scholarly journals An Efficient Hybrid Numerical Scheme for Nonlinear Multiterm Caputo Time and Riesz Space Fractional-Order Diffusion Equations with Delay

2021 ◽  
Vol 2021 ◽  
pp. 1-13
Author(s):  
A. K. Omran ◽  
M. A. Zaky ◽  
A. S. Hendy ◽  
V. G. Pimenov
Keyword(s):  
Fractional Order ◽  
Reaction Diffusion ◽  
Riesz Space ◽  
Reaction Diffusion Equation ◽  
Stability And Convergence ◽  
Energy Estimates ◽  
Discrete Energy ◽  
Fully Discrete ◽  
Exponential Rate Of Convergence ◽  
The Stability

In this paper, we construct and analyze a linearized finite difference/Galerkin–Legendre spectral scheme for the nonlinear multiterm Caputo time fractional-order reaction-diffusion equation with time delay and Riesz space fractional derivatives. The temporal fractional orders in the considered model are taken as 0 < β 0 < β 1 < β 2 < ⋯ < β m < 1 . The problem is first approximated by the L 1 difference method on the temporal direction, and then, the Galerkin–Legendre spectral method is applied on the spatial discretization. Armed by an appropriate form of discrete fractional Grönwall inequalities, the stability and convergence of the fully discrete scheme are investigated by discrete energy estimates. We show that the proposed method is stable and has a convergent order of 2 − β m in time and an exponential rate of convergence in space. We finally provide some numerical experiments to show the efficacy of the theoretical results.

scholarly journals Numerical treatment of singularly perturbed delay reaction-diffusion equations

2020 ◽  
Vol 12 (1) ◽  
pp. 15-24
Author(s):  
Gashu Gadisa Kiltu ◽  
Gemechis File Duressa ◽  
Tesfaye Aga Bullo
Keyword(s):  
Time Delay ◽  
Present Method ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Singularly Perturbed ◽  
Diffusion Equations ◽  
Reaction Diffusion Equation ◽  
Stability And Convergence ◽  
Numerical Treatment ◽  
The Stability

This paper presents a uniform convergent numerical method for solving singularly perturbed delay reaction-diffusion equations. The stability and convergence analysis are investigated. Numerical results are tabulated and the effect of the layer on the solution is examined. In a nutshell, the present method improves the findings of some existing numerical methods reported in the literature. Keywords: Singularly perturbed, Time delay, Reaction-diffusion equation, Layer

scholarly journals A Modified Techniques of Fractional-Order Cauchy-Reaction Diffusion Equation via Shehu Transform

2021 ◽  
Vol 2021 ◽  
pp. 1-15
Author(s):  
Mounirah Areshi ◽  
A. M. Zidan ◽  
Rasool Shah ◽  
Kamsing Nonlaopon
Keyword(s):  
Fractional Order ◽  
Reaction Diffusion ◽  
Approximate Solutions ◽  
Transformation Method ◽  
Reaction Diffusion Equations ◽  
Reaction Diffusion Equation ◽  
Closed Form Solutions ◽  
Homotopy Perturbation ◽  
In Series ◽  
The Given

In this article, the iterative transformation method and homotopy perturbation transformation method are applied to calculate the solution of time-fractional Cauchy-reaction diffusion equations. In this technique, Shehu transformation is combined of the iteration and the homotopy perturbation techniques. Four examples are examined to show validation and the efficacy of the present methods. The approximate solutions achieved by the suggested methods indicate that the approach is easy to apply to the given problems. Moreover, the solution in series form has the desire rate of convergence and provides closed-form solutions. It is noted that the procedure can be modified in other directions of fractional order problems. These solutions show that the current technique is very straightforward and helpful to perform in applied sciences.

Numerical Analysis for a Nonlocal Parabolic Problem

2016 ◽  
Vol 6 (4) ◽  
pp. 434-447 ◽  
Author(s):  
M. Mbehou ◽  
R. Maritz ◽  
P.M.D. Tchepmo
Keyword(s):  
Finite Element ◽  
Parabolic Problem ◽  
Reaction Diffusion ◽  
Finite Element Approximation ◽  
Reaction Diffusion Equation ◽  
Element Approximation ◽  
Galerkin Finite Element ◽  
Fully Discrete ◽  
Nonlinear Reaction Diffusion Equation ◽  
Nonlinear Parabolic

AbstractThis article is devoted to the study of the finite element approximation for a nonlocal nonlinear parabolic problem. Using a linearised Crank-Nicolson Galerkin finite element method for a nonlinear reaction-diffusion equation, we establish the convergence and error bound for the fully discrete scheme. Moreover, important results on exponential decay and vanishing of the solutions in finite time are presented. Finally, some numerical simulations are presented to illustrate our theoretical analysis.

scholarly journals A new parallel difference algorithm based on improved alternating segment Crank–Nicolson scheme for time fractional reaction–diffusion equation

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Xiaozhong Yang ◽  
Xu Dang
Keyword(s):  
Parallel Computing ◽  
Diffusion Equation ◽  
Difference Scheme ◽  
Reaction Diffusion ◽  
Difference Schemes ◽  
Reaction Diffusion Equation ◽  
Stability And Convergence ◽  
Parallel Difference ◽  
Fractional Reaction ◽  
Crank Nicolson

Abstract The fractional reaction–diffusion equation has profound physical and engineering background, and its rapid solution research is of important scientific significance and engineering application value. In this paper, we propose a parallel computing method of mixed difference scheme for time fractional reaction–diffusion equation and construct a class of improved alternating segment Crank–Nicolson (IASC–N) difference schemes. The class of parallel difference schemes constructed in this paper, based on the classical Crank–Nicolson (C–N) scheme and classical explicit and implicit schemes, combines with alternating segment techniques. We illustrate the unique existence, unconditional stability, and convergence of the parallel difference scheme solution theoretically. Numerical experiments verify the theoretical analysis, which shows that the IASC–N scheme has second order spatial accuracy and $2-\alpha $ 2 − α order temporal accuracy, and the computational efficiency is greatly improved compared with the implicit scheme and C–N scheme. The IASC–N scheme has ideal computation accuracy and obvious parallel computing properties, showing that the IASC–N parallel difference method is effective for solving time fractional reaction–diffusion equation.

scholarly journals Amplitude Modulation and Synchronization of Fractional-Order Memristor-Based Chua's Circuit

2013 ◽  
Vol 2013 ◽  
pp. 1-10 ◽  
Author(s):  
A. G. Radwan ◽  
K. Moaddy ◽  
I. Hashim
Keyword(s):  
Fractional Order ◽  
Amplitude Modulation ◽  
Lyapunov Stability Theory ◽  
Stability And Convergence ◽  
Chua’S Circuit ◽  
Nonlinear Controller ◽  
Chua's Circuit ◽  
Control Functions ◽  
Nonlinear Control Theory ◽  
The Stability

This paper presents a general synchronization technique and an amplitude modulation of chaotic generators. Conventional synchronization and antisynchronization are considered a very narrow subset from the proposed technique where the scale between the output response and the input response can be controlled via control functions and this scale may be either constant (positive, negative) or time dependent. The concept of the proposed technique is based on the nonlinear control theory and Lyapunov stability theory. The nonlinear controller is designed to ensure the stability and convergence of the proposed synchronization scheme. This technique is applied on the synchronization of two identical fractional-order Chua's circuit systems with memristor. Different examples are studied numerically with different system parameters, different orders, and with five alternative cases where the scaling functions are chosen to be positive/negative and constant/dynamic which covers all possible cases from conventional synchronization to the amplitude modulation cases to validate the proposed concept.

scholarly journals Linear θ-Method and Compact θ-Method for Generalised Reaction-Diffusion Equation with Delay

2018 ◽  
Vol 2018 ◽  
pp. 1-13
Author(s):  
Fengyan Wu ◽  
Qiong Wang ◽  
Xiujun Cheng ◽  
Xiaoli Chen
Keyword(s):  
Diffusion Equation ◽  
Necessary Conditions ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equation ◽  
Stability And Convergence ◽  
Asymptotically Stable ◽  
Sufficient And Necessary Conditions ◽  
Equation Solvability ◽  
Equation With Delay ◽  
Theoretical Results

This paper is concerned with the analysis of the linear θ-method and compact θ-method for solving delay reaction-diffusion equation. Solvability, consistence, stability, and convergence of the two methods are studied. When θ∈[0,1/2), sufficient and necessary conditions are given to show that the two methods are asymptotically stable. When θ∈[1/2,1], the two methods are proven to be unconditionally asymptotically stable. Finally, several examples are carried out to confirm the theoretical results.

scholarly journals A Numerical Study of Nonlinear Fractional Order Partial Integro-Differential Equation with a Weakly Singular Kernel

2021 ◽  
Vol 5 (3) ◽  
pp. 85
Author(s):  
Tayyaba Akram ◽  
Zeeshan Ali ◽  
Faranak Rabiei ◽  
Kamal Shah ◽  
Poom Kumam
Keyword(s):  
Differential Equation ◽  
Fractional Order ◽  
Numerical Study ◽  
Stability And Convergence ◽  
Weakly Singular Kernel ◽  
Integro Differential Equation ◽  
Suggested Technique ◽  
Collocation Approach ◽  
The Stability ◽  
Efficiency And Reliability

Fractional differential equations can present the physical pathways with the storage and inherited properties due to the memory factor of fractional order. The purpose of this work is to interpret the collocation approach for tackling the fractional partial integro-differential equation (FPIDE) by employing the extended cubic B-spline (ECBS). To determine the time approximation, we utilize the Caputo approach. The stability and convergence analysis have also been analyzed. The efficiency and reliability of the suggested technique are demonstrated by two numerical applications, which support the theoretical results and the effectiveness of the implemented algorithm.

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