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Legendre-Weighted Residual Methods for System of Fractional Order Differential Equations
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作者 Umme Ruman Md. Shafiqul Islam 《Journal of Applied Mathematics and Physics》 2024年第9期3163-3184,共22页
The numerical approach for finding the solution of fractional order systems of boundary value problems (BPVs) is derived in this paper. The implementation of the weighted residuals such as Galerkin, Least Square, and ... The numerical approach for finding the solution of fractional order systems of boundary value problems (BPVs) is derived in this paper. The implementation of the weighted residuals such as Galerkin, Least Square, and Collocation methods are included for solving fractional order differential equations, which is broadened to acquire the approximate solutions of fractional order systems with differentiable polynomials, namely Legendre polynomials, as basis functions. The algorithm of the residual formulations of matrix form can be coded efficiently. The interpretation of Caputo fractional derivatives is employed here. We have demonstrated these methods numerically through a few examples of linear and nonlinear BVPs. The results in absolute errors show that the present method efficiently finds the numerical solutions of fractional order systems of differential equations. 展开更多
关键词 fractional differential equations System of fractional order BVPs Weighted Residual Methods Modified Legendre Polynomials
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Application of Fractional Calculus to a Linear Third Order (Nonhomogeneous and Homogeneous) Ordinary Differential Equation
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作者 李春宜 《Journal of Southeast University(English Edition)》 EI CAS 1999年第2期115-120,共6页
In this paper we apply fractional calculus to solve the 3rd order ordinary differential equation of the following form: (z-a)(z-b)(z-c)φ 3+(βz 2+γz+D)φ 2+(α(2β-3α-3)z+αγ+α(α+1)(a+b+c))φ 1+α(α-... In this paper we apply fractional calculus to solve the 3rd order ordinary differential equation of the following form: (z-a)(z-b)(z-c)φ 3+(βz 2+γz+D)φ 2+(α(2β-3α-3)z+αγ+α(α+1)(a+b+c))φ 1+α(α-1)(β-2α-2)φ=f. 展开更多
关键词 fractional calculus third order differential equation LINEAR NONHOMOGENEOUS HOMOGENEOUS
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On Time Fractional Partial Differential Equations and Their Solution by Certain Formable Transform Decomposition Method
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作者 Rania Saadeh Ahmad Qazza +1 位作者 Aliaa Burqan Shrideh Al-Omari 《Computer Modeling in Engineering & Sciences》 SCIE EI 2023年第9期3121-3139,共19页
This paper aims to investigate a new efficient method for solving time fractional partial differential equations.In this orientation,a reliable formable transform decomposition method has been designed and developed,w... This paper aims to investigate a new efficient method for solving time fractional partial differential equations.In this orientation,a reliable formable transform decomposition method has been designed and developed,which is a novel combination of the formable integral transform and the decomposition method.Basically,certain accurate solutions for time-fractional partial differential equations have been presented.Themethod under concern demandsmore simple calculations and fewer efforts compared to the existingmethods.Besides,the posed formable transformdecompositionmethod has been utilized to yield a series solution for given fractional partial differential equations.Moreover,several interesting formulas relevant to the formable integral transform are applied to fractional operators which are performed as an excellent application to the existing theory.Furthermore,the formable transform decomposition method has been employed for finding a series solution to a time-fractional Klein-Gordon equation.Over and above,some numerical simulations are also provided to ensure reliability and accuracy of the new approach. 展开更多
关键词 caputo derivative fractional differential equations formable transform time-fractional klein-gordon equation decomposition method
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HIGH ACCURACY FINITE VOLUME ELEMENT METHOD FOR TWO-POINT BOUNDARY VALUE PROBLEM OF SECOND ORDER ORDINARY DIFFERENTIAL EQUATIONS 被引量:4
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作者 Wang Tongke(王同科) 《Numerical Mathematics A Journal of Chinese Universities(English Series)》 SCIE 2002年第2期213-225,共13页
In this paper, a high accuracy finite volume element method is presented for two-point boundary value problem of second order ordinary differential equation, which differs from the high order generalized difference me... In this paper, a high accuracy finite volume element method is presented for two-point boundary value problem of second order ordinary differential equation, which differs from the high order generalized difference methods. It is proved that the method has optimal order error estimate O(h3) in H1 norm. Finally, two examples show that the method is effective. 展开更多
关键词 SECOND order ordinary differential equation TWO-POINT boundary value problem high accuracy finite volume element method error estimate.
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Exact Solution to Nonlinear Differential Equations of Fractional Order via (<i>G’</i>/<i>G</i>)-Expansion Method 被引量:4
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作者 Muhammad Younis Asim Zafar 《Applied Mathematics》 2014年第1期1-6,共6页
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
关键词 EXACT Solution to Nonlinear differential equations of fractional order VIA (G’/G)-Expansion Method
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Solving the Nonlinear Variable Order Fractional Differential Equations by Using Euler Wavelets 被引量:1
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作者 Yanxin Wang Li Zhu Zhi Wang 《Computer Modeling in Engineering & Sciences》 SCIE EI 2019年第2期339-350,共12页
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
关键词 EULER WAVELETS variable order fractional differential equationS caputo fractional DERIVATIVES OPERATIONAL matrix convergence analysis.
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Existence and Uniqueness of Solution for a Fractional Order Integro-Differential Equation with Non-Local and Global Boundary Conditions 被引量:2
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作者 Mehran Fatemi Nihan Aliev Sedaghat Shahmorad 《Applied Mathematics》 2011年第10期1292-1296,共5页
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
关键词 fractional order Integro-differential equation NON-LOCAL BOUNDARY Conditions FUNDAMENTAL Solution
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EXISTENCE AND UNIQUENESS RESULTS FOR BOUNDARY VALUE PROBLEMS OF HIGHER ORDER FRACTIONAL INTEGRO-DIFFERENTIAL EQUATIONS INVOLVING GRONWALL'S INEQUALITY IN BANACH SPACES 被引量:2
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作者 Dimplekumar N. CHALISHAJAR K. KARTHIKEYAN 《Acta Mathematica Scientia》 SCIE CSCD 2013年第3期758-772,共15页
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
关键词 Boundary value problems fractional order integro-differential equations bound-ary value problems existence and uniqueness singular gronwall inequality fixed point theorem
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Solving a Nonlinear Multi-Order Fractional Differential Equation Using Legendre Pseudo-Spectral Method
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作者 Yin Yang 《Applied Mathematics》 2013年第1期113-118,共6页
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
关键词 LEGENDRE Pseudo-Spectral Method Multi-order fractional differential equationS caputo DERIVATIVE
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Existence of Periodic Solutions for Odd Order Ordinary Differential Equations via the Homotopy Method
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作者 刘停战 于波 《Northeastern Mathematical Journal》 CSCD 2004年第2期135-138,共4页
This paper deals with the problems of finding periodic solutions for the third order ordinary differential equations of the form (1) where T is a fixed positive number and f satisfies some additional conditions which ... This paper deals with the problems of finding periodic solutions for the third order ordinary differential equations of the form (1) where T is a fixed positive number and f satisfies some additional conditions which will be stated later.The periodicity problem has been one of main topics in the qualitative theory of ordinary 展开更多
关键词 homotopy method finding periodic solution odd order ordinary differential equations global convergence
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A UNIFORMLY CONVERGENT SECOND ORDER DIFFERENCE SCHEME FOR A SINGULARLY PERTURBED SELF-ADJOINT ORDINARY DIFFERENTIAL EQUATION IN CONSERVATION FORM
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作者 郭雯 林鹏程 《Applied Mathematics and Mechanics(English Edition)》 SCIE EI 1989年第3期231-241,共11页
In this paper, based on the idea of El-Mistikawy and Werle[1] we construct a difference scheme for a singularly perturbed self-adjoint ordinary differential equation in conservation form. We prove that it is a uniform... In this paper, based on the idea of El-Mistikawy and Werle[1] we construct a difference scheme for a singularly perturbed self-adjoint ordinary differential equation in conservation form. We prove that it is a uniformly convergent second order scheme. 展开更多
关键词 exp A UNIFORMLY CONVERGENT SECOND order DIFFERENCE SCHEME FOR A SINGULARLY PERTURBED SELF-ADJOINT ordinary differential equation IN CONSERVATION FORM
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Hojman's theorem of the third-order ordinary differential equation
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作者 吕洪升 张宏彬 顾书龙 《Chinese Physics B》 SCIE EI CAS CSCD 2009年第8期3135-3138,共4页
This paper extends Hojman's conservation law to the third-order differential equation. A new conserved quantity is constructed based on the Lie group of transformation generators of the equations of motion. The gener... This paper extends Hojman's conservation law to the third-order differential equation. A new conserved quantity is constructed based on the Lie group of transformation generators of the equations of motion. The generators contain variations of the time and generalized coordinates. Two independent non-trivial conserved quantities of the third-order ordinary differential equation are obtained. A simple example is presented to illustrate the applications of the results. 展开更多
关键词 third-order ordinary differential equation Lie symmetry Hojman's conservation law
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Criteria for System of Three Second-Order Ordinary Differential Equations to Be Reduced to a Linear System via Restricted Class of Point Transformation
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作者 Supaporn Suksern Nawee Sakdadech 《Applied Mathematics》 2014年第3期553-571,共19页
This paper is devoted to the study of the linearization problem of system of three second-order ordinary differential equations and . The necessary conditions for linearization by general point transformation and are ... This paper is devoted to the study of the linearization problem of system of three second-order ordinary differential equations and . The necessary conditions for linearization by general point transformation and are found. The sufficient conditions for linearization by restricted class of point transformation and are obtained. Moreover, the procedure for obtaining the linearizing transformation is provided in explicit forms. Examples demonstrating the procedure of using the linearization theorems are presented. 展开更多
关键词 LINEARIZATION Problem Point Transformation SYSTEM of THREE SECOND-order ordinary differential equation
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Solution of Second-Order Ordinary Differential Equations via Simulated Annealing
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作者 Abdulazeez Bilesanmi Ashiribo Senapon Wusu Akinwale Lewis Olutimo 《Open Journal of Optimization》 2019年第1期32-37,共6页
In this paper, we approach the problem of obtaining approximate solution of second-order initial value problems by converting it to an optimization problem. It is assumed that the solution can be approximated by a pol... In this paper, we approach the problem of obtaining approximate solution of second-order initial value problems by converting it to an optimization problem. It is assumed that the solution can be approximated by a polynomial. The coefficients of the polynomial are then optimized using simulated annealing technique. Numerical examples with good results show the accuracy of the proposed approach compared with some existing methods. 展开更多
关键词 SIMULATED ANNEALING SECOND-order ordinary differential equation POLYNOMIAL Optimization
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中立型Caputo分数阶泛函微分方程解的存在性和Hyers-Ulam稳定性
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作者 王奇 邓茜茜 +1 位作者 解晨曦 胡玉婷 《山东航空学院学报》 2024年第4期139-144,共6页
考虑具有无穷时滞和多个Caputo分数阶导数的中立型分数阶泛函微分方程解的存在性和Hyers-Ulam稳定性。利用压缩映射原理及无穷时滞的相空间理论得到方程解的存在性,并利用分数阶微积分的广义Gronwal型不等式及分数阶积分算子的单调性得... 考虑具有无穷时滞和多个Caputo分数阶导数的中立型分数阶泛函微分方程解的存在性和Hyers-Ulam稳定性。利用压缩映射原理及无穷时滞的相空间理论得到方程解的存在性,并利用分数阶微积分的广义Gronwal型不等式及分数阶积分算子的单调性得到解的Hyers-Ulam稳定性。 展开更多
关键词 caputo分数阶泛函微分方程 压缩映射原理 GRONWALL不等式 HYERS-ULAM稳定性
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CONVERGENCE ANALYSIS OF THE JACOBI SPECTRAL-COLLOCATION METHOD FOR FRACTIONAL INTEGRO-DIFFERENTIAL EQUATIONS 被引量:9
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作者 杨银 陈艳萍 黄云清 《Acta Mathematica Scientia》 SCIE CSCD 2014年第3期673-690,共18页
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
关键词 Spectral Jacobi-collocation method fractional order integro-differential equations caputo derivative
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EXISTENCE OF SOLUTION FOR BOUNDARY VALUE PROBLEM OF NONLINEAR FRACTIONAL DIFFERENTIAL EQUATION 被引量:10
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作者 Su Xinwei Liu Landong 《Applied Mathematics(A Journal of Chinese Universities)》 SCIE CSCD 2007年第3期291-298,共8页
This paper is concerned with the boundary value problem of a nonlinear fractional differential equation. By means of Schauder fixed-point theorem, an existence result of solution is obtained.
关键词 fractional differential equation boundary value problem caputo's fractional derivative Schauder fixed-point theorem.
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On the partial stability of nonlinear impulsive Caputo fractional systems
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作者 Boulbaba Ghanmi Saifeddine Ghnimi 《Applied Mathematics(A Journal of Chinese Universities)》 SCIE CSCD 2023年第2期166-179,共14页
In this work,stability with respect to part of the variables of nonlinear impulsive Caputo fractional differential equations is investigated.Some effective sufficient conditions of stability,uniform stability,asymptot... In this work,stability with respect to part of the variables of nonlinear impulsive Caputo fractional differential equations is investigated.Some effective sufficient conditions of stability,uniform stability,asymptotic uniform stability and Mittag Leffler stability.The approach presented is based on the specially introduced piecewise continuous Lyapunov functions.Furthermore,some numerical examples are given to show the effectiveness of our obtained theoretical results. 展开更多
关键词 impulsive fractional differential equations Mittag-Leffler function partial stability caputo derivative
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The Fractional Investigation of Fornberg-Whitham Equation Using an Efficient Technique
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作者 Hassan Khan Poom Kumam +2 位作者 Asif Nawaz Qasim Khan Shahbaz Khan 《Computer Modeling in Engineering & Sciences》 SCIE EI 2023年第4期259-273,共15页
In the last few decades,it has become increasingly clear that fractional calculus always plays a very significant role in various branches of applied sciences.For this reason,fractional partial differential equations(... In the last few decades,it has become increasingly clear that fractional calculus always plays a very significant role in various branches of applied sciences.For this reason,fractional partial differential equations(FPDEs)are of more importance to model the different physical processes in nature more accurately.Therefore,the analytical or numerical solutions to these problems are taken into serious consideration and several techniques or algorithms have been developed for their solution.In the current work,the idea of fractional calculus has been used,and fractional FornbergWhithamequation(FFWE)is represented in its fractional view analysis.Awell-knownmethod which is residual power series method(RPSM),is then implemented to solve FFWE.TheRPSMresults are discussed through graphs and tables which conform to the higher accuracy of the proposed technique.The solutions at different fractional orders are obtained and shown to be convergent toward an integer-order solution.Because the RPSM procedure is simple and straightforward,it can be extended to solve other FPDEs and their systems. 展开更多
关键词 caputo derivative fractional partial differential equations fornberg-whitham residual power series method
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Existence of positive solutions for integral boundary value problem of fractional differential equations 被引量:4
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作者 Xiping Liu Guiyun Wu 《上海师范大学学报(自然科学版)》 2014年第5期496-505,共10页
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
关键词 fractional differential equations Riemann-Liouville fractional derivative fixed point theorem fractional order linear derivative operator
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