This paper investigates the dynamics of the beam-like structures whose response manifests a strong scale effect. The space-Fractional Euler–Bernoulli beam (s-FEBB) and space-Fractional Timoshenko beam (s-FTB) models, which are suitable …
This paper investigates the dynamics of the beam-like structures whose response manifests a strong scale effect. The space-Fractional Euler–Bernoulli beam (s-FEBB) and space-Fractional Timoshenko beam (s-FTB) models, which are suitable for small-scale slender beams and small-scale thick beams, respectively, have been extended to a dynamic case. The study provides appropriate governing equations, numerical approximation, detailed analysis of free vibration, and experimental validation. The parametric study presents the influence of non-locality parameters on the frequencies and shape of modes delivering a depth insight into a dynamic response of small scale beams. The comparison of the s-FEBB and s-FTB models determines the applicability limit of s-FEBB and indicates that the model (also the classical one) without shear effect and rotational inertia can only be applied to beams significantly slender than in a static case. Furthermore, the validation has confirmed that the fractional beam model exhibits very good agreement with the experimental results existing in the literature—for both the static and the dynamic cases. Moreover, it has been proven that for fractional beams it is possible to establish constant parameters of non-locality related to the material and its microstructure, independent of beam geometry, the boundary conditions, and the type of analysis (with or without inertial forces).
This article presents a method for the approximate calculation of fractional Caputo derivatives, including a crucial aspect of the ability to handle arbitrary—even variable—terminals and order. The proposed method involves …
This article presents a method for the approximate calculation of fractional Caputo derivatives, including a crucial aspect of the ability to handle arbitrary—even variable—terminals and order. The proposed method involves rearranging the fractional operator as a series of higher-order derivatives considered at a specific point. We demonstrate the effect of the number of terms included in the series expansion on the solution accuracy and error analysis. The advantage of the method is its simplicity and ease of implementation. Additionally, the method allows for a quick estimation of the fractional derivative by using a few first terms of the expansion. The elaborated algorithm is tested against a comprehensive series of illustrative examples, providing very good agreement with the exact/reference solutions. Furthermore, the application of the proposed method to fractional boundary/initial value problems is included.
This article presents a method for the approximate calculation of fractional Caputo derivatives, including a crucial aspect of the ability to handle arbitrary—even variable—terminals and order. The proposed method involves …
This article presents a method for the approximate calculation of fractional Caputo derivatives, including a crucial aspect of the ability to handle arbitrary—even variable—terminals and order. The proposed method involves rearranging the fractional operator as a series of higher-order derivatives considered at a specific point. We demonstrate the effect of the number of terms included in the series expansion on the solution accuracy and error analysis. The advantage of the method is its simplicity and ease of implementation. Additionally, the method allows for a quick estimation of the fractional derivative by using a few first terms of the expansion. The elaborated algorithm is tested against a comprehensive series of illustrative examples, providing very good agreement with the exact/reference solutions. Furthermore, the application of the proposed method to fractional boundary/initial value problems is included.
This paper investigates the dynamics of the beam-like structures whose response manifests a strong scale effect. The space-Fractional Euler–Bernoulli beam (s-FEBB) and space-Fractional Timoshenko beam (s-FTB) models, which are suitable …
This paper investigates the dynamics of the beam-like structures whose response manifests a strong scale effect. The space-Fractional Euler–Bernoulli beam (s-FEBB) and space-Fractional Timoshenko beam (s-FTB) models, which are suitable for small-scale slender beams and small-scale thick beams, respectively, have been extended to a dynamic case. The study provides appropriate governing equations, numerical approximation, detailed analysis of free vibration, and experimental validation. The parametric study presents the influence of non-locality parameters on the frequencies and shape of modes delivering a depth insight into a dynamic response of small scale beams. The comparison of the s-FEBB and s-FTB models determines the applicability limit of s-FEBB and indicates that the model (also the classical one) without shear effect and rotational inertia can only be applied to beams significantly slender than in a static case. Furthermore, the validation has confirmed that the fractional beam model exhibits very good agreement with the experimental results existing in the literature—for both the static and the dynamic cases. Moreover, it has been proven that for fractional beams it is possible to establish constant parameters of non-locality related to the material and its microstructure, independent of beam geometry, the boundary conditions, and the type of analysis (with or without inertial forces).
Fractional continua is a generalisation of the classical continuum body. This new concept shows the application of fractional calculus in continuum mechanics. The advantage is that the obtained description is …
Fractional continua is a generalisation of the classical continuum body. This new concept shows the application of fractional calculus in continuum mechanics. The advantage is that the obtained description is non-local. This natural non-locality is inherently a consequence of fractional derivative definition which is based on the interval, thus variates from the classical approach where the definition is given in a point. In the paper, the application of fractional continua to one-dimensional problem of linear elasticity under small deformation assumption is presented.
Abstract Fractional continuum mechanics is the generalization of classical mechanics utilizing fractional calculus. Contrary to classical theory, the obtained description is non-local, which is inherently the consequence of the fractional …
Abstract Fractional continuum mechanics is the generalization of classical mechanics utilizing fractional calculus. Contrary to classical theory, the obtained description is non-local, which is inherently the consequence of the fractional derivative definition based on the interval. So, all fields obtained in the framework of this new formulation, such as temperature, thermal stresses, total stresses, displacements, etc., at the specific point of interest, depend on the information from its surroundings. The dimensions of these surroundings and the ways of influencing the results are governed by the fractional differential operator applied. In this article, the application of the fractional continuum mechanics to thermoelasticity is presented. A classical solution is obtained as a special case. Keywords: Fractional calculusNon-local modelsThermoelasticity Notes Color versions of one or more of the figures in the article can be found online at www.tandfonline.com/uths.
Purpose Recently, a new formulation has been introduced for non-local mechanics in terms of fractional calculus. Fractional calculus is a branch of mathematical analysis that studies the differential operators of …
Purpose Recently, a new formulation has been introduced for non-local mechanics in terms of fractional calculus. Fractional calculus is a branch of mathematical analysis that studies the differential operators of an arbitrary (real or complex) order and is used successfully in various fields such as mathematics, science and engineering. The purpose of this paper is to introduce a new fractional non-local theory which may be applicable in various simple or complex mechanical problems. Design/methodology/approach In this paper (by using fractional calculus), a fractional non-local theory based on the conformable fractional derivative (CFD) definition is presented, which is a generalized form of the Eringen non-local theory (ENT). The theory contains two free parameters: the fractional parameter which controls the stress gradient order in the constitutive relation and could be an integer and a non-integer and the non-local parameter to consider the small-scale effect in the micron and the sub-micron scales. The non-linear governing equation is solved by the Galerkin and the parameter expansion methods. The non-linearity of the governing equation is due to the presence of von-Kármán non-linearity and CFD definition. Findings The theory has been used to study linear and non-linear free vibration of the simply-supported (S-S) and the clamped-free (C-F) nano beams and then the influence of the fractional and the non-local parameters has been shown on the linear and non-linear frequency ratio. Originality/value A new parameter of the theory (the fractional parameter) makes the modeling more fixable – this model can conclude all of integer and non-integer operators and is not limited to special operators such as ENT. In other words, it allows us to use more sophisticated mathematics to model physical phenomena. On the other hand, in the comparison of classic fractional non-local theory, the theory applicable in various simple or complex mechanical problems may be used because of simpler forms of the governing equation owing to the use of CFD definition.
We present numerical algorithms calculating compositions of the left and right fractional integrals. We apply quadratic interpolation and obtain the fractional Simpson's rule. We estimate the local truncation error of …
We present numerical algorithms calculating compositions of the left and right fractional integrals. We apply quadratic interpolation and obtain the fractional Simpson's rule. We estimate the local truncation error of the proposed approximations, calculate errors generated by presented algorithms, and determine the experimental rate of convergence. Finally, we show examples of numerical evaluation of these operators.
Abstract. This paper considers discrete mass-spring structure identification in a nonlocal continuum space-fractional model, defined as an optimization task. Dynamic (eigenvalues and eigenvectors) and static (displacement field) solutions to discrete …
Abstract. This paper considers discrete mass-spring structure identification in a nonlocal continuum space-fractional model, defined as an optimization task. Dynamic (eigenvalues and eigenvectors) and static (displacement field) solutions to discrete and continuum theories are major constituents of the objective function. It is assumed that the masses in both descriptions are equal (and constant), whereas the spring stiffness distribution in a discrete system becomes a crucial unknown. The considerations include a variety of configurations of the nonlocal parameter and the order of the fractional model, which makes the study comprehensive, and for the first time provides insight into the possible properties (geometric and mechanical) of a discrete structure homogenized by a space-fractional formulation.
Abstract In this paper, we construct a method to find approximate solutions to fractional differential equations involving fractional derivatives with respect to another function. The method is based on an …
Abstract In this paper, we construct a method to find approximate solutions to fractional differential equations involving fractional derivatives with respect to another function. The method is based on an equivalence relation between the fractional differential equation and the Volterra–Stieltjes integral equation of the second kind. The generalized midpoint rule is applied to solve numerically the integral equation and an estimation for the error is given. Results of numerical experiments demonstrate that satisfactory and reliable results could be obtained by the proposed method.
The main goal of this paper is to study selected aspects of the space-Fractional Continuum Mechanical model (s-FCM) approximation. The considerations include the in-depth analysis of three generalized finite difference …
The main goal of this paper is to study selected aspects of the space-Fractional Continuum Mechanical model (s-FCM) approximation. The considerations include the in-depth analysis of three generalized finite difference schemes, which are based on: (i) The rectangle rule, (ii) the trapezoidal rule, and (iii) Simpson's (parabolas) rule, respectively. The analysis includes the dynamic (eigenvalues) response of the 1D s-FCM body. In the numerical study special emphasis is placed on the relationship between fundamental s-FCM model (material) parameters (the order of continua and the length scale) and the quality of approximation.
Variable-order fractional operators were conceived and mathematically formalized only in recent years. The possibility of formulating evolutionary governing equations has led to the successful application of these operators to the …
Variable-order fractional operators were conceived and mathematically formalized only in recent years. The possibility of formulating evolutionary governing equations has led to the successful application of these operators to the modelling of complex real-world problems ranging from mechanics, to transport processes, to control theory, to biology. Variable-order fractional calculus (VO-FC) is a relatively less known branch of calculus that offers remarkable opportunities to simulate interdisciplinary processes. Recognizing this untapped potential, the scientific community has been intensively exploring applications of VO-FC to the modelling of engineering and physical systems. This review is intended to serve as a starting point for the reader interested in approaching this fascinating field. We provide a concise and comprehensive summary of the progress made in the development of VO-FC analytical and computational methods with application to the simulation of complex physical systems. More specifically, following a short introduction of the fundamental mathematical concepts, we present the topic of VO-FC from the point of view of practical applications in the context of scientific modelling.
In this work, we combine conformable double Laplace transform and Adomian decomposition method and present a new approach for solving singular one-dimensional conformable pseudoparabolic equation and conformable coupled pseudoparabolic equation. …
In this work, we combine conformable double Laplace transform and Adomian decomposition method and present a new approach for solving singular one-dimensional conformable pseudoparabolic equation and conformable coupled pseudoparabolic equation. Furthermore, some examples are given to show the performance of the proposed method.
The current article is concerned with the investigation of the formation of the time-dependent three-dimensional distributions and redistributions of the stress and displacement fields of the rotating annular and circular …
The current article is concerned with the investigation of the formation of the time-dependent three-dimensional distributions and redistributions of the stress and displacement fields of the rotating annular and circular plates/disks under the nonuniform distributions of the dynamic thermomechanical loads or shocks. Redistribution of the stress and displacement fields happens due to the Gerasimov-Caputo-type relaxation kernel of the fractional viscoelastic model of the material in addition to the dynamic nature of the loading. All the mechanical and thermal material properties of the plate or disk may be tailored in both radial and transverse directions. The 3D thermoviscoelasticity theory is employed to develop the governing equations of motion (3D vibration) and heat transfer. Different thermal and mechanical boundary conditions are implemented. The resulting nonlinear time-dependent fractional-order thermoviscoelastic integro-differential equations are solved by proposing and implementing a special procedure that uses numerical evaluation of the singular-kernel Caputo-integral and second-order forward, backward, and central discretization of the spatial and time domains. Eventually, the influences of the bidirectional distribution of the material properties, 2D temperature distribution, fractional order and parameters of the viscoelasticity model, and thermomechanical boundary conditions are investigated on the distributions of the displacement and stress components rigorously and accompanied by 3D demonstrations.
Fractional calculus is now a well-established tool in engineering science, with very promising applications in materials modelling. Indeed, several studies have shown that fractional operators can successfully describe complex long-memory …
Fractional calculus is now a well-established tool in engineering science, with very promising applications in materials modelling. Indeed, several studies have shown that fractional operators can successfully describe complex long-memory and multiscale phenomena in materials, which can hardly be captured by standard mathematical approaches as, for instance, classical differential calculus. Furthermore, fractional calculus has recently proved to be an excellent framework for modelling non-conventional fractal and non-local media, opening valuable prospects on future engineered materials. The theme issue gathers cutting-edge theoretical, computational and experimental studies on advanced materials modelling via fractional calculus, with a focus on complex phenomena and non-conventional media. This article is part of the theme issue 'Advanced materials modelling via fractional calculus: challenges and perspectives'.
Abstract In this paper we consider the initial value problem for some impulsive differential equations with higher order Katugampola fractional derivative (fractional order $q \in (1,2]$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>q</mml:mi><mml:mo>∈</mml:mo><mml:mo>(</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mn>2</mml:mn><mml:mo>]</mml:mo></mml:math> ). The …
Abstract In this paper we consider the initial value problem for some impulsive differential equations with higher order Katugampola fractional derivative (fractional order $q \in (1,2]$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>q</mml:mi><mml:mo>∈</mml:mo><mml:mo>(</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mn>2</mml:mn><mml:mo>]</mml:mo></mml:math> ). The systems of impulsive higher order fractional differential equations can involve one or two kinds of impulses, and by analyzing the error between the approximate solution and exact solution it is found that these impulsive systems are equivalent to some integral equations with one or two undetermined constants correspondingly, which uncover the non-uniqueness of solution to these impulsive systems. Some numerical examples are offered to explain the obtained results.
In this study, a numerical technique is applied to investigate solutions of linear variable order differential equations(VODEs) in fluid mechanics. We investigate variable order(VO) in the Caputo sense. Our aim …
In this study, a numerical technique is applied to investigate solutions of linear variable order differential equations(VODEs) in fluid mechanics. We investigate variable order(VO) in the Caputo sense. Our aim is to apply Bernstein polynomials(Bps) as the basis functions and operational matrices. We calculate two types of operational matrices of Bps. Using these operational matrices the main equation can be transformed into a system of algebraic equations, which must then be solved to get approximate solutions. Some examples are provided to indicate the accuracy of the presented method.
The coronavirus infectious disease (COVID-19) is a novel respiratory disease reported in 2019 in China. The infection is very destructive to human lives and caused millions of deaths. Various approaches …
The coronavirus infectious disease (COVID-19) is a novel respiratory disease reported in 2019 in China. The infection is very destructive to human lives and caused millions of deaths. Various approaches have been made recently to understand the complex dynamics of COVID-19. The mathematical modeling approach is one of the considerable tools to study the disease spreading pattern. In this article, we develop a fractional order epidemic model for COVID-19 in the sense of Caputo operator. The model is based on the effective contacts among the population and environmental impact to analyze the disease dynamics. The fractional models are comparatively better in understanding the disease outbreak and providing deeper insights into the infectious disease dynamics. We first consider the classical integer model studied in recent literature and then we generalize it by introducing the Caputo fractional derivative. Furthermore, we explore some fundamental mathematical analysis of the fractional model, including the basic reproductive number R0 and equilibria stability utilizing the Routh-Hurwitz and the Lyapunov function approaches. Besides theoretical analysis, we also focused on the numerical solution. To simulate the model, we use the well-known generalized Adams–Bashforth Moulton Scheme. Finally, the influence of some of the model essential parameters on the dynamics of the disease is demonstrated graphically.
Click to increase image sizeClick to decrease image size Additional informationNotes on contributorsJames F. EppersonThe author received his Ph.D.—on the numerical solution of the Stefan Problem—from Carnegie-Mellon University (1980), under …
Click to increase image sizeClick to decrease image size Additional informationNotes on contributorsJames F. EppersonThe author received his Ph.D.—on the numerical solution of the Stefan Problem—from Carnegie-Mellon University (1980), under George Fix.
<abstract><p>We develop four numerical schemes to solve fractional differential equations involving the Caputo fractional derivative with arbitrary kernels. Firstly, we derive the four numerical schemes, namely, explicit product integration rectangular …
<abstract><p>We develop four numerical schemes to solve fractional differential equations involving the Caputo fractional derivative with arbitrary kernels. Firstly, we derive the four numerical schemes, namely, explicit product integration rectangular rule (forward Euler method), implicit product integration rectangular rule (backward Euler method), implicit product integration trapezoidal rule and Adam-type predictor-corrector method. In addition, the error estimation and stability for all four presented schemes are analyzed. To demonstrate the accuracy and effectiveness of the proposed methods, numerical examples are considered for various linear and nonlinear fractional differential equations with different kernels. The results show that theses numerical schemes are feasible in application.</p></abstract>
Fractional calculus plays an increasingly important role in mechanics research. This review investigates the progress of an interdisciplinary approach, fractional plasticity (FP), based on fractional derivative and classic plasticity since …
Fractional calculus plays an increasingly important role in mechanics research. This review investigates the progress of an interdisciplinary approach, fractional plasticity (FP), based on fractional derivative and classic plasticity since FP was proposed as an efficient alternative to modelling state-dependent nonassociativity without an additional plastic potential function. Firstly, the stress length scale (SLS) is defined to conduct fractional differential, which influences the direction and intensity of the nonassociated flow of geomaterials owing to the integral definition of the fractional operator. Based on the role of SLS, two branches of FP, respectively considering the past stress and future reference critical state can be developed. Merits and demerits of these approaches are then discussed, which leads to the definition of the third branch of FP, by considering the influences of both past and future stress states. In addition, some specific cases and potential applications of the third branch can be realised when specific SLS are adopted.
This paper presents a new class of fractional order Runge–Kutta (FORK) methods for numerically approximating the solution of fractional differential equations (FDEs). We construct explicit and implicit FORK methods for …
This paper presents a new class of fractional order Runge–Kutta (FORK) methods for numerically approximating the solution of fractional differential equations (FDEs). We construct explicit and implicit FORK methods for FDEs by using the Caputo generalized Taylor series formula. Due to the dependence of fractional derivatives on a fixed base point, in the proposed method, we had to modify the right-hand side of the given equation in all steps of the FORK methods. Some coefficients for explicit and implicit FORK schemes are presented. The convergence analysis of the proposed method is also discussed. Numerical experiments are presented to clarify the effectiveness and robustness of the method.
The fractional Legendre polynomials (FLPs) that we present as an effective method for solving fractional delay differential equations (FDDEs) are used in this work. The Liouville–Caputo sense is used to …
The fractional Legendre polynomials (FLPs) that we present as an effective method for solving fractional delay differential equations (FDDEs) are used in this work. The Liouville–Caputo sense is used to characterize fractional derivatives. This method uses the spectral collocation technique based on FLPs. The proposed method converts FDDEs into a set of algebraic equations. We lay out a study of the convergence analysis and figure out the upper bound on error for the approximate solution. Examples are provided to demonstrate the precision of the suggested approach.
Many fundamental physical problems are modeled using differential equations, describing time- and space-dependent variables from conservation laws. Practical problems, such as surface morphology, particle interactions, and memory effects, reveal the …
Many fundamental physical problems are modeled using differential equations, describing time- and space-dependent variables from conservation laws. Practical problems, such as surface morphology, particle interactions, and memory effects, reveal the limitations of traditional tools. Fractional calculus is a valuable tool for these issues, with applications ranging from membrane diffusion to electrical response of complex fluids, particularly electrolytic cells like liquid crystal cells. This paper presents the main fractional tools to formulate a diffusive model regarding time-fractional derivatives and modify the continuity equations stating the conservation laws. We explore two possible ways to introduce time-fractional derivatives to extend the continuity equations to the field of arbitrary-order derivatives. This investigation is essential, because while the mathematical description of neutral particle diffusion has been extensively covered by various authors, a comprehensive treatment of the problem for electrically charged particles remains in its early stages. For this reason, after presenting the appropriate mathematical tools based on fractional calculus, we demonstrate that generalizing the diffusion equation leads to a generalized definition of the displacement current. This modification has strong implications in defining the electrical impedance of electrolytic cells but, more importantly, in the formulation of the Maxwell equations in material systems.