scholarly journals Approximate Solutions of the Space-Fractional Diffusion Equations with Additive Noise

2021 ◽  
Author(s):  
Wael W. Mohammed ◽  
Hijaz Ahmad
Keyword(s):  
Additive Noise ◽  
Approximate Solutions ◽  
Fractional Diffusion ◽  
Diffusion Equations ◽  
Critical Dynamics ◽  
Ginzburg Landau ◽  
Fractional Diffusion Equations ◽  
Stochastic Space ◽  
Polynomial Term ◽  
Fractional Space

Abstract In this article we take into account a class of stochastic space diffusion equations with polynomials forced by additive noise. We derive rigorously limiting equations which de…ne the critical dynamics. Also, we approximate solutions of stochastic fractional space di¤usion equations with polynomial term by limiting equations, which are ordinary di¤er-ential equations. Moreover, we address the e¤ect of the noise on the solution’s stabilization. Finally, we apply our results to Fisher’s equation and Ginzburg–Landau models.

scholarly journals IMPACT OF THE SAME DEGENERATE ADDITIVE NOISE ON A COUPLED SYSTEM OF FRACTIONAL SPACE DIFFUSION EQUATIONS

Fractals ◽  
2021 ◽  
Author(s):  
WAEL W. MOHAMMED ◽  
NAVEED IQBAL
Keyword(s):  
Stochastic System ◽  
Additive Noise ◽  
Coupled System ◽  
Fractional Diffusion ◽  
Diffusion Equations ◽  
Volterra System ◽  
Fractional Diffusion Equations ◽  
Brusselator Model ◽  
The Stability ◽  
Fractional Space

In this paper, we present a class of stochastic system of fractional space diffusion equations forced by additive noise. Our goal here is to approximate the solutions of this system via a system of ordinary differential equations. Moreover, we study the influence of the same degenerate additive noise on the stability of the solutions of the stochastic system of fractional diffusion equations. We are interested in the systems that have nonlinear polynomial and give applications as Lotka–Volterra system from biology and the Brusselator model for the Belousov–Zhabotinsky chemical reaction from chemistry to illustrate our results.

scholarly journals Additive Noise Effects on the Stabilization of Fractional-Space Diffusion Equation Solutions

Mathematics ◽  
2022 ◽  
Vol 10 (1) ◽  
pp. 130
Author(s):  
Wael W. Mohammed ◽  
Naveed Iqbal ◽  
Thongchai Botmart
Keyword(s):  
Differential Equation ◽  
Diffusion Equation ◽  
Additive Noise ◽  
Diffusion Equations ◽  
Critical Dynamics ◽  
Ginzburg Landau ◽  
Noise Effects ◽  
Special Cases ◽  
Ginzburg Landau Equations ◽  
Fractional Space

This paper considers a class of stochastic fractional-space diffusion equations with polynomials. We establish a limiting equation that specifies the critical dynamics in a rigorous way. After this, we use the limiting equation, which is an ordinary differential equation, to approximate the solution of the stochastic fractional-space diffusion equation. This equation has never been studied before using a combination of additive noise and fractional-space, therefore we generalize some previously obtained results as special cases. Furthermore, we use Fisher’s and Ginzburg–Landau equations to illustrate our results. Finally, we look at how additive noise affects the stabilization of the solutions.

scholarly journals Solving a Higher-Dimensional Time-Fractional Diffusion Equation via the Fractional Reduced Differential Transform Method

2021 ◽  
Vol 5 (4) ◽  
pp. 168
Author(s):  
Salah Abuasad ◽  
Saleh Alshammari ◽  
Adil Al-rabtah ◽  
Ishak Hashim
Keyword(s):  
Approximate Solutions ◽  
Fractional Diffusion ◽  
Differential Transform Method ◽  
Diffusion Equations ◽  
Caputo Fractional Derivatives ◽  
Transform Method ◽  
Fractional Diffusion Equations ◽  
Higher Dimensional ◽  
Differential Transform ◽  
Reduced Differential Transform Method

In this study, exact and approximate solutions of higher-dimensional time-fractional diffusion equations were obtained using a relatively new method, the fractional reduced differential transform method (FRDTM). The exact solutions can be found with the benefit of a special function, and we applied Caputo fractional derivatives in this method. The numerical results and graphical representations specified that the proposed method is very effective for solving fractional diffusion equations in higher dimensions.

scholarly journals Modified Fractional Reduced Differential Transform Method for the Solution of Multiterm Time-Fractional Diffusion Equations

2019 ◽  
Vol 2019 ◽  
pp. 1-14 ◽  
Author(s):  
Salah Abuasad ◽  
Ishak Hashim ◽  
Samsul Ariffin Abdul Karim
Keyword(s):  
Initial Conditions ◽  
Approximate Solutions ◽  
Fractional Diffusion ◽  
Differential Transform Method ◽  
Diffusion Equations ◽  
Transform Method ◽  
Fractional Diffusion Equations ◽  
Fractional Equations ◽  
Differential Transform ◽  
Reduced Differential Transform Method

In this study, we introduce a new modification of fractional reduced differential transform method (m-FRDTM) to find exact and approximate solutions for nonhomogeneous linear multiterm time-fractional diffusion equations (MT-TFDEs) of constant coefficients in a bounded domain with suitable initial conditions. Different applications in two and three fractional order terms are given to illustrate our new modification. The approximate solutions are given in the form of series solutions. The results show that the m-FRDTM for MT-TFDEs is a powerful method and can be generalized to other types of multiterm time-fractional equations.

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