This article introduces the computational analytical approach to solve the m-dimensional space-time variable Caputo fractional order advection–dispersion equation with the Dirichlet boundary using the two-step Adomia...This article introduces the computational analytical approach to solve the m-dimensional space-time variable Caputo fractional order advection–dispersion equation with the Dirichlet boundary using the two-step Adomian decomposition method and obtain the exact solution in just one iteration.Moreover,with the help of fixed point theory,we study the existence and uniqueness conditions for the positive solution and prove some new results.Also,obtain the Ulam–Hyers stabilities for the proposed problem.Two gen-eralized examples are considered to show the method’s applicability and compared with other existing numerical methods.The present method performs exceptionally well in terms of efficiency and simplicity.Further,we solved both examples using the two most well-known numerical methods and compared them with the TSADM solution.展开更多
Fractional differential equations have recently been applied in various areas of engineering, science, finance, applied mathematics, bio-engineering and others. However, many researchers remain unaware of this field. ...Fractional differential equations have recently been applied in various areas of engineering, science, finance, applied mathematics, bio-engineering and others. However, many researchers remain unaware of this field. In this paper, an efficient numerical method for solving the fractional Advection-dispersion equation (ADE) is considered. The fractional derivative is described in the Caputo sense. The method is based on Chebyshev approximations. The properties of Chebyshev polynomials are used to reduce ADE to a system of ordinary differential equations, which are solved using the finite difference method (FDM). Moreover, the convergence analysis and an upper bound of the error for the derived formula are given. Numerical solutions of ADE are presented and the results are compared with the exact solution.展开更多
In this paper,we propose numerical methods for the Riesz space fractional advection-dispersion equations with delay(RFADED).We utilize the fractional backward differential formulas method of second order(FBDF2)and wei...In this paper,we propose numerical methods for the Riesz space fractional advection-dispersion equations with delay(RFADED).We utilize the fractional backward differential formulas method of second order(FBDF2)and weighted shifted Grünwald difference(WSGD)operators to approximate the Riesz fractional derivative and present the finite difference method for the RFADED.Firstly,the FBDF2 and the shifted Grünwald methods are introduced.Secondly,based on the FBDF2 method and the WSGD operators,the finite difference method is applied to the problem.We also show that our numerical schemes are conditionally stable and convergent with the accuracy of O(+h2)and O(2+h2)respectively.Thirdly we find the analytical solution for RFDED in terms Mittag-Leffler type functions.Finally,some numerical examples are given to show the efficacy of the numerical methods and the results are found to be in complete agreement with the analytical solution.展开更多
In this paper,a Riesz space fractional advection-dispersion equation with fractional Robin boundary condition is considered.By applying the fractional central di erence formula and the weighted and shifted Grunwald-L...In this paper,a Riesz space fractional advection-dispersion equation with fractional Robin boundary condition is considered.By applying the fractional central di erence formula and the weighted and shifted Grunwald-Letnikov formula,we derive a weighted implicit nite difference scheme with accuracy O(△t^2+h^2).The solvability,stability,and convergence of the proposed numerical scheme are proved.A numerical example is presented to confirm the accuracy and efficiency of the scheme.展开更多
In this article, a new application to find the exact solutions of nonlinear partial time-space fractional differential Equation has been discussed. Firstly, the fractional complex transformation has been implemented t...In this article, a new application to find the exact solutions of nonlinear partial time-space fractional differential Equation has been discussed. Firstly, the fractional complex transformation has been implemented to convert nonlinear partial fractional differential Equations into nonlinear ordinary differential Equations. Afterwards, the (G'/G)-expansion method has been implemented, to celebrate the exact solutions of these Equations, in the sense of modified Riemann-Liouville derivative. As application, the exact solutions of time-space fractional Burgers’ Equation have been discussed.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
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
An Euler wavelets method is proposed to solve a class of nonlinear variable order fractional differential equations in this paper.The properties of Euler wavelets and their operational matrix together with a family of...An Euler wavelets method is proposed to solve a class of nonlinear variable order fractional differential equations in this paper.The properties of Euler wavelets and their operational matrix together with a family of piecewise functions are first presented.Then they are utilized to reduce the problem to the solution of a nonlinear system of algebraic equations.And the convergence of the Euler wavelets basis is given.The method is computationally attractive and some numerical examples are provided to illustrate its high accuracy.展开更多
In this article, the modified simple equation method has been extended to celebrate the exact solutions of nonlinear partial time-space differential equations of fractional order. Firstly, the fractional complex trans...In this article, the modified simple equation method has been extended to celebrate the exact solutions of nonlinear partial time-space differential equations of fractional order. Firstly, the fractional complex transformation has been implemented to convert nonlinear partial fractional differential equations into nonlinear ordinary differential equations. Afterwards, modified simple equation method has been implemented, to find the exact solutions of these equations, in the sense of modified Riemann-Liouville derivative. For applications, the exact solutions of time-space fractional derivative Burgers’ equation and time-space fractional derivative foam drainage equation have been discussed. Moreover, it can also be concluded that the proposed method is easy, direct and concise as compared to other existing methods.展开更多
In this paper, two numerical methods are proposed for solving distributed-order fractional Bagley-Torvik equation.This equation is used in modeling the motion of a rigid plate immersed in a Newtonian fluid with respec...In this paper, two numerical methods are proposed for solving distributed-order fractional Bagley-Torvik equation.This equation is used in modeling the motion of a rigid plate immersed in a Newtonian fluid with respect to the nonnegative density function. Using the composite Boole's rule the distributedorder Bagley-Torvik equation is approximated by a multi-term time-fractional equation, which is then solved by the GrunwaldLetnikov method(GLM) and the fractional differential transform method(FDTM). Finally, we compared our results with the exact results of some cases and show the excellent agreement between the approximate result and the exact solution.展开更多
In this article, Crank-Nicolson method is used to study the variable order fractional cable equation. The variable order fractional derivatives are described in the Riemann- Liouville and the Griinwald-Letnikov sense....In this article, Crank-Nicolson method is used to study the variable order fractional cable equation. The variable order fractional derivatives are described in the Riemann- Liouville and the Griinwald-Letnikov sense. The stability analysis of the proposed technique is discussed. Numerical results are provided and compared with exact solutions to show the accuracy of the proposed technique.展开更多
We show for the Benjamin-Ono equation an existence uniqeness theorem in Sobolev spaces of arbitrary fractional order s greater-than-or-equal-to 2, provided the initial data is given in the same space.
Stochastic point kinetics equations(SPKEs) are a system of Ito? stochastic differential equations whose solution has been obtained by higher-order approximation.In this study, a fractional model of SPKEs has been anal...Stochastic point kinetics equations(SPKEs) are a system of Ito? stochastic differential equations whose solution has been obtained by higher-order approximation.In this study, a fractional model of SPKEs has been analyzed. The efficiency of the proposed higher-order approximation scheme has been discussed in the results section. The solutions of SPKEs in the presence of Newtonian temperature feedback have also been provided to further discuss the physical behavior of the fractional model.展开更多
In this paper, we apply the Legendre spectral-collocation method to obtain approximate solutions of nonlinear multi-order fractional differential equations (M-FDEs). The fractional derivative is described in the Caput...In this paper, we apply the Legendre spectral-collocation method to obtain approximate solutions of nonlinear multi-order fractional differential equations (M-FDEs). The fractional derivative is described in the Caputo sense. The study is conducted through illustrative example to demonstrate the validity and applicability of the presented method. The results reveal that the proposed method is very effective and simple. Moreover, only a small number of shifted Legendre polynomials are needed to obtain a satisfactory result.展开更多
The reduced-order finite element method (FEM) based on a proper orthogo- nal decomposition (POD) theory is applied to the time fractional Tricomi-type equation. The present method is an improvement on the general ...The reduced-order finite element method (FEM) based on a proper orthogo- nal decomposition (POD) theory is applied to the time fractional Tricomi-type equation. The present method is an improvement on the general FEM. It can significantly save mem- ory space and effectively relieve the computing load due to its reconstruction of POD basis functions. Furthermore, the reduced-order finite element (FE) scheme is shown to be un- conditionally stable, and error estimation is derived in detail. Two numerical examples are presented to show the feasibility and effectiveness of the method for time fractional differential equations (FDEs).展开更多
This paper is concerned with the following variable-order fractional Laplacian equations , where N ≥ 1 and N > 2s(x,y) for (x,y) ∈ Ω × Ω, Ω is a bounded domain in R<sup>N</sup>, s(⋅)...This paper is concerned with the following variable-order fractional Laplacian equations , where N ≥ 1 and N > 2s(x,y) for (x,y) ∈ Ω × Ω, Ω is a bounded domain in R<sup>N</sup>, s(⋅) ∈ C (R<sup>N</sup> × R<sup>N</sup>, (0,1)), (-Δ)<sup>s(⋅)</sup> is the variable-order fractional Laplacian operator, λ, μ > 0 are two parameters, V: Ω → [0, ∞) is a continuous function, f ∈ C(Ω × R) and q ∈ C(Ω). Under some suitable conditions on f, we obtain two solutions for this problem by employing the mountain pass theorem and Ekeland’s variational principle. Our result generalizes the related ones in the literature.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 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
In this paper, we use the fractional complex transform and the (G'/G)-expansion method to study the nonlinear fractional differential equations and find the exact solutions. The fractional complex transform is prop...In this paper, we use the fractional complex transform and the (G'/G)-expansion method to study the nonlinear fractional differential equations and find the exact solutions. The fractional complex transform is proposed to convert a partial fractional differential equation with Jumarie's modified Riemann-Liouville derivative into its ordinary differential equation. It is shown that the considered transform and method are very efficient and powerful in solving wide classes of nonlinear fractional order equations.展开更多
The reproducing kernel particle method (RKPM) has been efficiently applied to problems with large deformations, high gradients and high modal density. In this paper, it is extended to solve a nonlocal problem modele...The reproducing kernel particle method (RKPM) has been efficiently applied to problems with large deformations, high gradients and high modal density. In this paper, it is extended to solve a nonlocal problem modeled by a fractional advectiondiffusion equation (FADE), which exhibits a boundary layer with low regularity. We formulate this method on a moving least-square approach. Via the enrichment of fractional-order power functions to the traditional integer-order basis for RKPM, leading terms of the solution to the FADE can be exactly reproduced, which guarantees a good approximation to the boundary layer. Numerical tests are performed to verify the proposed approach.展开更多
文摘This article introduces the computational analytical approach to solve the m-dimensional space-time variable Caputo fractional order advection–dispersion equation with the Dirichlet boundary using the two-step Adomian decomposition method and obtain the exact solution in just one iteration.Moreover,with the help of fixed point theory,we study the existence and uniqueness conditions for the positive solution and prove some new results.Also,obtain the Ulam–Hyers stabilities for the proposed problem.Two gen-eralized examples are considered to show the method’s applicability and compared with other existing numerical methods.The present method performs exceptionally well in terms of efficiency and simplicity.Further,we solved both examples using the two most well-known numerical methods and compared them with the TSADM solution.
文摘Fractional differential equations have recently been applied in various areas of engineering, science, finance, applied mathematics, bio-engineering and others. However, many researchers remain unaware of this field. In this paper, an efficient numerical method for solving the fractional Advection-dispersion equation (ADE) is considered. The fractional derivative is described in the Caputo sense. The method is based on Chebyshev approximations. The properties of Chebyshev polynomials are used to reduce ADE to a system of ordinary differential equations, which are solved using the finite difference method (FDM). Moreover, the convergence analysis and an upper bound of the error for the derived formula are given. Numerical solutions of ADE are presented and the results are compared with the exact solution.
文摘In this paper,we propose numerical methods for the Riesz space fractional advection-dispersion equations with delay(RFADED).We utilize the fractional backward differential formulas method of second order(FBDF2)and weighted shifted Grünwald difference(WSGD)operators to approximate the Riesz fractional derivative and present the finite difference method for the RFADED.Firstly,the FBDF2 and the shifted Grünwald methods are introduced.Secondly,based on the FBDF2 method and the WSGD operators,the finite difference method is applied to the problem.We also show that our numerical schemes are conditionally stable and convergent with the accuracy of O(+h2)and O(2+h2)respectively.Thirdly we find the analytical solution for RFDED in terms Mittag-Leffler type functions.Finally,some numerical examples are given to show the efficacy of the numerical methods and the results are found to be in complete agreement with the analytical solution.
基金Supported by the Nation Natural Science Foundation of China(No.11271141)the Chongqing Science and Technology Commission(cstc2018jcyjAX0787)。
文摘In this paper,a Riesz space fractional advection-dispersion equation with fractional Robin boundary condition is considered.By applying the fractional central di erence formula and the weighted and shifted Grunwald-Letnikov formula,we derive a weighted implicit nite difference scheme with accuracy O(△t^2+h^2).The solvability,stability,and convergence of the proposed numerical scheme are proved.A numerical example is presented to confirm the accuracy and efficiency of the scheme.
文摘In this article, a new application to find the exact solutions of nonlinear partial time-space fractional differential Equation has been discussed. Firstly, the fractional complex transformation has been implemented to convert nonlinear partial fractional differential Equations into nonlinear ordinary differential Equations. Afterwards, the (G'/G)-expansion method has been implemented, to celebrate the exact solutions of these Equations, in the sense of modified Riemann-Liouville derivative. As application, the exact solutions of time-space fractional Burgers’ Equation have been discussed.
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.
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The authors are grateful to the editor,the associate editor and the anonymous reviewers for their constructive and helpful comments.This work was supported by the Zhejiang Provincial Natural Science Foundation of China(No.LY18A010026),the National Natural Science Foundation of China(No.11701304,11526117)Zhejiang Provincial Natural Science Foundation of China(No.LQ16A010006)+1 种基金the Natural Science Foundation of Ningbo City,China(No.2017A610143)the Natural Science Foundation of Ningbo City,China(2018A610195).
文摘An Euler wavelets method is proposed to solve a class of nonlinear variable order fractional differential equations in this paper.The properties of Euler wavelets and their operational matrix together with a family of piecewise functions are first presented.Then they are utilized to reduce the problem to the solution of a nonlinear system of algebraic equations.And the convergence of the Euler wavelets basis is given.The method is computationally attractive and some numerical examples are provided to illustrate its high accuracy.
文摘In this article, the modified simple equation method has been extended to celebrate the exact solutions of nonlinear partial time-space differential equations of fractional order. Firstly, the fractional complex transformation has been implemented to convert nonlinear partial fractional differential equations into nonlinear ordinary differential equations. Afterwards, modified simple equation method has been implemented, to find the exact solutions of these equations, in the sense of modified Riemann-Liouville derivative. For applications, the exact solutions of time-space fractional derivative Burgers’ equation and time-space fractional derivative foam drainage equation have been discussed. Moreover, it can also be concluded that the proposed method is easy, direct and concise as compared to other existing methods.
文摘In this paper, two numerical methods are proposed for solving distributed-order fractional Bagley-Torvik equation.This equation is used in modeling the motion of a rigid plate immersed in a Newtonian fluid with respect to the nonnegative density function. Using the composite Boole's rule the distributedorder Bagley-Torvik equation is approximated by a multi-term time-fractional equation, which is then solved by the GrunwaldLetnikov method(GLM) and the fractional differential transform method(FDTM). Finally, we compared our results with the exact results of some cases and show the excellent agreement between the approximate result and the exact solution.
文摘In this article, Crank-Nicolson method is used to study the variable order fractional cable equation. The variable order fractional derivatives are described in the Riemann- Liouville and the Griinwald-Letnikov sense. The stability analysis of the proposed technique is discussed. Numerical results are provided and compared with exact solutions to show the accuracy of the proposed technique.
文摘We show for the Benjamin-Ono equation an existence uniqeness theorem in Sobolev spaces of arbitrary fractional order s greater-than-or-equal-to 2, provided the initial data is given in the same space.
文摘Stochastic point kinetics equations(SPKEs) are a system of Ito? stochastic differential equations whose solution has been obtained by higher-order approximation.In this study, a fractional model of SPKEs has been analyzed. The efficiency of the proposed higher-order approximation scheme has been discussed in the results section. The solutions of SPKEs in the presence of Newtonian temperature feedback have also been provided to further discuss the physical behavior of the fractional model.
文摘In this paper, we apply the Legendre spectral-collocation method to obtain approximate solutions of nonlinear multi-order fractional differential equations (M-FDEs). The fractional derivative is described in the Caputo sense. The study is conducted through illustrative example to demonstrate the validity and applicability of the presented method. The results reveal that the proposed method is very effective and simple. Moreover, only a small number of shifted Legendre polynomials are needed to obtain a satisfactory result.
基金Project supported by the National Natural Science Foundation of China(Nos.11361035 and 11301258)the Natural Science Foundation of Inner Mongolia(Nos.2012MS0106 and 2012MS0108)
文摘The reduced-order finite element method (FEM) based on a proper orthogo- nal decomposition (POD) theory is applied to the time fractional Tricomi-type equation. The present method is an improvement on the general FEM. It can significantly save mem- ory space and effectively relieve the computing load due to its reconstruction of POD basis functions. Furthermore, the reduced-order finite element (FE) scheme is shown to be un- conditionally stable, and error estimation is derived in detail. Two numerical examples are presented to show the feasibility and effectiveness of the method for time fractional differential equations (FDEs).
文摘This paper is concerned with the following variable-order fractional Laplacian equations , where N ≥ 1 and N > 2s(x,y) for (x,y) ∈ Ω × Ω, Ω is a bounded domain in R<sup>N</sup>, s(⋅) ∈ C (R<sup>N</sup> × R<sup>N</sup>, (0,1)), (-Δ)<sup>s(⋅)</sup> is the variable-order fractional Laplacian operator, λ, μ > 0 are two parameters, V: Ω → [0, ∞) is a continuous function, f ∈ C(Ω × R) and q ∈ C(Ω). Under some suitable conditions on f, we obtain two solutions for this problem by employing the mountain pass theorem and Ekeland’s variational principle. Our result generalizes the related ones in the literature.
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.
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.
文摘In this paper, we use the fractional complex transform and the (G'/G)-expansion method to study the nonlinear fractional differential equations and find the exact solutions. The fractional complex transform is proposed to convert a partial fractional differential equation with Jumarie's modified Riemann-Liouville derivative into its ordinary differential equation. It is shown that the considered transform and method are very efficient and powerful in solving wide classes of nonlinear fractional order equations.
基金supported partly by the National Natural Science Foundation of China (Grant 11521202)support from the Chinese Scholarship Councilpartially support by an Army Research Office (Grant W911NF-15-10569)
文摘The reproducing kernel particle method (RKPM) has been efficiently applied to problems with large deformations, high gradients and high modal density. In this paper, it is extended to solve a nonlocal problem modeled by a fractional advectiondiffusion equation (FADE), which exhibits a boundary layer with low regularity. We formulate this method on a moving least-square approach. Via the enrichment of fractional-order power functions to the traditional integer-order basis for RKPM, leading terms of the solution to the FADE can be exactly reproduced, which guarantees a good approximation to the boundary layer. Numerical tests are performed to verify the proposed approach.