operators classes. To this end, we first develop Green’s function to … This paper primarily lies in presenting existence and uniqueness analysis of boundary fractional difference equations of a special Riemann–Liouville operators classes. To this end, we first develop Green’s function to the corresponding fractional boundary value problems and provide boundary conditions to find the constants. Then we study the existence of solutions and we examine the bounded of their solutions. Eventually, two numerical examples are given to demonstrate the efficiency and uniqueness behavior of the boundary value problem. Furthermore, such fractional problems can typically be converted to a direct fractional problem with certain boundary or initial conditions in transport in porous media.

Coauthor	Papers Together
Mustafa İnç	94
Abdullahi Yusuf	55
Devendra Kumar	53
Thabet Abdeljawad	48
Guo–Cheng Wu	47
Soheil Salahshour	45
Hossein Jafari	45
Jagdev Singh	44
Shahram Rezapour	42
Aliyu Isa Aliyu	41
Xiao‐Jun Yang	41
Kottakkaran Sooppy Nisar	41
Sami I. Muslih	40
Ali Akgül	40
Amin Jajarmi	40
Pshtiwan Othman Mohammed	39
Muhammad Rafiq	39
H. M. Srivastava	38
Alireza Khalili Golmankhaneh	35
Ali Saleh Alshomrani	33
Fahd Jarad	30
Eqab M. Rabei	29
Hassan Khan	28
J. A. Tenreiro Machado	27
K. Hosseini	26
J.F. Gómez‐Aguilar	26
Ali Ahmadian	26
Nauman Ahmed	25
Rabha W. Ibrahim	24
Octavian G. Mustafa	23
M.S. Osman	22
Ali Raza	22
Rasool Shah	21
Babak Shiri	21
Aly R. Seadawy	21
Yu–Ming Chu	21
M. Mallika Arjunan	20
Sunıl Dutt Purohıt	20
Zulqurnain Sabir	19
Ahmed A. El‐Deeb	19
Mir Sajjad Hashemi	18
Muhammad Asif Zahoor Raja	18
Arran Fernandez	18
Ravi P. Agarwal	18
Maysaa Al Qurashi	18
Muhammad Arif	17
Hasib Khan	17
Saima Rashid	17
Poom Kumam	17
N. H. Sweilam	17

Theory and Applications of Fractional Differential Equations

2006-01-01

Anatoly A. Kilbas , H. M. Srivastava , Juan J. Trujillo

New fractional derivatives with nonlocal and non-singular kernel: Theory and application to heat transfer model

2016-01-01

Abdon Atangana , Dumitru Băleanu

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Fractional Differential Equations - An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of their Solution and some of their Applications

1999-01-01

Applications of Fractional Calculus in Physics

2000-01-01

R. Hilfer

An introduction to fractional calculus, P.L. Butzer & U. Westphal fractional time evolution, R. Hilfer fractional powers of infinitesimal generators of semigroups, U. Westphal fractional differences, derivatives and fractal time … An introduction to fractional calculus, P.L. Butzer & U. Westphal fractional time evolution, R. Hilfer fractional powers of infinitesimal generators of semigroups, U. Westphal fractional differences, derivatives and fractal time series, B.J. West and P. Grigolini fractional kinetics of Hamiltonian chaotic systems, G.M. Zaslavsky polymer science applications of path integration, integral equations, and fractional calculus, J.F. Douglas applications to problems in polymer physics and rheology, H. Schiessel et al applications of fractional calculus and regular variation in thermodynamics, R. Hilfer.

Fractional Calculus in Bioengineering

2007-02-12

Bruce J. West

Applications of Fractional Calculus in Physics

2000-03-01

R. Hilfer

Formulation of Euler–Lagrange equations for fractional variational problems

2002-08-01

Om P. Agrawal

Fractional Calculus

2011-11-20

Dumitru Băleanu , Kai Diethelm , Enrico Scalas +1 more

Theory and Applications of Fractional Differential Equations, Volume 204 (North-Holland Mathematics Studies)

2006-02-01

Anatoly A. Kilbas , H. M. Srivastava , Juan J. Trujillo

A new definition of fractional derivative

2014-01-16

Roshdi Khalil , Mohammed Al Horani , A. Yousef +1 more

We give a new definition of fractional derivative and fractional integral. The form of the definition shows that it is the most natural definition, and the most fruitful one. The … We give a new definition of fractional derivative and fractional integral. The form of the definition shows that it is the most natural definition, and the most fruitful one. The definition for 0≤α<1 coincides with the classical definitions on polynomials (up to a constant). Further, if α=1, the definition coincides with the classical definition of first derivative. We give some applications to fractional differential equations.

Lagrangian Formulation of Classical Fields within Riemann-Liouville Fractional Derivatives

2005-01-01

Dumitru Băleanu , Sami I. Muslih

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On conformable fractional calculus

2014-10-29

Thabet Abdeljawad

Hamiltonian formulation of systems with linear velocities within Riemann–Liouville fractional derivatives

2005-01-29

Sami I. Muslih , Dumitru Băleanu

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Mechanics with fractional derivatives

1997-03-01

Fred Riewe

Lagrangian and Hamiltonian mechanics can be formulated to include derivatives of fractional order [F. Riewe, Phys. Rev. 53, 1890 (1996)]. Lagrangians with fractional derivatives lead directly to equations of motion … Lagrangian and Hamiltonian mechanics can be formulated to include derivatives of fractional order [F. Riewe, Phys. Rev. 53, 1890 (1996)]. Lagrangians with fractional derivatives lead directly to equations of motion with nonconservative classical forces such as friction. The present work continues the development of fractional-derivative mechanics by deriving a modified Hamilton's principle, introducing two types of canonical transformations, and deriving the Hamilton-Jacobi equation using generalized mechanics with fractional and higher-order derivatives. The method is illustrated with a frictional force proportional to velocity. In contrast to conventional mechanics with integer-order derivatives, quantization of a fractional-derivative Hamiltonian cannot generally be achieved by the traditional replacement of momenta with coordinate derivatives. Instead, a quantum-mechanical wave equation is proposed that follows from the Hamilton-Jacobi equation by application of the correspondence principle.

The Hamilton formalism with fractional derivatives

2006-06-07

Eqab M. Rabei , Khaled I. Nawafleh , R. S. Hijjawi +2 more

Initial value problems in discrete fractional calculus

2008-09-10

Ferhan M. Atıcı , Paul W. Eloe

This paper is devoted to the study of discrete fractional calculus; the particular goal is to define and solve well-defined discrete fractional difference equations. For this purpose we first carefully … This paper is devoted to the study of discrete fractional calculus; the particular goal is to define and solve well-defined discrete fractional difference equations. For this purpose we first carefully develop the commutativity properties of the fractional sum and the fractional difference operators. Then a $\nu$-th ($0 < \nu \leq 1$) order fractional difference equation is defined. A nonlinear problem with an initial condition is solved and the corresponding linear problem with constant coefficients is solved as an example. Further, the half-order linear problem with constant coefficients is solved with a method of undetermined coefficients and with a transform method.

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Analysis of Fractional Differential Equations

2002-01-01

Kai Diethelm , Neville J. Ford

New Trends in Nanotechnology and Fractional Calculus Applications

2009-12-10

Dumitru Băleanu , Zi̇ya B. Güvenç , J. A. Tenreiro Machado

Physics of Fractal Operators

2003-01-01

Bruce J. West , Mauro Bologna , Paolo Grigolini

On the new fractional derivative and application to nonlinear Fisher’s reaction–diffusion equation

2015-11-12

Abdon Atangana

Chaos in a simple nonlinear system with Atangana–Baleanu derivatives with fractional order

2016-03-09

Abdon Atangana , İlknur Koca

A Hamiltonian Formulation and a Direct Numerical Scheme for Fractional Optimal Control Problems

2007-09-01

Om P. Agrawal , Dumitru Băleanu

Recent history of fractional calculus

2010-06-01

J. A. Tenreiro Machado , Virginia Kiryakova , Francesco Mainardi

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The Analysis of Fractional Differential Equations

2010-01-01

Kai Diethelm

Lagrangians with linear velocities within Riemann-Liouville fractional derivatives

2004-01-01

Dumitru Băleanu , T. Avkar

Lagrangians linear in velocities were analyzed using the fractional calculus and the Euler-Lagrange equations were derived. Two examples were investigated in details, the explicit solutions of Euler-Lagrange equations were obtained … Lagrangians linear in velocities were analyzed using the fractional calculus and the Euler-Lagrange equations were derived. Two examples were investigated in details, the explicit solutions of Euler-Lagrange equations were obtained and the recovery of the classical results was discussed.

Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission

2002-11-01

P. van den Driessche , James Watmough

On Riemann and Caputo fractional differences

2011-04-14

Thabet Abdeljawad

Modeling with fractional difference equations

2010-02-11

Ferhan M. Atıcı , Sevgi Şengül Ayan

Advances in Fractional Calculus

2007-01-01

Jocelyn Sabatier , Om P. Agrawal , J. A. Tenreiro Machado

Fractional Calculus

2012-01-01

Dumitru Băleanu , Kai Diethelm , Enrico Scalas +1 more

Fractional relaxation-oscillation and fractional diffusion-wave phenomena

1996-09-01

Francesco Mainardi

Fractional hamilton formalism within caputo’s derivative

2006-10-01

Dumitru Băleanu , Om P. Agrawal

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The Fractional Calculus - Theory and Applications of Differentiation and Integration to Arbitrary Order

1974-01-01

Keith B. Oldham , Jerome Spanier

Integration by parts and its applications of a new nonlocal fractional derivative with Mittag-Leffler nonsingular kernel

2017-03-13

Thabet Abdeljawad , Dumitru Băleanu

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A fractional epidemiological model for computer viruses pertaining to a new fractional derivative

2017-09-13

Jagdev Singh , Devendra Kumar , Zakia Hammouch +1 more

A General Formulation and Solution Scheme for Fractional Optimal Control Problems

2004-12-01

Om P. Agrawal

The random walk's guide to anomalous diffusion: a fractional dynamics approach

2000-12-01

Ralf Metzler , J. Klafter

Analysis of the Keller–Segel Model with a Fractional Derivative without Singular Kernel

2015-06-23

Abdon Atangana , Badr Saad T. Alkahtani

Using some investigations based on information theory, the model proposed by Keller and Segel was extended to the concept of fractional derivative using the derivative with fractional order without singular … Using some investigations based on information theory, the model proposed by Keller and Segel was extended to the concept of fractional derivative using the derivative with fractional order without singular kernel recently proposed by Caputo and Fabrizio. We present in detail the existence of the coupled-solutions using the fixed-point theorem. A detailed analysis of the uniqueness of the coupled-solutions is also presented. Using an iterative approach, we derive special coupled-solutions of the modified system and we present some numerical simulations to see the effect of the fractional order.

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Dynamic Equations on Time Scales

2001-01-01

Martin Böhner , Allan Peterson

On becoming familiar with difference equations and their close re lation to differential equations, I was in hopes that the theory of difference equations could be brought completely abreast with … On becoming familiar with difference equations and their close re lation to differential equations, I was in hopes that the theory of difference equations could be brought completely abreast with that

On some new properties of fractional derivatives with Mittag-Leffler kernel

2017-12-06

Dumitru Băleanu , Arran Fernandez

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Applications of New Time and Spatial Fractional Derivatives with Exponential Kernels

2016-01-01

Michèle Caputo , Mauro Fabrizio

In the paper, we present some applications and features related with the new notions of fractional derivatives with a time exponential kernel and with spatial Gauss kernel for gradient and … In the paper, we present some applications and features related with the new notions of fractional derivatives with a time exponential kernel and with spatial Gauss kernel for gradient and Laplacian operators. Specifically, for these new mode ls we have proved the coherence with the thermodynamic laws. Hence, we have revised the standard linear solid of Zener within continuum mechanics and the model of Cole and Cole inside electromagnetism by these new fractional operators. Moreover, by the Gaussian fractional gradient and through numerical simulations, we have studied the bell shaped filtering effects comparing the results with exponential and Caputo kernel.

Some existence results on nonlinear fractional differential equations

2013-04-02

Dumitru Băleanu , Shahram Rezapour , Hakimeh Mohammadi

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A new study on the mathematical modelling of human liver with Caputo–Fabrizio fractional derivative

2020-02-25

Dumitru Băleanu , Amin Jajarmi , Hakimeh Mohammadi +1 more

Advances in fractional calculus: theoretical developments and applications in physics and Engineering

2007-06-01

Jocelyn Sabatier , Om P. Agrawal , J. A. Tenreiro Machado

In the last two decades, fractional (or non integer) differentiation has played a very important role in various fields such as mechanics, electricity, chemistry, biology, economics, control theory and signal … In the last two decades, fractional (or non integer) differentiation has played a very important role in various fields such as mechanics, electricity, chemistry, biology, economics, control theory and signal and image processing. For example, in the last three fields, some important considerations such as modelling, curve fitting, filtering, pattern recognition, edge detection, identification, stability, controllability, observability and robustness are now linked to long-range dependence phenomena. Similar progress has been made in other fields listed here. The scope of the book is thus to present the state of the art in the study of fractional systems and the application of fractional differentiation. As this volume covers recent applications of fractional calculus, it will be of interest to engineers, scientists, and applied mathematicians.

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On the concept of solution for fractional differential equations with uncertainty

2009-11-18

Ravi P. Agarwal , V. Lakshmikantham , Juan J. Nieto

Discrete fractional logistic map and its chaos

2013-09-19

Guo–Cheng Wu , Dumitru Băleanu

Homotopy perturbation technique

1999-08-01

Ji‐Huan He

Chaos, fractional kinetics, and anomalous transport

2002-12-01

G. M. Zaslavsky

Local Fractional Fokker-Planck Equation

1998-01-12

Kiran M. Kolwankar , Anil D. Gangal

We propose a new class of differential equations, which we call local fractional differential equations. They involve local fractional derivatives and appear to be suitable to deal with phenomena taking … We propose a new class of differential equations, which we call local fractional differential equations. They involve local fractional derivatives and appear to be suitable to deal with phenomena taking place in fractal space and time. A local fractional analog of the Fokker-Planck equation has been derived starting from the Chapman-Kolmogorov condition. We solve the equation with a specific choice of the transition probability and show how subdiffusive behavior can arise.

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Discrete-time fractional variational problems

2010-05-11

Nuno R. O. Bastos , Rui A. C. Ferreira , Delfim F. M. Torres

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