Let X be a complex Banach space and let B and C be two closed linear operators on X satisfying the condition D(B)?D(C),and let d∈L^(1)(R_(+))and 0≤β<α≤2.We characterize the well-posedness of the fractional int...Let X be a complex Banach space and let B and C be two closed linear operators on X satisfying the condition D(B)?D(C),and let d∈L^(1)(R_(+))and 0≤β<α≤2.We characterize the well-posedness of the fractional integro-differential equations D^(α)u(t)+CD^(β)u(t)=Bu(t)+∫_(-∞)td(t-s)Bu(s)ds+f(t),(0≤t≤2π)on periodic Lebesgue-Bochner spaces L^(p)(T;X)and periodic Besov spaces B_(p,q)^(s)(T;X).展开更多
In this article, we develop a fully Discrete Galerkin(DG) method for solving ini- tial value fractional integro-differential equations(FIDEs). We consider Generalized Jacobi polynomials(CJPs) with indexes corres...In this article, we develop a fully Discrete Galerkin(DG) method for solving ini- tial value fractional integro-differential equations(FIDEs). We consider Generalized Jacobi polynomials(CJPs) with indexes corresponding to the number of homogeneous initial conditions as natural basis functions for the approximate solution. The fractional derivatives are used in the Caputo sense. The numerical solvability of algebraic system obtained from implementation of proposed method for a special case of FIDEs is investigated. We also provide a suitable convergence analysis to approximate solutions under a more general regularity assumption on the exact solution. Numerical results are presented to demonstrate the effectiveness of the proposed method.展开更多
We propose and analyze a spectral Jacobi-collocation approximation for fractional order integro-differential equations of Volterra type. The fractional derivative is described in the Caputo sense. We provide a rigorou...We propose and analyze a spectral Jacobi-collocation approximation for fractional order integro-differential equations of Volterra type. The fractional derivative is described in the Caputo sense. We provide a rigorous error analysis for the collection method, which shows that the errors of the approximate solution decay exponentially in L^∞ norm and weighted L^2-norm. The numerical examples are given to illustrate the theoretical results.展开更多
In this paper, we prove an important existence and uniqueness theorem for a fractional order Fredholm – Volterra integro-differential equation with non-local and global boundary conditions by converting it to the cor...In this paper, we prove an important existence and uniqueness theorem for a fractional order Fredholm – Volterra integro-differential equation with non-local and global boundary conditions by converting it to the corresponding well known Fredholm integral equation of second kind. The considered in this paper has been solved already numerically in [1].展开更多
We study boundary value problems for fractional integro-differential equations involving Caputo derivative of order α∈ (n-1, n) in Banach spaces. Existence and uniqueness results of solutions are established by vi...We study boundary value problems for fractional integro-differential equations involving Caputo derivative of order α∈ (n-1, n) in Banach spaces. Existence and uniqueness results of solutions are established by virtue of the Holder's inequality, a suitable singular Cronwall's inequality and fixed point theorem via a priori estimate method. At last, examples are given to illustrate the results.展开更多
The aim of this paper is to obtain the numerical solutions of fractional Volterra integrodifferential equations by the Jacobi spectral collocation method using the Jacobi-Gauss collocation points.We convert the fracti...The aim of this paper is to obtain the numerical solutions of fractional Volterra integrodifferential equations by the Jacobi spectral collocation method using the Jacobi-Gauss collocation points.We convert the fractional order integro-differential equation into integral equation by fractional order integral,and transfer the integro equations into a system of linear equations by the Gausssian quadrature.We furthermore perform the convergence analysis and prove the spectral accuracy of the proposed method in L∞norm.Two numerical examples demonstrate the high accuracy and fast convergence of the method at last.展开更多
In this manuscript,we analyze the solution for class of linear and nonlinear Caputo fractional Volterra Fredholm integro-differential equations with nonlinear time varying delay.Also,we demonstrate the stability analy...In this manuscript,we analyze the solution for class of linear and nonlinear Caputo fractional Volterra Fredholm integro-differential equations with nonlinear time varying delay.Also,we demonstrate the stability analysis for these equations.Our paper provides a convergence of semi-analytical approximate method for these equations.It would be desirable to point out approximate results.展开更多
In this article, variational iteration method (VIM) and homotopy perturbation method (HPM) solve the nonlinear initial value problems of first-order fractional quadratic integro-differential equations (FQIDEs). We use...In this article, variational iteration method (VIM) and homotopy perturbation method (HPM) solve the nonlinear initial value problems of first-order fractional quadratic integro-differential equations (FQIDEs). We use the Caputo sense in this article to describe the fractional derivatives. The solutions of the problems are derived by infinite convergent series, and the results show that both methods are most convenient and effective.展开更多
The fundamental objective of this work is to construct a comparative study of some modified methods with Sumudu transform on fractional delay integro-differential equation.The existed solution of the equation is very ...The fundamental objective of this work is to construct a comparative study of some modified methods with Sumudu transform on fractional delay integro-differential equation.The existed solution of the equation is very accurately computed.The aforesaid methods are presented with an illustrative example.展开更多
In this paper,we establish the new forms of Riemann-type fractional integral and derivative operators.The novel fractional integral operator is proved to be bounded in Lebesgue space and some classical fractional inte...In this paper,we establish the new forms of Riemann-type fractional integral and derivative operators.The novel fractional integral operator is proved to be bounded in Lebesgue space and some classical fractional integral and differential operators are obtained as special cases.The properties of new operators like semi-group,inverse and certain others are discussed and its weighted Laplace transform is evaluated.Fractional integro-differential freeelectron laser(FEL)and kinetic equations are established.The solutions to these new equations are obtained by using the modified weighted Laplace transform.The Cauchy problem and a growth model are designed as applications along with graphical representation.Finally,the conclusion section indicates future directions to the readers.展开更多
In this paper we study the self-similar solution to a class of nonlinear integro-differential equations which correspond to fractional order time derivative and interpolate nonlinear heat and wave equation. Using the ...In this paper we study the self-similar solution to a class of nonlinear integro-differential equations which correspond to fractional order time derivative and interpolate nonlinear heat and wave equation. Using the space-time estimates which were established by Hirata and Miao in [1] we prove the global existence of self-similar solution of Cauchy problem for the nonlinear integro-differential equation in C*([0,∞];B^8pp,∞(R^n).展开更多
The equations of motion governing the quasi-static and dynamical behavior of a viscoelastic Timoshenko beam are derived. The viscoelastic material is assumed to obey a three-dimensional fractional derivative constitut...The equations of motion governing the quasi-static and dynamical behavior of a viscoelastic Timoshenko beam are derived. The viscoelastic material is assumed to obey a three-dimensional fractional derivative constitutive relation. ne quasi-static behavior of the viscoelastic Timoshenko beam under step loading is analyzed and the analytical solution is obtained. The influence of material parameters on the deflection is investigated. The dynamical response of the viscoelastic Timoshenko beam subjected to a periodic excitation is studied by means of mode shape functions. And the effect of both transverse shear and rotational inertia on the vibration of the beam is discussed.展开更多
In this paper, we study the existence and uniqueness of pseudo S-asymptotically ω-periodic mild solutions of class r for fractional integro-differential neutral equations. An example is presented to illustrate the ap...In this paper, we study the existence and uniqueness of pseudo S-asymptotically ω-periodic mild solutions of class r for fractional integro-differential neutral equations. An example is presented to illustrate the application of the abstract results.展开更多
In this paper,a compact finite difference scheme for the nonlinear fractional integro-differential equation with weak singularity at the initial time is developed,with O(N^(-(2-α))+M^(-4))accuracy order,where N;M den...In this paper,a compact finite difference scheme for the nonlinear fractional integro-differential equation with weak singularity at the initial time is developed,with O(N^(-(2-α))+M^(-4))accuracy order,where N;M denote the numbers of grids in temporal and spatial direction,α ∈(0,1)is the fractional order.To recover the full accuracy based on the regularity requirement of the solution,we adopt the L1 method and the trapezoidal product integration(PI)rule with graded meshes to discretize the Caputo derivative and the Riemann-Liouville integral,respectively,further handle the nonlinear term carefully by the Newton linearized method.Based on the discrete fractional Gr¨onwall inequality and preserved discrete coefficients of Riemann-Liouville fractional integral,the stability and convergence of the proposed scheme are analyzed by the energy method.Theoretical results are also confirmed by a numerical example.展开更多
The dynamic stability of simple supported viscoelastic column, subjected to a periodic axial force, is investigated. The viscoelastic material was assumed to obey the fractional derivative constitutive relation. The g...The dynamic stability of simple supported viscoelastic column, subjected to a periodic axial force, is investigated. The viscoelastic material was assumed to obey the fractional derivative constitutive relation. The governing equation of motion was derived as a weakly singular Volterra integro-partial-differential equation, and it was simplified into weakly singular Volterra integro-ordinary-differential equation by the Galerkin method. In terms of the averaging method, the dynamical stability was analyzed. A new numerical method is proposed to avoid storing all history data. Numerical examples are presented and the numerical results agree with the analytical ones.展开更多
In this paper,we study a class of nonlinear fractional integro-differential equations,the fractional derivative is described in the Caputo sense.Using the properties of the Caputo derivative,we convert the fractional ...In this paper,we study a class of nonlinear fractional integro-differential equations,the fractional derivative is described in the Caputo sense.Using the properties of the Caputo derivative,we convert the fractional integro-differential equations into equivalent integral-differential equations of Volterra type with singular kernel,then we propose and analyze a spectral Jacobi-collocation approximation for nonlinear integro-differential equations of Volterra type.We provide a rigorous error analysis for the spectral methods,which shows that both the errors of approximate solutions and the errors of approximate fractional derivatives of the solutions decay exponentially in L^(∞)-norm and weighted L^(2)-norm.展开更多
This study aims to apply the two-step Adomian decomposition method(TSADM)to find an analytical solution of integro-differential equations for fractional order without discretization/approximation with less number of i...This study aims to apply the two-step Adomian decomposition method(TSADM)to find an analytical solution of integro-differential equations for fractional order without discretization/approximation with less number of iterations and reduce the computational efforts.Moreover,we have established the results for the existence and uniqueness of a solution with the help of some fixed point theorems and the Banach contraction principle.Furthermore,the method is demonstrated on different test examples arising in real life situations.It is concluded that the TSADM provides exact solution of the fractional integro-differential equations in one iteration.At the same time,the other existing methods furnish an approximate solution and require lots of computation to solve the problem applying discretization/approximation on fractional operators.展开更多
In this paper,we give a new numerical method for solving a linear system of fractional integro-differential equations.The fractional derivative is considered in the Caputo sense.The proposed method is least squares me...In this paper,we give a new numerical method for solving a linear system of fractional integro-differential equations.The fractional derivative is considered in the Caputo sense.The proposed method is least squares method aid of Hermite polynomials.The suggested method reduces this type of systems to the solution of systems of linear algebraic equations.To demonstrate the accuracy and applicability of the presented method some test examples are provided.Numerical results show that this approach is easy to implement and accurate when applied to integro-differential equations.We show that the solutions approach to classical solutions as the order of the fractional derivatives approach.展开更多
This paper mainly considers the optimal convergence analysis of the q-Maruyama method for stochastic Volterra integro-differential equations(SVIDEs)driven by Riemann-Liouville fractional Brownian motion under the glob...This paper mainly considers the optimal convergence analysis of the q-Maruyama method for stochastic Volterra integro-differential equations(SVIDEs)driven by Riemann-Liouville fractional Brownian motion under the global Lipschitz and linear growth conditions.Firstly,based on the contraction mapping principle,we prove the well-posedness of the analytical solutions of the SVIDEs.Secondly,we show that the q-Maruyama method for the SVIDEs can achieve strong first-order convergence.In particular,when the q-Maruyama method degenerates to the explicit Euler-Maruyama method,our result improves the conclusion that the convergence rate is H+1/2,H∈(0,1/2)by Yang et al.,J.Comput.Appl.Math.,383(2021),113156.Finally,the numerical experiment verifies our theoretical results.展开更多
This paper is concerned with the approximate controllability of nonlinear fractional impulsive neutral stochastic integro-differential equations with nonlocal conditions and infinite delay in Hilbert spaces under the ...This paper is concerned with the approximate controllability of nonlinear fractional impulsive neutral stochastic integro-differential equations with nonlocal conditions and infinite delay in Hilbert spaces under the assumptions that the corresponding linear system is approximately controllable. By the Krasnoselskii-Schaefer-type fixed point theorem and stochastic analysis theory, some sufficient conditions are given for the approximate controllability of the system. At the end, an example is given to illustrate the application of our result.展开更多
基金the NSF of China(12171266,12171062)the NSF of Chongqing(CSTB2022NSCQ-JQX0004)。
文摘Let X be a complex Banach space and let B and C be two closed linear operators on X satisfying the condition D(B)?D(C),and let d∈L^(1)(R_(+))and 0≤β<α≤2.We characterize the well-posedness of the fractional integro-differential equations D^(α)u(t)+CD^(β)u(t)=Bu(t)+∫_(-∞)td(t-s)Bu(s)ds+f(t),(0≤t≤2π)on periodic Lebesgue-Bochner spaces L^(p)(T;X)and periodic Besov spaces B_(p,q)^(s)(T;X).
文摘In this article, we develop a fully Discrete Galerkin(DG) method for solving ini- tial value fractional integro-differential equations(FIDEs). We consider Generalized Jacobi polynomials(CJPs) with indexes corresponding to the number of homogeneous initial conditions as natural basis functions for the approximate solution. The fractional derivatives are used in the Caputo sense. The numerical solvability of algebraic system obtained from implementation of proposed method for a special case of FIDEs is investigated. We also provide a suitable convergence analysis to approximate solutions under a more general regularity assumption on the exact solution. Numerical results are presented to demonstrate the effectiveness of the proposed method.
基金supported by NSFC Project(11301446,11271145)China Postdoctoral Science Foundation Grant(2013M531789)+3 种基金Specialized Research Fund for the Doctoral Program of Higher Education(2011440711009)Program for Changjiang Scholars and Innovative Research Team in University(IRT1179)Project of Scientific Research Fund of Hunan Provincial Science and Technology Department(2013RS4057)the Research Foundation of Hunan Provincial Education Department(13B116)
文摘We propose and analyze a spectral Jacobi-collocation approximation for fractional order integro-differential equations of Volterra type. The fractional derivative is described in the Caputo sense. We provide a rigorous error analysis for the collection method, which shows that the errors of the approximate solution decay exponentially in L^∞ norm and weighted L^2-norm. The numerical examples are given to illustrate the theoretical results.
文摘In this paper, we prove an important existence and uniqueness theorem for a fractional order Fredholm – Volterra integro-differential equation with non-local and global boundary conditions by converting it to the corresponding well known Fredholm integral equation of second kind. The considered in this paper has been solved already numerically in [1].
基金supported by Grant In Aid research fund of Virginia Military Instittue, USA
文摘We study boundary value problems for fractional integro-differential equations involving Caputo derivative of order α∈ (n-1, n) in Banach spaces. Existence and uniqueness results of solutions are established by virtue of the Holder's inequality, a suitable singular Cronwall's inequality and fixed point theorem via a priori estimate method. At last, examples are given to illustrate the results.
基金This work is supported by the National Natural Science Foundation of China(Grant Nos.11701358,11774218)。
文摘The aim of this paper is to obtain the numerical solutions of fractional Volterra integrodifferential equations by the Jacobi spectral collocation method using the Jacobi-Gauss collocation points.We convert the fractional order integro-differential equation into integral equation by fractional order integral,and transfer the integro equations into a system of linear equations by the Gausssian quadrature.We furthermore perform the convergence analysis and prove the spectral accuracy of the proposed method in L∞norm.Two numerical examples demonstrate the high accuracy and fast convergence of the method at last.
文摘In this manuscript,we analyze the solution for class of linear and nonlinear Caputo fractional Volterra Fredholm integro-differential equations with nonlinear time varying delay.Also,we demonstrate the stability analysis for these equations.Our paper provides a convergence of semi-analytical approximate method for these equations.It would be desirable to point out approximate results.
文摘In this article, variational iteration method (VIM) and homotopy perturbation method (HPM) solve the nonlinear initial value problems of first-order fractional quadratic integro-differential equations (FQIDEs). We use the Caputo sense in this article to describe the fractional derivatives. The solutions of the problems are derived by infinite convergent series, and the results show that both methods are most convenient and effective.
文摘The fundamental objective of this work is to construct a comparative study of some modified methods with Sumudu transform on fractional delay integro-differential equation.The existed solution of the equation is very accurately computed.The aforesaid methods are presented with an illustrative example.
文摘In this paper,we establish the new forms of Riemann-type fractional integral and derivative operators.The novel fractional integral operator is proved to be bounded in Lebesgue space and some classical fractional integral and differential operators are obtained as special cases.The properties of new operators like semi-group,inverse and certain others are discussed and its weighted Laplace transform is evaluated.Fractional integro-differential freeelectron laser(FEL)and kinetic equations are established.The solutions to these new equations are obtained by using the modified weighted Laplace transform.The Cauchy problem and a growth model are designed as applications along with graphical representation.Finally,the conclusion section indicates future directions to the readers.
基金NSF of China,Special Funds for Major State Basic Research Projects of ChinaNSF of Chinese Academy of Engineering Physics
文摘In this paper we study the self-similar solution to a class of nonlinear integro-differential equations which correspond to fractional order time derivative and interpolate nonlinear heat and wave equation. Using the space-time estimates which were established by Hirata and Miao in [1] we prove the global existence of self-similar solution of Cauchy problem for the nonlinear integro-differential equation in C*([0,∞];B^8pp,∞(R^n).
文摘The equations of motion governing the quasi-static and dynamical behavior of a viscoelastic Timoshenko beam are derived. The viscoelastic material is assumed to obey a three-dimensional fractional derivative constitutive relation. ne quasi-static behavior of the viscoelastic Timoshenko beam under step loading is analyzed and the analytical solution is obtained. The influence of material parameters on the deflection is investigated. The dynamical response of the viscoelastic Timoshenko beam subjected to a periodic excitation is studied by means of mode shape functions. And the effect of both transverse shear and rotational inertia on the vibration of the beam is discussed.
基金supported by National Natural Science Foundation of China(Grant Nos.11271379 and 11671406)
文摘In this paper, we study the existence and uniqueness of pseudo S-asymptotically ω-periodic mild solutions of class r for fractional integro-differential neutral equations. An example is presented to illustrate the application of the abstract results.
基金supported by the National Natural Science Foundation of China(No.11701103,11801095)Young Top-notch Talent Program of Guangdong Province(No.2017GC010379)+2 种基金Natural Science Foundation of Guangdong Province(No.2022A1515012147,2019A1515010876,2017A030310538)the Project of Science and Technology of Guangzhou(No.201904010341,202102020704)the Opening Project of Guangdong Province Key Laboratory of Computational Science at the Sun Yat-sen University(2021023)。
文摘In this paper,a compact finite difference scheme for the nonlinear fractional integro-differential equation with weak singularity at the initial time is developed,with O(N^(-(2-α))+M^(-4))accuracy order,where N;M denote the numbers of grids in temporal and spatial direction,α ∈(0,1)is the fractional order.To recover the full accuracy based on the regularity requirement of the solution,we adopt the L1 method and the trapezoidal product integration(PI)rule with graded meshes to discretize the Caputo derivative and the Riemann-Liouville integral,respectively,further handle the nonlinear term carefully by the Newton linearized method.Based on the discrete fractional Gr¨onwall inequality and preserved discrete coefficients of Riemann-Liouville fractional integral,the stability and convergence of the proposed scheme are analyzed by the energy method.Theoretical results are also confirmed by a numerical example.
文摘The dynamic stability of simple supported viscoelastic column, subjected to a periodic axial force, is investigated. The viscoelastic material was assumed to obey the fractional derivative constitutive relation. The governing equation of motion was derived as a weakly singular Volterra integro-partial-differential equation, and it was simplified into weakly singular Volterra integro-ordinary-differential equation by the Galerkin method. In terms of the averaging method, the dynamical stability was analyzed. A new numerical method is proposed to avoid storing all history data. Numerical examples are presented and the numerical results agree with the analytical ones.
基金The author would like to thank the referees for the helpful suggestions.The work was supported by NSFC Project(Nos.11671342,91430213,11671157 and 11771369)Project of Scientific Research Fund of Hunan Provincial Science and Technology Department(No.2018JJ2374)Key Project of Hunan Provincial Department of Education(No.17A210).
文摘In this paper,we study a class of nonlinear fractional integro-differential equations,the fractional derivative is described in the Caputo sense.Using the properties of the Caputo derivative,we convert the fractional integro-differential equations into equivalent integral-differential equations of Volterra type with singular kernel,then we propose and analyze a spectral Jacobi-collocation approximation for nonlinear integro-differential equations of Volterra type.We provide a rigorous error analysis for the spectral methods,which shows that both the errors of approximate solutions and the errors of approximate fractional derivatives of the solutions decay exponentially in L^(∞)-norm and weighted L^(2)-norm.
文摘This study aims to apply the two-step Adomian decomposition method(TSADM)to find an analytical solution of integro-differential equations for fractional order without discretization/approximation with less number of iterations and reduce the computational efforts.Moreover,we have established the results for the existence and uniqueness of a solution with the help of some fixed point theorems and the Banach contraction principle.Furthermore,the method is demonstrated on different test examples arising in real life situations.It is concluded that the TSADM provides exact solution of the fractional integro-differential equations in one iteration.At the same time,the other existing methods furnish an approximate solution and require lots of computation to solve the problem applying discretization/approximation on fractional operators.
文摘In this paper,we give a new numerical method for solving a linear system of fractional integro-differential equations.The fractional derivative is considered in the Caputo sense.The proposed method is least squares method aid of Hermite polynomials.The suggested method reduces this type of systems to the solution of systems of linear algebraic equations.To demonstrate the accuracy and applicability of the presented method some test examples are provided.Numerical results show that this approach is easy to implement and accurate when applied to integro-differential equations.We show that the solutions approach to classical solutions as the order of the fractional derivatives approach.
基金supported by the National Natural Science Foundation of China(No.12071403).
文摘This paper mainly considers the optimal convergence analysis of the q-Maruyama method for stochastic Volterra integro-differential equations(SVIDEs)driven by Riemann-Liouville fractional Brownian motion under the global Lipschitz and linear growth conditions.Firstly,based on the contraction mapping principle,we prove the well-posedness of the analytical solutions of the SVIDEs.Secondly,we show that the q-Maruyama method for the SVIDEs can achieve strong first-order convergence.In particular,when the q-Maruyama method degenerates to the explicit Euler-Maruyama method,our result improves the conclusion that the convergence rate is H+1/2,H∈(0,1/2)by Yang et al.,J.Comput.Appl.Math.,383(2021),113156.Finally,the numerical experiment verifies our theoretical results.
文摘This paper is concerned with the approximate controllability of nonlinear fractional impulsive neutral stochastic integro-differential equations with nonlocal conditions and infinite delay in Hilbert spaces under the assumptions that the corresponding linear system is approximately controllable. By the Krasnoselskii-Schaefer-type fixed point theorem and stochastic analysis theory, some sufficient conditions are given for the approximate controllability of the system. At the end, an example is given to illustrate the application of our result.