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An Application of Multiple Erdélyi–Kober Fractional Integral Operators to Establish New Inequalities Involving a General Class of Functions

2024-07-25
Asifa Tassaddiq , Rekha Srivastava , Rabab Alharbi +2 more
This research aims to develop generalized fractional integral inequalities by utilizing multiple Erdélyi–Kober (E–K) fractional integral operators. Using a set of j, with (j∈N) positively continuous and decaying functions in … This research aims to develop generalized fractional integral inequalities by utilizing multiple Erdélyi–Kober (E–K) fractional integral operators. Using a set of j, with (j∈N) positively continuous and decaying functions in the finite interval a≤t≤x, the Fox-H function is involved in establishing new and novel fractional integral inequalities. Since the Fox-H function is the most general special function, the obtained inequalities are therefore sufficiently widespread and significant in comparison to the current literature to yield novel and unique results.

Blow‐up and decay of solutions for a viscoelastic Kirchhoff‐type equation with distributed delay and variable exponents

2024-02-15
Abdelbaki Choucha , Salah Boulaaras , Rashid Jan +1 more
In this article, we focussed on a nonlinear viscoelastic Kirchhoff‐type equation with distributed delay and variable‐exponents. The blow‐up solutions of the problem are proved under suitable hypothesis, and by using … In this article, we focussed on a nonlinear viscoelastic Kirchhoff‐type equation with distributed delay and variable‐exponents. The blow‐up solutions of the problem are proved under suitable hypothesis, and by using an integral inequality due to Komornik, the general decay outcome is obtained in the case .

New Inequalities Using Multiple Erdélyi–Kober Fractional Integral Operators

2024-03-22
Asifa Tassaddiq , Rekha Srivastava , Rabab Alharbi +2 more
The role of fractional integral inequalities is vital in fractional calculus to develop new models and techniques in the most trending sciences. Taking motivation from this fact, we use multiple … The role of fractional integral inequalities is vital in fractional calculus to develop new models and techniques in the most trending sciences. Taking motivation from this fact, we use multiple Erdélyi–Kober (M-E-K) fractional integral operators to establish Minkowski fractional inequalities. Several other new and novel fractional integral inequalities are also established. Compared to the existing results, these fractional integral inequalities are more general and substantial enough to create new and novel results. M-E-K fractional integral operators have been previously applied for other purposes but have never been applied to the subject of this paper. These operators generalize a popular class of fractional integrals; therefore, this approach will open an avenue for new research. The smart properties of these operators urge us to investigate more results using them.
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Stability for Weakly Coupled Wave Equations with a General Internal Control of Diffusive Type

2023-01-02
Abderrahmane Beniani , Noureddine Bahri , Rabab Alharbi +2 more
The present paper deals with well-posedness and asymptotic stability for weakly coupled wave equations with a more general internal control of diffusive type. Owing to the semigroup theory of linear … The present paper deals with well-posedness and asymptotic stability for weakly coupled wave equations with a more general internal control of diffusive type. Owing to the semigroup theory of linear operator, the well-posedness of system is proved. Furthermore, we show a general decay rate result. The method is based on the frequency domain approach combined with multiplier technique.
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Operational Calculus of the Quantum Statistical Fermi–Dirac and Bose–Einstein Functions Leading to the Novel Fractional Kinetic Equations

2024-12-19
Asifa Tassaddiq , Carlo Cattani , Rabab Alharbi +2 more
The sun is a fundamental element of the natural environment, and kinetic equations are crucial mathematical models for determining how quickly the chemical composition of a star like the sun … The sun is a fundamental element of the natural environment, and kinetic equations are crucial mathematical models for determining how quickly the chemical composition of a star like the sun is changing. Taking motivation from these facts, we develop and solve a novel fractional kinetic equation containing Fermi–Dirac (FD) and Bose–Einstein (BE) functions. Several distributional properties of these functions and their proposed new generalizations are investigated in this article. In fact, it is proved that these functions belong to distribution space D′ while their Fourier transforms belong to Z′. Fourier and Laplace transforms of these functions are computed by using their distributional representation. Thanks to them, we can compute various new fractional calculus formulae and a new relation involving the Fox–Wright function. Some fractional kinetic equations containing the FD and BE functions are also formulated and solved.

Existence of solutions for impulsive wave equations

2023-01-01
Svetlin G. Georgiev , Khaled Zennir , Keltoum Bouhali +3 more
<abstract><p>We study a class of initial value problems for impulsive nonlinear wave equations. A new topological approach is applied to prove the existence of at least one and at least … <abstract><p>We study a class of initial value problems for impulsive nonlinear wave equations. A new topological approach is applied to prove the existence of at least one and at least two nonnegative classical solutions. To prove our main results we give a suitable integral representation of the solutions of the considered problem. Then, we construct two operators so that any fixed point of their sum is a solution.</p></abstract>

Complex Generalized Representation of Gamma Function Leading to the Distributional Solution of a Singular Fractional Integral Equation

2023-11-10
Asifa Tassaddiq , Rekha Srivastava , R. Md. Kasmani +1 more
Firstly, a basic question to find the Laplace transform using the classical representation of gamma function makes no sense because the singularity at the origin nurtures so rapidly that Γze−sz … Firstly, a basic question to find the Laplace transform using the classical representation of gamma function makes no sense because the singularity at the origin nurtures so rapidly that Γze−sz cannot be integrated over positive real numbers. Secondly, Dirac delta function is a linear functional under which every function f is mapped to f(0). This article combines both functions to solve the problems that have remained unsolved for many years. For instance, it has been demonstrated that the power law feature is ubiquitous in theory but challenging to observe in practice. Since the fractional derivatives of the delta function are proportional to the power law, we express the gamma function as a complex series of fractional derivatives of the delta function. Therefore, a unified approach is used to obtain a large class of ordinary, fractional derivatives and integral transforms. All kinds of q-derivatives of these transforms are also computed. The most general form of the fractional kinetic integrodifferential equation available in the literature is solved using this particular representation. We extend the models that were valid only for a class of locally integrable functions to a class of singular (generalized) functions. Furthermore, we solve a singular fractional integral equation whose coefficients have infinite number of singularities, being the poles of gamma function. It is interesting to note that new solutions were obtained using generalized functions with complex coefficients.
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Global existence and general decay of solutions for a wave equation with memory, fractional boundary damping terms and logarithmic non-linearity

2024-05-29
Mohammed Said Touati Brahim , Nadjat Doudi , Rafik Guefaifia +3 more
This study examines the general decomposition and global existence of solutions of the nonlinear wave equation, considering the incorporation of fractional derivative boundary conditions and memory components. This study examines the general decomposition and global existence of solutions of the nonlinear wave equation, considering the incorporation of fractional derivative boundary conditions and memory components.

On the strong convergence of the solution of a generalized non-Newtonian fluid with Coulomb law in a thin film

2023-01-01
Hana Taklit Lahlah , Hamid Benseridi , Bahri Cherif +3 more
<abstract><p>The goal of this paper is to examine the strong convergence of the velocity of a non-Newtonian incompressible fluid whose viscosity follows the power law with Coulomb friction. We assume … <abstract><p>The goal of this paper is to examine the strong convergence of the velocity of a non-Newtonian incompressible fluid whose viscosity follows the power law with Coulomb friction. We assume that the fluid coefficients of the thin layer vary with respect to the thin layer parameter $ \varepsilon $. We give in a first step the description of the problem and basic equations. Then, we present the functional framework. The following paragraph is reserved for the main convergence results. Finally, we give the detail of the proofs of these results.</p></abstract>

Novel series representation of degenerate gamma function with formulation of new generalized kinetic equation

2025-03-20
Asifa Tassaddiq , Rabab Alharbi , R. Md. Kasmani +1 more

New Fractional Integral Inequalities Involving the Fox-H and Meijer-G Functions for Convex and Synchronous Functions

2025-04-17
Asifa Tassaddiq , Carlo Cattani , Rabab Alharbi +2 more
On the one hand, convex functions are important to derive rigorous convergence rates, and on the other, synchronous functions are significant to solve statistical problems using Chebyshev inequalities. Therefore, fractional … On the one hand, convex functions are important to derive rigorous convergence rates, and on the other, synchronous functions are significant to solve statistical problems using Chebyshev inequalities. Therefore, fractional integral inequalities involving such functions play a crucial role in creating new models and methods. Although a large class of fractional operators have been used to establish inequalities, nevertheless, these operators having the Fox-H and the Meijer-G functions in their kernel have been applied to establish fractional integral inequalities for such important classes of functions. Taking motivation from these facts, the primary objective of this work is to develop fractional inequalities involving the Fox-H function for convex and synchronous functions. Since the Fox-H function generalizes several important special functions of fractional calculus, our results are significant to innovate the existing literature. The inventive features of these functions compel researchers to formulate deeper results involving them. Therefore, compared with the ongoing research in this field, our results are general enough to yield novel and inventive fractional inequalities. For instance, new inequalities involving the Meijer-G function are obtained as the special cases of these outcomes, and certain generalizations of Chebyshev inequality are also included in this article.

Fractal generation and analysis using modified fixed-point iteration

2025-01-01
Asifa Tassaddiq , Muhammad Tanveer , Muhammad Arshad +2 more

New Fractional Integral Inequalities Involving the Fox-H and Meijer-G Functions for Convex and Synchronous Functions

2025-04-17
Asifa Tassaddiq , Carlo Cattani , Rabab Alharbi +2 more
On the one hand, convex functions are important to derive rigorous convergence rates, and on the other, synchronous functions are significant to solve statistical problems using Chebyshev inequalities. Therefore, fractional … On the one hand, convex functions are important to derive rigorous convergence rates, and on the other, synchronous functions are significant to solve statistical problems using Chebyshev inequalities. Therefore, fractional integral inequalities involving such functions play a crucial role in creating new models and methods. Although a large class of fractional operators have been used to establish inequalities, nevertheless, these operators having the Fox-H and the Meijer-G functions in their kernel have been applied to establish fractional integral inequalities for such important classes of functions. Taking motivation from these facts, the primary objective of this work is to develop fractional inequalities involving the Fox-H function for convex and synchronous functions. Since the Fox-H function generalizes several important special functions of fractional calculus, our results are significant to innovate the existing literature. The inventive features of these functions compel researchers to formulate deeper results involving them. Therefore, compared with the ongoing research in this field, our results are general enough to yield novel and inventive fractional inequalities. For instance, new inequalities involving the Meijer-G function are obtained as the special cases of these outcomes, and certain generalizations of Chebyshev inequality are also included in this article.

Novel series representation of degenerate gamma function with formulation of new generalized kinetic equation

2025-03-20
Asifa Tassaddiq , Rabab Alharbi , R. Md. Kasmani +1 more

Fractal generation and analysis using modified fixed-point iteration

2025-01-01
Asifa Tassaddiq , Muhammad Tanveer , Muhammad Arshad +2 more

Operational Calculus of the Quantum Statistical Fermi–Dirac and Bose–Einstein Functions Leading to the Novel Fractional Kinetic Equations

2024-12-19
Asifa Tassaddiq , Carlo Cattani , Rabab Alharbi +2 more
The sun is a fundamental element of the natural environment, and kinetic equations are crucial mathematical models for determining how quickly the chemical composition of a star like the sun … The sun is a fundamental element of the natural environment, and kinetic equations are crucial mathematical models for determining how quickly the chemical composition of a star like the sun is changing. Taking motivation from these facts, we develop and solve a novel fractional kinetic equation containing Fermi–Dirac (FD) and Bose–Einstein (BE) functions. Several distributional properties of these functions and their proposed new generalizations are investigated in this article. In fact, it is proved that these functions belong to distribution space D′ while their Fourier transforms belong to Z′. Fourier and Laplace transforms of these functions are computed by using their distributional representation. Thanks to them, we can compute various new fractional calculus formulae and a new relation involving the Fox–Wright function. Some fractional kinetic equations containing the FD and BE functions are also formulated and solved.

An Application of Multiple Erdélyi–Kober Fractional Integral Operators to Establish New Inequalities Involving a General Class of Functions

2024-07-25
Asifa Tassaddiq , Rekha Srivastava , Rabab Alharbi +2 more
This research aims to develop generalized fractional integral inequalities by utilizing multiple Erdélyi–Kober (E–K) fractional integral operators. Using a set of j, with (j∈N) positively continuous and decaying functions in … This research aims to develop generalized fractional integral inequalities by utilizing multiple Erdélyi–Kober (E–K) fractional integral operators. Using a set of j, with (j∈N) positively continuous and decaying functions in the finite interval a≤t≤x, the Fox-H function is involved in establishing new and novel fractional integral inequalities. Since the Fox-H function is the most general special function, the obtained inequalities are therefore sufficiently widespread and significant in comparison to the current literature to yield novel and unique results.

Global existence and general decay of solutions for a wave equation with memory, fractional boundary damping terms and logarithmic non-linearity

2024-05-29
Mohammed Said Touati Brahim , Nadjat Doudi , Rafik Guefaifia +3 more
This study examines the general decomposition and global existence of solutions of the nonlinear wave equation, considering the incorporation of fractional derivative boundary conditions and memory components. This study examines the general decomposition and global existence of solutions of the nonlinear wave equation, considering the incorporation of fractional derivative boundary conditions and memory components.

New Inequalities Using Multiple Erdélyi–Kober Fractional Integral Operators

2024-03-22
Asifa Tassaddiq , Rekha Srivastava , Rabab Alharbi +2 more
The role of fractional integral inequalities is vital in fractional calculus to develop new models and techniques in the most trending sciences. Taking motivation from this fact, we use multiple … The role of fractional integral inequalities is vital in fractional calculus to develop new models and techniques in the most trending sciences. Taking motivation from this fact, we use multiple Erdélyi–Kober (M-E-K) fractional integral operators to establish Minkowski fractional inequalities. Several other new and novel fractional integral inequalities are also established. Compared to the existing results, these fractional integral inequalities are more general and substantial enough to create new and novel results. M-E-K fractional integral operators have been previously applied for other purposes but have never been applied to the subject of this paper. These operators generalize a popular class of fractional integrals; therefore, this approach will open an avenue for new research. The smart properties of these operators urge us to investigate more results using them.
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Blow‐up and decay of solutions for a viscoelastic Kirchhoff‐type equation with distributed delay and variable exponents

2024-02-15
Abdelbaki Choucha , Salah Boulaaras , Rashid Jan +1 more
In this article, we focussed on a nonlinear viscoelastic Kirchhoff‐type equation with distributed delay and variable‐exponents. The blow‐up solutions of the problem are proved under suitable hypothesis, and by using … In this article, we focussed on a nonlinear viscoelastic Kirchhoff‐type equation with distributed delay and variable‐exponents. The blow‐up solutions of the problem are proved under suitable hypothesis, and by using an integral inequality due to Komornik, the general decay outcome is obtained in the case .

Complex Generalized Representation of Gamma Function Leading to the Distributional Solution of a Singular Fractional Integral Equation

2023-11-10
Asifa Tassaddiq , Rekha Srivastava , R. Md. Kasmani +1 more
Firstly, a basic question to find the Laplace transform using the classical representation of gamma function makes no sense because the singularity at the origin nurtures so rapidly that Γze−sz … Firstly, a basic question to find the Laplace transform using the classical representation of gamma function makes no sense because the singularity at the origin nurtures so rapidly that Γze−sz cannot be integrated over positive real numbers. Secondly, Dirac delta function is a linear functional under which every function f is mapped to f(0). This article combines both functions to solve the problems that have remained unsolved for many years. For instance, it has been demonstrated that the power law feature is ubiquitous in theory but challenging to observe in practice. Since the fractional derivatives of the delta function are proportional to the power law, we express the gamma function as a complex series of fractional derivatives of the delta function. Therefore, a unified approach is used to obtain a large class of ordinary, fractional derivatives and integral transforms. All kinds of q-derivatives of these transforms are also computed. The most general form of the fractional kinetic integrodifferential equation available in the literature is solved using this particular representation. We extend the models that were valid only for a class of locally integrable functions to a class of singular (generalized) functions. Furthermore, we solve a singular fractional integral equation whose coefficients have infinite number of singularities, being the poles of gamma function. It is interesting to note that new solutions were obtained using generalized functions with complex coefficients.
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Stability for Weakly Coupled Wave Equations with a General Internal Control of Diffusive Type

2023-01-02
Abderrahmane Beniani , Noureddine Bahri , Rabab Alharbi +2 more
The present paper deals with well-posedness and asymptotic stability for weakly coupled wave equations with a more general internal control of diffusive type. Owing to the semigroup theory of linear … The present paper deals with well-posedness and asymptotic stability for weakly coupled wave equations with a more general internal control of diffusive type. Owing to the semigroup theory of linear operator, the well-posedness of system is proved. Furthermore, we show a general decay rate result. The method is based on the frequency domain approach combined with multiplier technique.
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Existence of solutions for impulsive wave equations

2023-01-01
Svetlin G. Georgiev , Khaled Zennir , Keltoum Bouhali +3 more
<abstract><p>We study a class of initial value problems for impulsive nonlinear wave equations. A new topological approach is applied to prove the existence of at least one and at least … <abstract><p>We study a class of initial value problems for impulsive nonlinear wave equations. A new topological approach is applied to prove the existence of at least one and at least two nonnegative classical solutions. To prove our main results we give a suitable integral representation of the solutions of the considered problem. Then, we construct two operators so that any fixed point of their sum is a solution.</p></abstract>

On the strong convergence of the solution of a generalized non-Newtonian fluid with Coulomb law in a thin film

2023-01-01
Hana Taklit Lahlah , Hamid Benseridi , Bahri Cherif +3 more
<abstract><p>The goal of this paper is to examine the strong convergence of the velocity of a non-Newtonian incompressible fluid whose viscosity follows the power law with Coulomb friction. We assume … <abstract><p>The goal of this paper is to examine the strong convergence of the velocity of a non-Newtonian incompressible fluid whose viscosity follows the power law with Coulomb friction. We assume that the fluid coefficients of the thin layer vary with respect to the thin layer parameter $ \varepsilon $. We give in a first step the description of the problem and basic equations. Then, we present the functional framework. The following paragraph is reserved for the main convergence results. Finally, we give the detail of the proofs of these results.</p></abstract>
Coauthor Papers Together
Asifa Tassaddiq 7
R. Md. Kasmani 6
Rekha Srivastava 3
Sania Qureshi 3
Rashid Jan 2
Keltoum Bouhali 2
Carlo Cattani 2
Salah Boulaaras 2
Khaled Zennir 2
Svetlin G. Georgiev 1
Mohammed Said Touati Brahim 1
Noureddine Bahri 1
Mourad Dilmi 1
Muhammad Arshad 1
Hana Taklit Lahlah 1
Yousif Altayeb 1
Muhammad Tanveer 1
Salah Boulaaras 1
D. K. Almutairi 1
and Ruhaila Md Kasmani 1
Abdelbaki Choucha 1
Nadjat Doudi 1
Kwara Nantomah 1
Abderrahmane Beniani 1
Bahri Cherif 1
Hamid Benseridi 1
Mohamed Biomy 1
Rafik Guefaifia 1

Fractional distributional representation of gamma function and the generalized kinetic equation

2023-10-21
Asifa Tassaddiq , Carlo Cattani
Motivated by the recent research about non-integer order derivatives of Dirac delta function, we use a fractional Taylor series to explore a fractional distributional representation of gamma function. It proved … Motivated by the recent research about non-integer order derivatives of Dirac delta function, we use a fractional Taylor series to explore a fractional distributional representation of gamma function. It proved significant to develop a novel fractional kinetic equation, which is an essential mathematical model to measure the degree of transformation in the chemical structure of stars (sun) as a key component of our environment. In addition to the classical solution of the formulated fractional kinetic equation, the distributional solution is also attained which is not possible without using fractional derivatives of Dirac delta function. Novel identities involving the gamma function are obtained using several fractional transforms. Convergence of new series is proved using the theory of distributions leading to further new applications.

Unified Approach to Fractional Calculus Images of Special Functions—A Survey

2020-12-21
Virginia Kiryakova
Evaluation of images of special functions under operators of fractional calculus has become a hot topic with hundreds of recently published papers. These are growing daily and we are able … Evaluation of images of special functions under operators of fractional calculus has become a hot topic with hundreds of recently published papers. These are growing daily and we are able to comment here only on a few of them, including also some of the latest of 2019–2020, just for the purpose of illustrating our unified approach. Many authors are producing a flood of results for various operators of fractional order integration and differentiation and their generalizations of different special (and elementary) functions. This effect is natural because there are great varieties of special functions, respectively, of operators of (classical and generalized) fractional calculus, and thus, their combinations amount to a large number. As examples, we mentioned only two such operators from thousands of results found by a Google search. Most of the mentioned works use the same formal and standard procedures. Furthermore, in such results, often the originals and the images are special functions of different kinds, or the images are not recognized as known special functions, and thus are not easy to use. In this survey we present a unified approach to fulfill the mentioned task at once in a general setting and in a well visible form: for the operators of generalized fractional calculus (including also the classical operators of fractional calculus); and for all generalized hypergeometric functions such as pΨq and pFq, Fox H- and Meijer G-functions, thus incorporating wide classes of special functions. In this way, a great part of the results in the mentioned publications are well predicted and appear as very special cases of ours. The proposed general scheme is based on a few basic classical results (from the Bateman Project and works by Askey, Lavoie–Osler–Tremblay, etc.) combined with ideas and developments from more than 30 years of author’s research, and reflected in the cited recent works. The main idea is as follows: From one side, the operators considered by other authors are cases of generalized fractional calculus and so, are shown to be (m-times) compositions of weighted Riemann–Lioville, i.e., Erdélyi–Kober operators. On the other side, from each generalized hypergeometric function pΨq or pFq (p≤q or p=q+1) we can reach, from the final number of applications of such operators, one of the simplest cases where the classical results are known, for example: to 0Fq−p (hyper-Bessel functions, in particular trigonometric functions of order (q−p)), 0F0 (exponential function), or 1F0 (beta-distribution of form (1−z)αzβ). The final result, written explicitly, is that any GFC operator (of multiplicity m≥1) transforms a generalized hypergeometric function into the same kind of special function with indices p and q increased by m.
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Complex Generalized Representation of Gamma Function Leading to the Distributional Solution of a Singular Fractional Integral Equation

2023-11-10
Asifa Tassaddiq , Rekha Srivastava , R. Md. Kasmani +1 more
Firstly, a basic question to find the Laplace transform using the classical representation of gamma function makes no sense because the singularity at the origin nurtures so rapidly that Γze−sz … Firstly, a basic question to find the Laplace transform using the classical representation of gamma function makes no sense because the singularity at the origin nurtures so rapidly that Γze−sz cannot be integrated over positive real numbers. Secondly, Dirac delta function is a linear functional under which every function f is mapped to f(0). This article combines both functions to solve the problems that have remained unsolved for many years. For instance, it has been demonstrated that the power law feature is ubiquitous in theory but challenging to observe in practice. Since the fractional derivatives of the delta function are proportional to the power law, we express the gamma function as a complex series of fractional derivatives of the delta function. Therefore, a unified approach is used to obtain a large class of ordinary, fractional derivatives and integral transforms. All kinds of q-derivatives of these transforms are also computed. The most general form of the fractional kinetic integrodifferential equation available in the literature is solved using this particular representation. We extend the models that were valid only for a class of locally integrable functions to a class of singular (generalized) functions. Furthermore, we solve a singular fractional integral equation whose coefficients have infinite number of singularities, being the poles of gamma function. It is interesting to note that new solutions were obtained using generalized functions with complex coefficients.
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A Survey of Some Recent Developments on Higher Transcendental Functions of Analytic Number Theory and Applied Mathematics

2021-12-02
H. M. Srivastava
Often referred to as special functions or mathematical functions, the origin of many members of the remarkably vast family of higher transcendental functions can be traced back to such widespread … Often referred to as special functions or mathematical functions, the origin of many members of the remarkably vast family of higher transcendental functions can be traced back to such widespread areas as (for example) mathematical physics, analytic number theory and applied mathematical sciences. Here, in this survey-cum-expository review article, we aim at presenting a brief introductory overview and survey of some of the recent developments in the theory of several extensively studied higher transcendental functions and their potential applications. For further reading and researching by those who are interested in pursuing this subject, we have chosen to provide references to various useful monographs and textbooks on the theory and applications of higher transcendental functions. Some operators of fractional calculus, which are associated with higher transcendental functions, together with their applications, have also been considered. Many of the higher transcendental functions, especially those of the hypergeometric type, which we have investigated in this survey-cum-expository review article, are known to display a kind of symmetry in the sense that they remain invariant when the order of the numerator parameters or when the order of the denominator parameters is arbitrarily changed.
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Fractional integral inequalities involving Marichev–Saigo–Maeda fractional integral operator

2020-07-08
Asifa Tassaddiq , Aftab Khan , Gauhar Rahman +3 more
Abstract The aim of this present investigation is establishing Minkowski fractional integral inequalities and certain other fractional integral inequalities by employing the Marichev–Saigo–Maeda (MSM) fractional integral operator. The inequalities presented … Abstract The aim of this present investigation is establishing Minkowski fractional integral inequalities and certain other fractional integral inequalities by employing the Marichev–Saigo–Maeda (MSM) fractional integral operator. The inequalities presented in this paper are more general than the existing classical inequalities cited.
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Operators of fractional integration and their applications

2001-02-01
H. M. Srivastava , R. K. Saxena

An Introductory Overview of Fractional-Calculus Operators Based Upon the Fox-Wright and Related Higher Transcendental Functions

2021-09-30
H. M. Srivastava
This survey-cum-expository review article is motivated essentially by the widespread usages of the operators of fractional calculus (that is, fractional-order integrals and fractional-order derivatives) in the modeling and analysis of … This survey-cum-expository review article is motivated essentially by the widespread usages of the operators of fractional calculus (that is, fractional-order integrals and fractional-order derivatives) in the modeling and analysis of a remarkably large variety of applied scientific and real-world problems in mathematical, physical, biological, engineering and statistical sciences, and in other scientific disciplines. Here, in this article, we present a brief introductory overview of the theory and applications of the fractional-calculus operators which are based upon the general Fox-Wright function and its such specialized forms as (for example) the widely- and extensively investigated and potentially useful Mittag-Leffter type functions.This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium provided the original work is properly cited.
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A remark on integral operators involving the Gauss hypergeometric functions

1978-10-01
Megumi Saigo

On new fractional integral inequalities using Marichev–Saigo–Maeda operator

2022-08-10
Meenakshi Singhal , Ekta Mittal
Generalizations of fractional integral inequalities were introduced by many authors. The aim of our investigation is to establish some new fractional integral inequalities using Marichev–Saigo–Maeda (MSM) fractional integral operator for … Generalizations of fractional integral inequalities were introduced by many authors. The aim of our investigation is to establish some new fractional integral inequalities using Marichev–Saigo–Maeda (MSM) fractional integral operator for convex function. Further, we obtain some more fractional integral inequalities of Grüss type using MSM operator.

Solution of Abel-type integral equation involving the Appell hypergeometric function

2009-11-13
R. K. Raina
Abstract The present paper gives a formal solution of a certain Abel-type integral equation involving the Appell hypergeometric function in the kernel. The integral equation and its solution give rise … Abstract The present paper gives a formal solution of a certain Abel-type integral equation involving the Appell hypergeometric function in the kernel. The integral equation and its solution give rise to new forms of generalized fractional calculus operators (viz. the generalized fractional integrals and generalized fractional derivatives). These and their various consequences are also mentioned. The concluding remarks briefly point out possibilities of further work concerning the operators studied in this paper. Keywords: Abel-type integral equationfractional derivatives and fractional integralsAppell hypergeometric functionanalytic functionsSaigo-type fractional integrals and derivatives 2000 Mathematics Subject Classification : 26A3330C4533A3545E10 Acknowledgements The author is thankful to the referee for his comments and suggestions. Additional informationNotes on contributorsR. K. Raina Present address: 10/11 Ganpati Vihar, Opposite Sector 5, Udaipur 313002, Rajasthan, India

Some integral inequalities involving Saigo fractional integral operators

2018-04-03
Mohamed Houas
In this paper, the Saigo fractional integral operator is used to generate some new integral inequalities. By using the Saigo fractional integral operator, we also generate new classes of integral … In this paper, the Saigo fractional integral operator is used to generate some new integral inequalities. By using the Saigo fractional integral operator, we also generate new classes of integral inequalities using a family of n positive functions defined on [a, b].

Positivity of certain classes of functions related to the Fox H-functions with applications

2021-05-30
Khaled Mehrez

Completely Monotonic Gamma Ratio and Infinitely Divisible H-Function of Fox

2015-07-21
Dmitrii Karp , E. G. Prilepkina
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New integral representations for the Fox–Wright functions and its applications

2018-08-29
Khaled Mehrez
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General escape criteria for the generation of fractals in extended Jungck–Noor orbit

2022-01-13
Asifa Tassaddiq

Fourier transform and distributional representation of the gamma function leading to some new identities

2004-01-01
M. A. Chaudhry , Asghar Qadir
We present a Fourier transform representation of the gamma functions, which leads naturally to a distributional representation for them. Both of these representations lead to new identities for the integrals … We present a Fourier transform representation of the gamma functions, which leads naturally to a distributional representation for them. Both of these representations lead to new identities for the integrals of gamma functions multiplied by other functions, which are also presented here.
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New inequalities via Caputo-Fabrizio integral operator with applications

2023-01-01
Hong Yang , Shahid Qaisar , Arslan Munir +1 more
<abstract><p>Fractional integral inequalities have become one of the most useful and expansive tools for the development of many fields of pure and applied mathematics over the past few years. Many … <abstract><p>Fractional integral inequalities have become one of the most useful and expansive tools for the development of many fields of pure and applied mathematics over the past few years. Many authors have just recently introduced various generalized inequalities that involved the fractional integral operators. The main goal of the present study is to incorporate the concept of strongly $ \left(s, m\right) $-convex functions and Hermite-Hadamard inequality with Caputo-Fabrizio integral operator. Also, we consider a new identity for twice differentiable mapping in the context of Caputo-Fabrizio fractional integral operator. Then, considering this identity as an auxiliary result, new mid-point version using well known inequalities like Hölder, power-mean, Young are presented. Moreover, some graphs of obtained inequalities are given for better understanding by the reader. Finally, we discussed some applications to matrix inequalities and spacial means.</p></abstract>

Fourier transform and distributional representation of the generalized gamma function with some applications

2011-04-15
Asifa Tassaddiq , Asghar Qadir

Generalized Ostrowski type inequalities for local fractional integrals

2016-12-07
Mehmet Zeki Sarıkaya , Hüseyin Budak
First, we establish the generalized Ostrowski inequality for local fractional integrals on fractal sets<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper R Superscript alpha"><mml:semantics><mml:msup><mml:mi>R</mml:mi><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi>α</mml:mi></mml:mrow></mml:msup><mml:annotation encoding="application/x-tex">R^{\alpha }</mml:annotation></mml:semantics></mml:math></inline-formula><inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis 0 greater-than alpha less-than-or-equal-to … First, we establish the generalized Ostrowski inequality for local fractional integrals on fractal sets<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper R Superscript alpha"><mml:semantics><mml:msup><mml:mi>R</mml:mi><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi>α</mml:mi></mml:mrow></mml:msup><mml:annotation encoding="application/x-tex">R^{\alpha }</mml:annotation></mml:semantics></mml:math></inline-formula><inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis 0 greater-than alpha less-than-or-equal-to 1 right-parenthesis"><mml:semantics><mml:mrow><mml:mo>(</mml:mo><mml:mn>0</mml:mn><mml:mo>&gt;</mml:mo><mml:mi>α</mml:mi><mml:mo>≤</mml:mo><mml:mn>1</mml:mn><mml:mo>)</mml:mo></mml:mrow><mml:annotation encoding="application/x-tex">\left ( 0&gt;\alpha \leq 1\right )</mml:annotation></mml:semantics></mml:math></inline-formula>of real line numbers. Secondly, we obtain some new inequalities using the generalized convex function on fractal sets<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper R Superscript alpha"><mml:semantics><mml:msup><mml:mi>R</mml:mi><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi>α</mml:mi></mml:mrow></mml:msup><mml:annotation encoding="application/x-tex">R^{\alpha }</mml:annotation></mml:semantics></mml:math></inline-formula>.

New Results Involving the Generalized Krätzel Function with Application to the Fractional Kinetic Equations

2023-02-20
Asifa Tassaddiq , Rekha Srivastava
Sun is a basic component of the natural environment and kinetic equations are important mathematical models to assess the rate of change of chemical composition of a star such as … Sun is a basic component of the natural environment and kinetic equations are important mathematical models to assess the rate of change of chemical composition of a star such as the sun. In this article, a new fractional kinetic equation is formulated and solved using generalized Krätzel integrals because the nuclear reaction rate in astrophysics is represented in terms of these integrals. Furthermore, new identities involving Fox–Wright function are discussed and used to simplify the results. We compute new fractional calculus formulae involving the Krätzel function by using Kiryakova’s fractional integral and derivative operators which led to several new identities for a variety of other classic fractional transforms. A number of new identities for the generalized Krätzel function are then analyzed in relation to the H-function. The closed form of such results is also expressible in terms of Mittag-Leffler function. Distributional representation of Krätzel function and its Laplace transform has been essential in achieving the goals of this work.
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Wave energy decay under fractional derivative controls

2005-10-29
B. Mbodje
Journal Article Wave energy decay under fractional derivative controls Get access Brahima Mbodje Brahima Mbodje Real Results Tutoring Service, 13000 F York Road PMB 176, Charlotte, NC 28278-7602, USA **Present … Journal Article Wave energy decay under fractional derivative controls Get access Brahima Mbodje Brahima Mbodje Real Results Tutoring Service, 13000 F York Road PMB 176, Charlotte, NC 28278-7602, USA **Present address: Division of Mathematics and Sciences, Rust College, 150 Rust Avenue, Holly Springs, MS 38635, USA Search for other works by this author on: Oxford Academic Google Scholar IMA Journal of Mathematical Control and Information, Volume 23, Issue 2, June 2006, Pages 237–257, https://doi.org/10.1093/imamci/dni056 Published: 01 June 2006

New Results Involving Riemann Zeta Function Using Its Distributional Representation

2022-05-06
Asifa Tassaddiq , Rekha Srivastava
The relation of special functions with fractional integral transforms has a great influence on modern science and research. For example, an old special function, namely, the Mittag–Leffler function, became the … The relation of special functions with fractional integral transforms has a great influence on modern science and research. For example, an old special function, namely, the Mittag–Leffler function, became the queen of fractional calculus because its image under the Laplace transform is known to a large audience only in this century. By taking motivation from these facts, we use distributional representation of the Riemann zeta function to compute its Laplace transform, which has played a fundamental role in applying the operators of generalized fractional calculus to this well-studied function. Hence, similar new images under various other popular fractional transforms can be obtained as special cases. A new fractional kinetic equation involving the Riemann zeta function is formulated and solved. Thereafter, a new relation involving the Laplace transform of the Riemann zeta function and the Fox–Wright function is explored, which proved to significantly simplify the results. Various new distributional properties are also derived.
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Fourier transform representation of the generalized hypergeometric functions with applications to the confluent and Gauss hypergeometric functions

2015-05-21
Mohammed H. Al-Lail , Asghar Qadir

Certain generalized fractional integral inequalities

2020-01-01
Kottakkaran Sooppy Nisar , Gauhar Rahman , Aftab Khan +2 more
The principal aim of this article is to establish certain generalized fractional integral inequalities by utilizing the Marichev-Saigo-Maeda (MSM) fractional integral operator. Some new classes of generalized fractional integral inequalities … The principal aim of this article is to establish certain generalized fractional integral inequalities by utilizing the Marichev-Saigo-Maeda (MSM) fractional integral operator. Some new classes of generalized fractional integral inequalities for a class of <i>n</i> (<i>n</i> ∈ $\mathbb{N}$) positive continuous and decreasing functions on [<i>a, b</i>] by using the MSM fractional integral operator also derived.

New Inequalities Using Multiple Erdélyi–Kober Fractional Integral Operators

2024-03-22
Asifa Tassaddiq , Rekha Srivastava , Rabab Alharbi +2 more
The role of fractional integral inequalities is vital in fractional calculus to develop new models and techniques in the most trending sciences. Taking motivation from this fact, we use multiple … The role of fractional integral inequalities is vital in fractional calculus to develop new models and techniques in the most trending sciences. Taking motivation from this fact, we use multiple Erdélyi–Kober (M-E-K) fractional integral operators to establish Minkowski fractional inequalities. Several other new and novel fractional integral inequalities are also established. Compared to the existing results, these fractional integral inequalities are more general and substantial enough to create new and novel results. M-E-K fractional integral operators have been previously applied for other purposes but have never been applied to the subject of this paper. These operators generalize a popular class of fractional integrals; therefore, this approach will open an avenue for new research. The smart properties of these operators urge us to investigate more results using them.
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Unified Fractional Kinetic Equation and a Fractional Diffusion Equation

2004-01-01
R. K. Saxena , A. M. Mathai , H. J. Haubold
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Why Do Big Data and Machine Learning Entail the Fractional Dynamics?

2021-02-28
Haoyu Niu , YangQuan Chen , Bruce J. West
Fractional-order calculus is about the differentiation and integration of non-integer orders. Fractional calculus (FC) is based on fractional-order thinking (FOT) and has been shown to help us to understand complex … Fractional-order calculus is about the differentiation and integration of non-integer orders. Fractional calculus (FC) is based on fractional-order thinking (FOT) and has been shown to help us to understand complex systems better, improve the processing of complex signals, enhance the control of complex systems, increase the performance of optimization, and even extend the enabling of the potential for creativity. In this article, the authors discuss the fractional dynamics, FOT and rich fractional stochastic models. First, the use of fractional dynamics in big data analytics for quantifying big data variability stemming from the generation of complex systems is justified. Second, we show why fractional dynamics is needed in machine learning and optimal randomness when asking: "is there a more optimal way to optimize?". Third, an optimal randomness case study for a stochastic configuration network (SCN) machine-learning method with heavy-tailed distributions is discussed. Finally, views on big data and (physics-informed) machine learning with fractional dynamics for future research are presented with concluding remarks.
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New Fractional Inequalities Involving Saigo Fractional Integral Operator

2014-08-25
Vaijanath L. Chinchane , Deepak B‎. ‎Pachpatte
The aim of this paper is to obtain some new results related to Minkowski fractional integral inequality and other integra l inequalities using Saigo fractional integral . The aim of this paper is to obtain some new results related to Minkowski fractional integral inequality and other integra l inequalities using Saigo fractional integral .

Chebyshev type inequalities for the Saigo fractional integrals and their q-analogues

2013-01-01
Sunıl Dutt Purohıt , Ravinder Krishna Raina
The aim of the present paper is to obtain certain new integral inequalities involving the Saigo fractional integral operator.It is also shown how the various inequalities considered in this paper … The aim of the present paper is to obtain certain new integral inequalities involving the Saigo fractional integral operator.It is also shown how the various inequalities considered in this paper admit themselves of q -extensions which are capable of yielding various results in the theory of q -integral inequalities.
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On Minkowski and Hermite-Hadamard integral inequalities via fractional integration

2010-01-01
Zoubir Dahmani
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On Weighted Gruss Type Inequalities via Fractional Integrals

2010-10-01
Dahmani

Grüss-Type Inequalities by Means of Generalized Fractional Integrals

2019-04-02
J. Vanterler da C. Sousa , D. S. Oliveira , E. Capelas de Oliveira

Operational Calculus of the Quantum Statistical Fermi–Dirac and Bose–Einstein Functions Leading to the Novel Fractional Kinetic Equations

2024-12-19
Asifa Tassaddiq , Carlo Cattani , Rabab Alharbi +2 more
The sun is a fundamental element of the natural environment, and kinetic equations are crucial mathematical models for determining how quickly the chemical composition of a star like the sun … The sun is a fundamental element of the natural environment, and kinetic equations are crucial mathematical models for determining how quickly the chemical composition of a star like the sun is changing. Taking motivation from these facts, we develop and solve a novel fractional kinetic equation containing Fermi–Dirac (FD) and Bose–Einstein (BE) functions. Several distributional properties of these functions and their proposed new generalizations are investigated in this article. In fact, it is proved that these functions belong to distribution space D′ while their Fourier transforms belong to Z′. Fourier and Laplace transforms of these functions are computed by using their distributional representation. Thanks to them, we can compute various new fractional calculus formulae and a new relation involving the Fox–Wright function. Some fractional kinetic equations containing the FD and BE functions are also formulated and solved.

Asymptotic stability of non-Newtonian flows with large perturbation in R2

2005-06-10
Bo‐Qing Dong , Zhi‐Min Chen

Asymptotic behavior of a non-Newtonian flow in a thin domain with Navier law on a rough boundary

2015-02-14
Francisco J. Suárez‐Grau

Non-uniform stability for bounded semi-groups on Banach spaces

2008-10-25
C. J. K. Batty , Thomas Duyckaerts

Fractional master equations and fractal time random walks

1995-02-01
R. Hilfer , L. Anton
Fractional master equations containing fractional time derivatives of order 0\ensuremath{\le}1 are introduced on the basis of a recent classification of time generators in ergodic theory. It is shown that fractional … Fractional master equations containing fractional time derivatives of order 0\ensuremath{\le}1 are introduced on the basis of a recent classification of time generators in ergodic theory. It is shown that fractional master equations are contained as a special case within the traditional theory of continuous time random walks. The corresponding waiting time density \ensuremath{\psi}(t) is obtained exactly as \ensuremath{\psi}(t)=(${\mathit{t}}^{\mathrm{\ensuremath{\omega}}\mathrm{\ensuremath{-}}1}$/C)${\mathit{E}}_{\mathrm{\ensuremath{\omega}},\mathrm{\ensuremath{\omega}}}$(-${\mathit{t}}^{\mathrm{\ensuremath{\omega}}}$/C), where ${\mathit{E}}_{\mathrm{\ensuremath{\omega}},\mathrm{\ensuremath{\omega}}}$(x) is the generalized Mittag-Leffler function. This waiting time distribution is singular both in the long time as well as in the short time limit.

Certain Chebyshev Type Integral Inequalities Involving Hadamard’s Fractional Operators

2014-01-01
Sotiris K. Ntouyas , Sunıl Dutt Purohıt , Jessada Tariboon
We establish certain new fractional integral inequalities for the differentiable functions whose derivatives belong to the space<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M1"><mml:msub><mml:mrow><mml:mi>L</mml:mi></mml:mrow><mml:mrow><mml:mi>p</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mo stretchy="false">[</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mi>∞</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="false">)</mml:mo></mml:math>, related to the weighted version of the … We establish certain new fractional integral inequalities for the differentiable functions whose derivatives belong to the space<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M1"><mml:msub><mml:mrow><mml:mi>L</mml:mi></mml:mrow><mml:mrow><mml:mi>p</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mo stretchy="false">[</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mi>∞</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="false">)</mml:mo></mml:math>, related to the weighted version of the Chebyshev functional, involving Hadamard’s fractional integral operators. As an application, particular results have been also established.
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Comparison results for systems of impulse parabolic equations with applications to population dynamics

1997-01-01
Mokhtar Kirane , Yuri V. Rogovchenko

Applications of fractional calculus to parabolic starlike and uniformly convex functions

2000-02-01
H. M. Srivastava , Akshaya Mishra

Tauberian theorems and stability of one-parameter semigroups

1988-01-01
Wolfgang Arendt , C. J. K. Batty
The main result is the following stability theorem: Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper T equals left-parenthesis upper T left-parenthesis t right-parenthesis right-parenthesis Subscript t greater-than-or-slanted-equals 0"> <mml:semantics> <mml:mrow> … The main result is the following stability theorem: Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper T equals left-parenthesis upper T left-parenthesis t right-parenthesis right-parenthesis Subscript t greater-than-or-slanted-equals 0"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">T</mml:mi> </mml:mrow> <mml:mo>=</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">(</mml:mo> <mml:mi>T</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:msub> <mml:mo stretchy="false">)</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>t</mml:mi> <mml:mo>⩾<!-- ⩾ --></mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:msub> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {T} = {(T(t))_{t \geqslant 0}}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a bounded <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C 0"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>C</mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">{C_0}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-semigroup on a reflexive space <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X"> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding="application/x-tex">X</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Denote by <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A"> <mml:semantics> <mml:mi>A</mml:mi> <mml:annotation encoding="application/x-tex">A</mml:annotation> </mml:semantics> </mml:math> </inline-formula> the generator of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper T"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">T</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {T}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and by <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sigma left-parenthesis upper A right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>σ<!-- σ --></mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>A</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\sigma (A)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> the spectrum of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A"> <mml:semantics> <mml:mi>A</mml:mi> <mml:annotation encoding="application/x-tex">A</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. If <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sigma left-parenthesis upper A right-parenthesis intersection i bold upper R"> <mml:semantics> <mml:mrow> <mml:mi>σ<!-- σ --></mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>A</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>∩<!-- ∩ --></mml:mo> <mml:mi>i</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">R</mml:mi> </mml:mrow> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">\sigma (A) \cap i{\mathbf {R}}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is countable and no eigenvalue of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A"> <mml:semantics> <mml:mi>A</mml:mi> <mml:annotation encoding="application/x-tex">A</mml:annotation> </mml:semantics> </mml:math> </inline-formula> lies on the imaginary axis, then <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="limit Underscript t right-arrow normal infinity Endscripts upper T left-parenthesis t right-parenthesis x equals 0"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:munder> <mml:mo movablelimits="true" form="prefix">lim</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>t</mml:mi> <mml:mo stretchy="false">→<!-- → --></mml:mo> <mml:mi mathvariant="normal">∞<!-- ∞ --></mml:mi> </mml:mrow> </mml:munder> </mml:mrow> <mml:mi>T</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mi>x</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">{\lim _{t \to \infty }}T(t)x = 0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for all <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="x element-of upper X"> <mml:semantics> <mml:mrow> <mml:mi>x</mml:mi> <mml:mo>∈<!-- ∈ --></mml:mo> <mml:mi>X</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">x \in X</mml:annotation> </mml:semantics> </mml:math> </inline-formula>.
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Global solutions for impulsive abstract partial differential equations

2008-03-25
Eduardo Hernández , Sueli M. Tanaka Aki , Hernán R. Henrı́quez

Impulsive effects on global existence of solutions of semilinear heat equations

1996-05-01
C. Y. Chan , Keng Deng

Several interesting integral inequalities

2009-01-01
Wenjun Liu , Quốc Anh Ngô , Vu Nhat Huy
In this paper, several interesting integral inequalities are presented and some open problems are proposed later on. In this paper, several interesting integral inequalities are presented and some open problems are proposed later on.
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Asymptotic analysis of a non-Newtonian fluid in a thin domain with Tresca law

2004-09-17
Mahdi Boukrouche , Rachid Mir

Fractals via Ishikawa Iteration

2011-01-01
Bhagwati Prasad , Kuldip Katiyar

Semigroups of Linear Operators and Applications to Partial Differential Equations

1983-01-01
A. Pazy

Existence and nonexistence of global solutions of some system of semilinear wave equations

2003-06-30
Meng-Rong Li , Long-Yi Tsai

On "Equation missing" <!-- No EquationSource Format="TEX", only image and EquationSource Format="MATHML" -->-equivalence of impulsive differential equations and its applications to partial impulsive differential equations

2012-08-14
Atanaska Georgieva , S. I. Kostadinov , GT Stamov +1 more
Abstract By means of the Schauder-Tychonoff principle, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>L</mml:mi> <mml:mi>p</mml:mi> </mml:msub> <mml:mo>(</mml:mo> <mml:mi>k</mml:mi> <mml:mo>)</mml:mo> </mml:math> -equivalence are established between linear and nonlinear perturbed impulsive differential equations with an … Abstract By means of the Schauder-Tychonoff principle, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>L</mml:mi> <mml:mi>p</mml:mi> </mml:msub> <mml:mo>(</mml:mo> <mml:mi>k</mml:mi> <mml:mo>)</mml:mo> </mml:math> -equivalence are established between linear and nonlinear perturbed impulsive differential equations with an unbounded linear part in an arbitrary Banach space. The feasibility of our theoretical results is illustrated by an example involving partial impulsive differential equations of the parabolic type. MSC: 34A37, 35Q72, 47H10.
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Existence of Solutions for Impulsive Parabolic Partial Differential Equations

2015-04-01
Abdelkader Boucherif , A.S. Alqahtani , B. Chanane
We discuss the propagation of heat along a homogeneous rod of length A under the influence of a nonlinear heat source and impulsive effects at fixed times. This problem is … We discuss the propagation of heat along a homogeneous rod of length A under the influence of a nonlinear heat source and impulsive effects at fixed times. This problem is described by an initial-boundary value problem for a nonlinear parabolic partial differential equation subjected to impulsive effects at fixed times. Using Green's function, we convert the problem into a nonlinear integral equation. Sufficient conditions are provided that enable the application of fixed point theorems to prove existence and uniqueness of solutions.