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LOCAL GAUSSIAN-COLLOCATION SCHEME TO APPROXIMATE THE SOLUTION OF nonlinear fractional differential equationS USING VOLTERRA INTEGRAL equationS

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Journal of Computational Mathematics 2021年第2期39卷 261-282页

作者： Pouria Assari Fatemeh Asadi-Mehregan Mehdi Dehghan Department of Mathematics Faculty of SciencesBu-Ali Sina UniversityHamedan 65178Iran Department of Applied Mathematics Faculty of Mathematics and Computer SciencesAmirkabir University of TechnologyNo.424Hafez Ave.Tehran 15914Iran

This work describes an accurate and effective method for numerically solving a class of nonlinear fractional differential *** start the method,we equivalently convert these types of differential equations to nonlinear fractional Volterra integral equations of the second kind by integrating from both sides of ***,the solution of the mentioned Volterra integral equations can be estimated using the collocation method based on locally supported Gaussian *** local Gaussian-collocation scheme estimates the unknown function utilizing a small set of data instead of all points in the solution domain,so the proposed method uses much less computer memory and volume computing in comparison with global *** apply the composite non-uniform Gauss-Legendre quadrature formula to estimate singular-fractional integrals in the *** of the fact that the proposed scheme requires no cell structures on the domain,it is a meshless ***,we obtain the error analysis of the proposed method and demon-strate that the convergence rate of the approach is arbitrarily *** examples clearly show the reliability and efficiency of the new technique and confirm the theoretical error estimates.

关键词： nonlinear fractional differential equation Volterra integral equation Gaussian-collocation method Meshless method Error analysis

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Finite-Time Stability for nonlinear fractional differential equations with Time Delay

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Journal of Donghua University(English Edition) 2022年第5期39卷 446-453页

作者： HE Huazhen KOU Chunhai College of Science Donghua UniversityShanghai 201620China

The finite-time stability and the finite-time contractive stability of solutions for nonlinear fractional differential equations with bounded delay are investigated. The derivative of Lyapunov function along solutions of the considered system is defined in terms of the Caputo fractional Dini derivative. Based on the Lyapunov-Razumikhin method, several sufficient criteria are established to guarantee the finite-time stability and the finite-time contractive stability of solutions for the related systems. An example is provided to illustrate the effectiveness of the obtained results.

关键词： finite-time stability nonlinear fractional differential equation time delay Caputo fractional Dini derivative Lyapunov-Razumikhin method

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Solving fractional differential equations via Fixed Points of Chatterjea Maps

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Computer Modeling in Engineering & Sciences 2023年第6期135卷 2617-2648页

作者： Nawab Hussain Saud M.Alsulami Hind Alamri Department of Mathematics King Abdulaziz UniversityP.O.Box 80203Jeddah21589Saudi Arabia Department of Mathematics College of ScienceTaif UniversityP.O.Box 11099Taif21944Saudi Arabia

In this paper,we present the existence and uniqueness of fixed points and common fixed points for Reich and Chatterjea pairs of self-maps in complete metric ***,we study fixed point theorems for Reich and Chatterjea nonexpansive mappings in a Banach space using the Krasnoselskii-Ishikawa iteration method associated withSλand consider some applications of our results to prove the existence of solutions for nonlinear integral and nonlinear fractional differential *** also establish certain interesting examples to illustrate the usability of our results.

关键词： Common fixed points Reich and Chatterjea mappings Krasnoselskii-Ishikawa iteration complete metric space Banach space integral equation nonlinear fractional differential equation

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Existence of Positive Solutions of Three-point Boundary Value Problem for Higher Order nonlinear fractional differential equations

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Chinese Quarterly Journal of Mathematics 2012年第4期27卷 516-525页

作者：韩仁基葛建生蒋威 Department of Basic Courses Huishang Vocational College School of Mathematical Sciences Anhui University

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.

关键词： nonlinear fractional differential equation three-point boundary value problem positive solutions green’s function banach contraction mapping fixed point theorem in cones

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ADI Finite Element Method for 2D nonlinear Time fractional Reaction-Subdiffusion equation

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American Journal of Computational Mathematics 2016年第4期6卷 336-356页

作者： Peng Zhu Shenglan Xie Department of Mathematics Jiaxing University Jiaxing China Nanhu College Jiaxing University Jiaxing China

In this paper, an alternating direction Galerkin finite element method is presented for solving 2D time fractional reaction sub-diffusion equation with nonlinear source term. Firstly, one order implicit-explicit method is used for time discretization, then Galerkin finite element method is adopted for spatial discretization and obtain a fully discrete linear system. Secondly, Galerkin alternating direction procedure for the system is derived by adding an extra term. Finally, the stability and convergence of the method are analyzed rigorously. Numerical results confirm the accuracy and efficiency of the proposed method.

关键词： nonlinear fractional differential equation Alternating Direction Implicit Method Finite Element Method Riemann-Liouville fractional Derivative

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POSITIVE SOLUTIONS TO BOUNDARY VALUE PROBLEMS OF fractional differential equationS WITH PARAMETER

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Annals of differential equations 2014年第2期30卷 191-198页

作者： Qi Wang Dicheng Lu Rui Wang Yayun Fang School of Mathematical Sciences Anhui University

In this paper, we consider a class of nonlinear fractional differential equation boun- dary value problem with a parameter. By some fixed point theorems, sufficient con- ditions for the existence, nonexistence and mul... 详细信息

关键词： nonlinear fractional differential equation boundary value problem positive solution fixed point theorem

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