A New Approach and Solution Technique to Solve Time Fractional Nonlinear Reaction-Diffusion Equations

Sungu Inci Cilingir; Demir Huseyin

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A New Approach and Solution Technique to Solve Time Fractional Nonlinear Reaction-Diffusion Equations

机译：求解时间分数阶非线性反应扩散方程的新方法和求解技术

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A new application of the hybrid generalized differential transform and finite difference method is proposed by solving time fractional nonlinear reaction-diffusion equations. This method is a combination of the multi-time-stepping temporal generalized differential transform and the spatial finite difference methods. The procedure first converts the time-evolutionary equations into Poisson equations which are then solved using the central difference method. The temporal differential transform method as used in the paper takes care of stability and the finite difference method on the resulting equation results in a system of diagonally dominant linear algebraic equations. The Gauss-Seidel iterative procedure then used to solve the linear system thus has assured convergence. To have optimized convergence rate, numerical experiments were done by using a combination of factors involving multi-time-stepping, spatial step size, and degree of the polynomial fit in time. It is shown that the hybrid technique is reliable, accurate, and easy to apply.

机译：通过求解时间分数阶非线性反应扩散方程，提出了混合广义差分变换与有限差分方法的新应用。该方法是多时间步时间广义差分变换和空间有限差分方法的组合。该过程首先将时间演化方程转换为泊松方程，然后使用中心差法求解。本文中使用的时间微分变换方法要考虑稳定性，而对所得方程的有限差分法将导致对角线优势线性代数方程组。因此，用于求解线性系统的高斯-赛德尔迭代过程确保了收敛。为了获得最佳的收敛速度，通过使用涉及多个时间步长，空间步长大小和多项式时间拟合度的因素的组合来进行数值实验。结果表明，混合技术可靠，准确，易于应用。

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《Mathematical Problems in Engineering》 |2015年第12期|457013.1-457013.13|共13页
作者
Sungu Inci Cilingir; Demir Huseyin;
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Ondokuz Mayis Univ, Fac Educ, Sch Math Educ, Dept Elementary, TR-55139 Samsun, Turkey.;

Ondokuz Mayis Univ, Arts & Sci Fac, Dept Math, TR-55139 Samsun, Turkey.;

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1. A New Approach and Solution Technique to Solve Time Fractional Nonlinear Reaction-Diffusion Equations [J] . InciCilingir Sungu, HuseyinDemir Mathematical Problems in Engineering: Theory, Methods and Applications . 2015,第5期

机译：求解时间分数阶非线性反应扩散方程的新方法和求解技术
2. In this paper, we implement the fractional complex transform method to convert the nonlinear fractional Klein-Gordon equation (FKGE) to an ordinary differential equation. We use the variational iteration method (VIM) to solve the resulting ODE. The fractional derivatives are presented in terms of the Caputo sense. Some numerical examples are presented to validate the proposed techniques. Finally, a comparison with the numerical solution using Runge-Kutta of order four is given. [J] . M. M. Khader, M. Adel Nonlinear engineering: Modeling and application . 2016,第3期

机译：在本文中，我们采用分数阶复杂变换方法将非线性分数阶Klein-Gordon方程（FKGE）转换为常微分方程。我们使用变分迭代方法（VIM）来解决所得的ODE。分数导数以Caputo形式表示。提出了一些数值例子来验证所提出的技术。最后，与使用四阶Runge-Kutta的数值解进行了比较。
3. A high-order numerical scheme for solving nonlinear time fractional reaction-diffusion equations with initial singularity [J] . Haiyu Liu, Shujuan Lue Applied numerical mathematics . 2021,第Nova期

机译：具有初始奇异性的非线性时间分数反应扩散方程的高阶数值方案
4. Computationally efficient methods for solving time-variable-order time-space fractional reaction-diffusion equation [C] . Qianqian Yang, Timothy Moroney, Fawang Liu, Proceedings of the Fifth symposium on fractional differentiation and its applications . 2012

机译：求解时变阶时空分数反应扩散方程的计算有效方法
5. Approximate Techniques for the Solution of Unconfined Groundwater Flow Equations and a Derivation of a Two-Sided Fractional Conservation of Mass Equation. [D] . Olsen, Jeffrey S. 2017

机译：无限制地下水流方程解的近似技术和质量方程的双向分数守恒推导。
6. Analytical Solutions for Multi-Time Scale Fractional Stochastic Differential Equations Driven by Fractional Brownian Motion and Their Applications [O] . Xiao-Li Ding, Juan J. Nieto 2018

机译：分数褐色运动及其应用驱动的多时间尺度分数随机微分方程的分析解
7. A New Approach and Solution Technique to Solve Time Fractional Nonlinear Reaction-Diffusion Equations [O] . Inci Cilingir Sungu, Huseyin Demir 2015

机译：一种解决时间分数非线性反应扩散方程的新方法和解决方案技术

A New Approach and Solution Technique to Solve Time Fractional Nonlinear Reaction-Diffusion Equations

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