Fractional differential equations(FDEs)provide a powerful tool for modeling systems with memory and non-local effects,but understanding their underlying structure remains a significant challenge.While numerous numeric...Fractional differential equations(FDEs)provide a powerful tool for modeling systems with memory and non-local effects,but understanding their underlying structure remains a significant challenge.While numerous numerical and semi-analytical methods exist to find solutions,new approaches are needed to analyze the intrinsic properties of the FDEs themselves.This paper introduces a novel computational framework for the structural analysis of FDEs involving iterated Caputo derivatives.The methodology is based on a transformation that recasts the original FDE into an equivalent higher-order form,represented as the sum of a closed-form,integer-order component G(y)and a residual fractional power seriesΨ(x).This transformed FDE is subsequently reduced to a first-order ordinary differential equation(ODE).The primary novelty of the proposed methodology lies in treating the structure of the integer-order component G(y)not as fixed,but as a parameterizable polynomial whose coefficients can be determined via global optimization.Using particle swarm optimization,the framework identifies an optimal ODE architecture by minimizing a dual objective that balances solution accuracy against a high-fidelity reference and the magnitude of the truncated residual series.The effectiveness of the approach is demonstrated on both a linear FDE and a nonlinear fractional Riccati equation.Results demonstrate that the framework successfully identifies an optimal,low-degree polynomial ODE architecture that is not necessarily identical to the forcing function of the original FDE.This work provides a new tool for analyzing the underlying structure of FDEs and gaining deeper insights into the interplay between local and non-local dynamics in fractional systems.展开更多
In this paper, the (G′/G)-expansion method is extended to solve fractional partial differential equations in the sense of modified Riemann-Liouville derivative. Based on a nonlinear fractional complex transformation,...In this paper, the (G′/G)-expansion method is extended to solve fractional partial differential equations in the sense of modified Riemann-Liouville derivative. Based on a nonlinear fractional complex transformation, a certain fractional partial differential equation can be turned into another ordinary differential equation of integer order. For illustrating the validity of this method, we apply it to the space-time fractional generalized Hirota-Satsuma coupled KdV equations and the time-fractional fifth-order Sawada-Kotera equation. As a result, some new exact solutions for them are successfully established.展开更多
In this paper, we study a boundary value problem of nonlinear fractional dif- ferential equations of order q （1 〈 q 〈 2） with non-separated integral boundary conditions. Some new existence and uniqueness results a...In this paper, we study a boundary value problem of nonlinear fractional dif- ferential equations of order q （1 〈 q 〈 2） with non-separated integral boundary conditions. Some new existence and uniqueness results are obtained by using some standard fixed point theorems and Leray-Schauder degree theory. Some illustrative examples are also presented. We extend previous results even in the integer case q = 2.展开更多
In this article, the existence and uniqueness of positive solution for a class of nonlinear fractional differential equations is proved by constructing the upper and lower control functions of the nonlinear term witho...In this article, the existence and uniqueness of positive solution for a class of nonlinear fractional differential equations is proved by constructing the upper and lower control functions of the nonlinear term without any monotone requirement. Our main method to the problem is the method of upper and lower solutions and Schauder fixed point theorem. Finally, we give an example to illuminate our results.展开更多
Mathematical simulation of nonlinear physical and abstract systems is a very vital process for predicting the solution behavior of fractional partial differential equations(FPDEs)corresponding to different application...Mathematical simulation of nonlinear physical and abstract systems is a very vital process for predicting the solution behavior of fractional partial differential equations(FPDEs)corresponding to different applications in science and engineering. In this paper, an attractive reliable analytical technique, the conformable residual power series, is implemented for constructing approximate series solutions for a class of nonlinear coupled FPDEs arising in fluid mechanics and fluid flow, which are often designed to demonstrate the behavior of weakly nonlinear and long waves and describe the interaction of shallow water waves. In the proposed technique the n-truncated representation is substituted into the original system and it is assumed the(n-1) conformable derivative of the residuum is zero. This allows us to estimate coefficients of truncation and successively add the subordinate terms in the multiple fractional power series with a rapidly convergent form. The influence, capacity, and feasibility of the presented approach are verified by testing some real-world applications. Finally, highlights and some closing comments are attached.展开更多
In this paper, first we obtain some new fractional integral inequalities. Then using these inequalities and fixed point theorems, we prove the existence of solutions for two different classes of functional fractional ...In this paper, first we obtain some new fractional integral inequalities. Then using these inequalities and fixed point theorems, we prove the existence of solutions for two different classes of functional fractional differential equations.展开更多
In this paper,we concern ourselves with the existence of positive solutions for a type of integral boundary value problem of fractional differential equations with the fractional order linear derivative operator. By u...In this paper,we concern ourselves with the existence of positive solutions for a type of integral boundary value problem of fractional differential equations with the fractional order linear derivative operator. By using the fixed point theorem in cone,the existence of positive solutions for the boundary value problem is obtained. Some examples are also presented to illustrate the application of our main results.展开更多
This article investigates the fractional derivative order identification, the coefficient identification, and the source identification in the fractional diffusion problems. If 1 〈 α〈 2, we prove the unique determi...This article investigates the fractional derivative order identification, the coefficient identification, and the source identification in the fractional diffusion problems. If 1 〈 α〈 2, we prove the unique determination of the fractional derivative order and the dif- fusion coefficient p（x） by fo u（0, s）ds, 0 〈 t 〈 T for one-dimensional fractional diffusion-wave equations. Besides, if 0 〈 α 〈 1, we show the unique determination of the source term f（x, y） by U（0, 0, t）, 0 〈 t 〈 T for two-dimensional fractional diffusion equations. Here, a denotes the fractional derivative order over t.展开更多
Nonlinear delay Caputo fractional differential equations with non-instantaneous impulses are studied and we consider the general case of delay,depending on both the time and the state variable.The case when the lower ...Nonlinear delay Caputo fractional differential equations with non-instantaneous impulses are studied and we consider the general case of delay,depending on both the time and the state variable.The case when the lower limit of the Caputo fractional derivative is fixed at the initial time,and the case when the lower limit of the fractional derivative is changed at the end of each interval of action of the impulse are studied.Practical stability properties,based on the modified Razumikhin method are investigated.Several examples are given in this paper to illustrate the results.展开更多
In this paper, we consider the positive solutions of fractional three-point boundary value problem of the form Dο^α＋u（t）＋f（t,u（t）,u＇（t）,…,u^（n-3）（5）,u^（n-2）（t））=0,u^（i）（0）=0,0≤i≤n-2,u^（n-...In this paper, we consider the positive solutions of fractional three-point boundary value problem of the form Dο^α＋u（t）＋f（t,u（t）,u＇（t）,…,u^（n-3）（5）,u^（n-2）（t））=0,u^（i）（0）=0,0≤i≤n-2,u^（n-2）（1）-βu^（n-2）（ξ）=0,where 0〈t〈1,n-1〈α≤n,n≥2,ξ Е（0,1）,βξ^a-n〈1. We first transform it into another equivalent boundary value problem. Then, we derive the Green＇s function for the equivalent boundary value problem and show that it satisfies certain properties. At last, by using some fixed-point theorems, we obtain the existence of positive solution for this problem. Example is given to illustrate the effectiveness of our result.展开更多
A class of time fractional partial differential equations is considered, which in- cludes a time fractional diffusion equation, a time fractional reaction-diffusion equation, a time fractional advection-diffusion equa...A class of time fractional partial differential equations is considered, which in- cludes a time fractional diffusion equation, a time fractional reaction-diffusion equation, a time fractional advection-diffusion equation, and their corresponding integer-order partial differential equations. The fundamental solutions to the Cauchy problem in a whole-space domain and the signaling problem in a half-space domain are obtained by using Fourier- Laplace transforms and their inverse transforms. The appropriate structures of the Green functions are provided. On the other hand, the solutions in the form of a series to the initial and boundary value problems in a bounded-space domain are derived by the sine- Laplace or cosine-Laplace transforms. Two examples are presented to show applications of the present technique.展开更多
In this paper, a new numerical method for solving fractional differential equations(FDEs) is presented. The method is based upon the fractional Taylor basis approximations. The operational matrix of the fractional int...In this paper, a new numerical method for solving fractional differential equations(FDEs) is presented. The method is based upon the fractional Taylor basis approximations. The operational matrix of the fractional integration for the fractional Taylor basis is introduced. This matrix is then utilized to reduce the solution of the fractional differential equations to a system of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of this technique.展开更多
In this article,we establish new and more general traveling wave solutions of space-time fractional Klein–Gordon equation with quadratic nonlinearity and the space-time fractional breaking soliton equations using the...In this article,we establish new and more general traveling wave solutions of space-time fractional Klein–Gordon equation with quadratic nonlinearity and the space-time fractional breaking soliton equations using the modified simple equation method.The proposed method is so powerful and effective to solve nonlinear space-time fractional differential equations by with modified Riemann–Liouville derivative.展开更多
In this article, two numerical techniques, namely, the homotopy perturbation and the matrix approach methods have been proposed and implemented to obtain an approximate solution of the linear fractional differential e...In this article, two numerical techniques, namely, the homotopy perturbation and the matrix approach methods have been proposed and implemented to obtain an approximate solution of the linear fractional differential equation. To test the effectiveness of these methods, two numerical examples with known exact solution are illustrated. Numerical experiments show that the accuracy of these methods is in a good agreement with the exact solution. However, a comparison between these methods shows that the matrix approach method provides more accurate results.展开更多
This paper presents a method to solve the problems of solutions for integer differential and partial differential equations using the convergence of Adomian's Method. In this paper, we firstly use the convergence ...This paper presents a method to solve the problems of solutions for integer differential and partial differential equations using the convergence of Adomian's Method. In this paper, we firstly use the convergence of Adomian's Method to derive the solutions of high order linear fractional equations, and then the numerical solutions for nonlinear fractional equations. we also get the solutions of two fractional reaction-diffusion equations.We can see the advantage of this method to deal with fractional differential equations.展开更多
In this paper,by using the coincidence degree theory,the existence of solutions for a coupled system of fractional p-Laplacian differential equations at resonance is studied.The result obtained in this paper extends s...In this paper,by using the coincidence degree theory,the existence of solutions for a coupled system of fractional p-Laplacian differential equations at resonance is studied.The result obtained in this paper extends some known results.An example is given to illustrate our result.展开更多
In this paper, we study existence and uniqueness of solutions to nonlinear three point boundary value problems for fractional differential equation of the type c D δ 0＋ u（t） = f （t, u（t）, c D σ 0＋ u（t））, t...In this paper, we study existence and uniqueness of solutions to nonlinear three point boundary value problems for fractional differential equation of the type c D δ 0＋ u（t） = f （t, u（t）, c D σ 0＋ u（t））, t ∈ [0, T ], u（0） = αu（η）, u（T ） = βu（η）, where 1 〈 δ 〈 2, 0 〈 σ 〈 1, α, β∈ R, η∈ （0, T ）, αη（1 -β） ＋ （1-α）（T βη） = 0 and c D δ 0＋ , c D σ 0＋ are the Caputo fractional derivatives. We use Schauder fixed point theorem and contraction mapping principle to obtain existence and uniqueness results. Examples are also included to show the applicability of our results.展开更多
In this paper,two classes of Riesz space fractional partial differential equations including space-fractional and space-time-fractional ones are considered.These two models can be regarded as the generalization of the...In this paper,two classes of Riesz space fractional partial differential equations including space-fractional and space-time-fractional ones are considered.These two models can be regarded as the generalization of the classical wave equation in two space dimensions.Combining with the Crank-Nicolson method in temporal direction,efficient alternating direction implicit Galerkin finite element methods for solving these two fractional models are developed,respectively.The corresponding stability and convergence analysis of the numerical methods are discussed.Numerical results are provided to verify the theoretical analysis.展开更多
An initial value problem was considered for a coupled differential system with multi-term Caputo type fractional derivatives. By means of nonlinear alternative of Leray-Schauder and Banach contraction principle,the ex...An initial value problem was considered for a coupled differential system with multi-term Caputo type fractional derivatives. By means of nonlinear alternative of Leray-Schauder and Banach contraction principle,the existence and uniqueness of solutions for the system were derived. Using a fractional predictorcorrector method, a numerical method was presented for the specified system. An example was given to illustrate the obtained results.展开更多
In this article, the fractional derivatives in the sense of modified Riemann-Liouville derivative and the Exp-function method are employed for constructing the exact solutions of nonlinear time fractional partial diff...In this article, the fractional derivatives in the sense of modified Riemann-Liouville derivative and the Exp-function method are employed for constructing the exact solutions of nonlinear time fractional partial differential equations in mathematical physics. As a result, some new exact solutions for them are successfully established. It is indicated that the solutions obtained by the Exp-function method are reliable, straightforward and effective method for strongly nonlinear fractional partial equations with modified Riemann-Liouville derivative by Jumarie's. This approach can also be applied to other nonlinear time and space fractional differential equations.展开更多
基金Research Council of Lithuania(LMTLT),agreement No.S-PD-24-120Research Council of Lithuania(LMTLT),agreement No.S-PD-24-120funded by the Research Council of Lithuania.
文摘Fractional differential equations(FDEs)provide a powerful tool for modeling systems with memory and non-local effects,but understanding their underlying structure remains a significant challenge.While numerous numerical and semi-analytical methods exist to find solutions,new approaches are needed to analyze the intrinsic properties of the FDEs themselves.This paper introduces a novel computational framework for the structural analysis of FDEs involving iterated Caputo derivatives.The methodology is based on a transformation that recasts the original FDE into an equivalent higher-order form,represented as the sum of a closed-form,integer-order component G(y)and a residual fractional power seriesΨ(x).This transformed FDE is subsequently reduced to a first-order ordinary differential equation(ODE).The primary novelty of the proposed methodology lies in treating the structure of the integer-order component G(y)not as fixed,but as a parameterizable polynomial whose coefficients can be determined via global optimization.Using particle swarm optimization,the framework identifies an optimal ODE architecture by minimizing a dual objective that balances solution accuracy against a high-fidelity reference and the magnitude of the truncated residual series.The effectiveness of the approach is demonstrated on both a linear FDE and a nonlinear fractional Riccati equation.Results demonstrate that the framework successfully identifies an optimal,low-degree polynomial ODE architecture that is not necessarily identical to the forcing function of the original FDE.This work provides a new tool for analyzing the underlying structure of FDEs and gaining deeper insights into the interplay between local and non-local dynamics in fractional systems.
文摘In this paper, the (G′/G)-expansion method is extended to solve fractional partial differential equations in the sense of modified Riemann-Liouville derivative. Based on a nonlinear fractional complex transformation, a certain fractional partial differential equation can be turned into another ordinary differential equation of integer order. For illustrating the validity of this method, we apply it to the space-time fractional generalized Hirota-Satsuma coupled KdV equations and the time-fractional fifth-order Sawada-Kotera equation. As a result, some new exact solutions for them are successfully established.
文摘In this paper, we study a boundary value problem of nonlinear fractional dif- ferential equations of order q （1 〈 q 〈 2） with non-separated integral boundary conditions. Some new existence and uniqueness results are obtained by using some standard fixed point theorems and Leray-Schauder degree theory. Some illustrative examples are also presented. We extend previous results even in the integer case q = 2.
基金supported by Science and Technology Project of Chongqing Municipal Education Committee (kJ110501) of ChinaNatural Science Foundation Project of CQ CSTC (cstc2012jjA20016) of ChinaNational Natural Science Foundation of China (11101298)
文摘In this article, the existence and uniqueness of positive solution for a class of nonlinear fractional differential equations is proved by constructing the upper and lower control functions of the nonlinear term without any monotone requirement. Our main method to the problem is the method of upper and lower solutions and Schauder fixed point theorem. Finally, we give an example to illuminate our results.
基金Authors gratefully acknowledge Ajman University for providing facilities for our research under Grant Ref.No.2019-IRG-HBS-11.
文摘Mathematical simulation of nonlinear physical and abstract systems is a very vital process for predicting the solution behavior of fractional partial differential equations(FPDEs)corresponding to different applications in science and engineering. In this paper, an attractive reliable analytical technique, the conformable residual power series, is implemented for constructing approximate series solutions for a class of nonlinear coupled FPDEs arising in fluid mechanics and fluid flow, which are often designed to demonstrate the behavior of weakly nonlinear and long waves and describe the interaction of shallow water waves. In the proposed technique the n-truncated representation is substituted into the original system and it is assumed the(n-1) conformable derivative of the residuum is zero. This allows us to estimate coefficients of truncation and successively add the subordinate terms in the multiple fractional power series with a rapidly convergent form. The influence, capacity, and feasibility of the presented approach are verified by testing some real-world applications. Finally, highlights and some closing comments are attached.
文摘In this paper, first we obtain some new fractional integral inequalities. Then using these inequalities and fixed point theorems, we prove the existence of solutions for two different classes of functional fractional differential equations.
基金supported by Natural Science Foundation of China（No.11171220） Support Projects of University of Shanghai for Science and Technology（No.14XPM01）
文摘In this paper,we concern ourselves with the existence of positive solutions for a type of integral boundary value problem of fractional differential equations with the fractional order linear derivative operator. By using the fixed point theorem in cone,the existence of positive solutions for the boundary value problem is obtained. Some examples are also presented to illustrate the application of our main results.
基金supported by the National Natural Science Foundation of China (11226166 and 11001033)Scientific Research Fund of Hunan Provinical Education (11C0052)
文摘This article investigates the fractional derivative order identification, the coefficient identification, and the source identification in the fractional diffusion problems. If 1 〈 α〈 2, we prove the unique determination of the fractional derivative order and the dif- fusion coefficient p（x） by fo u（0, s）ds, 0 〈 t 〈 T for one-dimensional fractional diffusion-wave equations. Besides, if 0 〈 α 〈 1, we show the unique determination of the source term f（x, y） by U（0, 0, t）, 0 〈 t 〈 T for two-dimensional fractional diffusion equations. Here, a denotes the fractional derivative order over t.
基金supported by Portuguese funds through the CIDMA-Center for Research and Development in Mathematics and Applicationsthe Portuguese Foundation for Science and Technology(FCT-Fundação para a Ciência e a Tecnologia),within project UIDB/04106/2020Fund Scientific Research MU21FMI007,University of Plovdiv"Paisii Hilendarski".
文摘Nonlinear delay Caputo fractional differential equations with non-instantaneous impulses are studied and we consider the general case of delay,depending on both the time and the state variable.The case when the lower limit of the Caputo fractional derivative is fixed at the initial time,and the case when the lower limit of the fractional derivative is changed at the end of each interval of action of the impulse are studied.Practical stability properties,based on the modified Razumikhin method are investigated.Several examples are given in this paper to illustrate the results.
基金Supported by the National Nature Science Foundation of China(11071001)Supported by the Key Program of Ministry of Education of China(205068)
文摘In this paper, we consider the positive solutions of fractional three-point boundary value problem of the form Dο^α＋u（t）＋f（t,u（t）,u＇（t）,…,u^（n-3）（5）,u^（n-2）（t））=0,u^（i）（0）=0,0≤i≤n-2,u^（n-2）（1）-βu^（n-2）（ξ）=0,where 0〈t〈1,n-1〈α≤n,n≥2,ξ Е（0,1）,βξ^a-n〈1. We first transform it into another equivalent boundary value problem. Then, we derive the Green＇s function for the equivalent boundary value problem and show that it satisfies certain properties. At last, by using some fixed-point theorems, we obtain the existence of positive solution for this problem. Example is given to illustrate the effectiveness of our result.
基金supported by the Fundamental Research Funds for the Central Universities, South China University of Technology (No.2009ZM0050)the Research Foundation for the Doctoral Program of Higher Education of China (No. 20070561040)the Natural Science Foundation of Guangdong Province of China (No.07300823)
文摘A class of time fractional partial differential equations is considered, which in- cludes a time fractional diffusion equation, a time fractional reaction-diffusion equation, a time fractional advection-diffusion equation, and their corresponding integer-order partial differential equations. The fundamental solutions to the Cauchy problem in a whole-space domain and the signaling problem in a half-space domain are obtained by using Fourier- Laplace transforms and their inverse transforms. The appropriate structures of the Green functions are provided. On the other hand, the solutions in the form of a series to the initial and boundary value problems in a bounded-space domain are derived by the sine- Laplace or cosine-Laplace transforms. Two examples are presented to show applications of the present technique.
文摘In this paper, a new numerical method for solving fractional differential equations(FDEs) is presented. The method is based upon the fractional Taylor basis approximations. The operational matrix of the fractional integration for the fractional Taylor basis is introduced. This matrix is then utilized to reduce the solution of the fractional differential equations to a system of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of this technique.
文摘In this article,we establish new and more general traveling wave solutions of space-time fractional Klein–Gordon equation with quadratic nonlinearity and the space-time fractional breaking soliton equations using the modified simple equation method.The proposed method is so powerful and effective to solve nonlinear space-time fractional differential equations by with modified Riemann–Liouville derivative.
文摘In this article, two numerical techniques, namely, the homotopy perturbation and the matrix approach methods have been proposed and implemented to obtain an approximate solution of the linear fractional differential equation. To test the effectiveness of these methods, two numerical examples with known exact solution are illustrated. Numerical experiments show that the accuracy of these methods is in a good agreement with the exact solution. However, a comparison between these methods shows that the matrix approach method provides more accurate results.
基金the National Natural Science Foundation of China(Nos.11601003,11371027)Natural Science Research Project of Colleges of Anhui Province(No.KJ2016A023)+1 种基金Natural Science Foundation of Anhui Province(No.1508085MA01)College Students’Scientific Research Training Plan of Anhui University(No.KYXL2014006)
文摘This paper presents a method to solve the problems of solutions for integer differential and partial differential equations using the convergence of Adomian's Method. In this paper, we firstly use the convergence of Adomian's Method to derive the solutions of high order linear fractional equations, and then the numerical solutions for nonlinear fractional equations. we also get the solutions of two fractional reaction-diffusion equations.We can see the advantage of this method to deal with fractional differential equations.
基金The Key NSF(KJ2015A196)of Anhui Higher Educationthe Young Foundation(2015QN19)of Hefei Normal University
文摘In this paper,by using the coincidence degree theory,the existence of solutions for a coupled system of fractional p-Laplacian differential equations at resonance is studied.The result obtained in this paper extends some known results.An example is given to illustrate our result.
文摘In this paper, we study existence and uniqueness of solutions to nonlinear three point boundary value problems for fractional differential equation of the type c D δ 0＋ u（t） = f （t, u（t）, c D σ 0＋ u（t））, t ∈ [0, T ], u（0） = αu（η）, u（T ） = βu（η）, where 1 〈 δ 〈 2, 0 〈 σ 〈 1, α, β∈ R, η∈ （0, T ）, αη（1 -β） ＋ （1-α）（T βη） = 0 and c D δ 0＋ , c D σ 0＋ are the Caputo fractional derivatives. We use Schauder fixed point theorem and contraction mapping principle to obtain existence and uniqueness results. Examples are also included to show the applicability of our results.
基金supported by the Guangxi Natural Science Foundation[grant numbers 2018GXNSFBA281020,2018GXNSFAA138121]the Doctoral Starting up Foundation of Guilin University of Technology[grant number GLUTQD2016044].
文摘In this paper,two classes of Riesz space fractional partial differential equations including space-fractional and space-time-fractional ones are considered.These two models can be regarded as the generalization of the classical wave equation in two space dimensions.Combining with the Crank-Nicolson method in temporal direction,efficient alternating direction implicit Galerkin finite element methods for solving these two fractional models are developed,respectively.The corresponding stability and convergence analysis of the numerical methods are discussed.Numerical results are provided to verify the theoretical analysis.
基金National Natural Science Foundation of China(No.11371087)
文摘An initial value problem was considered for a coupled differential system with multi-term Caputo type fractional derivatives. By means of nonlinear alternative of Leray-Schauder and Banach contraction principle,the existence and uniqueness of solutions for the system were derived. Using a fractional predictorcorrector method, a numerical method was presented for the specified system. An example was given to illustrate the obtained results.
文摘In this article, the fractional derivatives in the sense of modified Riemann-Liouville derivative and the Exp-function method are employed for constructing the exact solutions of nonlinear time fractional partial differential equations in mathematical physics. As a result, some new exact solutions for them are successfully established. It is indicated that the solutions obtained by the Exp-function method are reliable, straightforward and effective method for strongly nonlinear fractional partial equations with modified Riemann-Liouville derivative by Jumarie's. This approach can also be applied to other nonlinear time and space fractional differential equations.
