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 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 Released power in a vortex-antivortex pairs annihilation process

2020 ◽  
Vol 20 (1) ◽  
Author(s):  
Cristian Aguirre-Tellez ◽  
Miryam Rincón-Joya ◽  
José José Barba-Ortega
Keyword(s):  
Magnetic Field ◽  
External Magnetic Field ◽  
Diffusion Equation ◽  
Power Dissipation ◽  
Time Dependent ◽  
Thermal Dissipation ◽  
Annihilation Process ◽  
Ginzburg Landau ◽  
Dissipation Process ◽  
Ginzburg Landau Equations

In this paper, we studied the power dissipation process of a Shubnikov vortex-antivortex pair in a mesoscopic superconducting square sample with a concentric square defect in presence of an oscillatory external magnetic field. The time-dependent Ginzburg-Landau equations and the diffusion equation were numerically solved. The significant result is that the thermal dissipation is associated with a sizeable relaxation of the superconducting electrons, so that the power released in this kind of process might become calculated as a function of the time. Also, we analyzed the effect that the Ginbzurg-Landau κand deformation τparameters have on the magnetization, dissipate power and super-electrons density.

scholarly journals A Novel Method for Solving the Bagley-Torvik Equation as Ordinary Differential Equation

2019 ◽  
Vol 14 (8) ◽  
Author(s):  
Yong Xu ◽  
Qixian Liu ◽  
Jike Liu ◽  
Yanmao Chen
Keyword(s):  
Differential Equation ◽  
Boundary Condition ◽  
Ordinary Differential Equation ◽  
Diffusion Equation ◽  
Fractional Derivative ◽  
Separation Of Variables ◽  
Computational Cost ◽  
Diffusion Equations ◽  
Novel Method ◽  
High Computational Efficiency

We present a novel method to solve the Bagley-Torvik equation by transforming it into ordinary differential equations (ODEs). This method is based on the equivalence between the Caputo-type fractional derivative (FD) of order 3/2 and the solution of a diffusion equation subjected to certain initial and boundary conditions. The key procedure is to approximate the infinite boundary condition by a finite one, so that the diffusion equation can be solved by separation of variables. By this procedure, the Bagley-Torvik and the diffusion equations together are transformed to be a set of ODEs, which can be integrated numerically by the Runge-Kutta scheme. The presented method is tested by various numerical cases including linear, nonlinear, nonsmooth, or multidimensional equations, respectively. Importantly, high computational efficiency is achieved as this method is at the expense of linearly increasing computational cost with the solution domain being enlarged.

scholarly journals From continuous time random walks to the generalized diffusion equation

2018 ◽  
Vol 21 (1) ◽  
pp. 10-28 ◽  
Author(s):  
Trifce Sandev ◽  
Ralf Metzler ◽  
Aleksei Chechkin
Keyword(s):  
Diffusion Equation ◽  
Continuous Time ◽  
Fractional Diffusion ◽  
Distribution Functions ◽  
Diffusion Equations ◽  
Fractional Diffusion Equations ◽  
Probability Distribution Functions ◽  
Special Cases ◽  
Continuous Time Random Walks ◽  
Random Walk Theory

AbstractWe obtain a generalized diffusion equation in modified or Riemann-Liouville form from continuous time random walk theory. The waiting time probability density function and mean squared displacement for different forms of the equation are explicitly calculated. We show examples of generalized diffusion equations in normal or Caputo form that encode the same probability distribution functions as those obtained from the generalized diffusion equation in modified form. The obtained equations are general and many known fractional diffusion equations are included as special cases.

scholarly journals A Convergent Collocation Approach for Generalized Fractional Integro-Differential Equations Using Jacobi Poly-Fractonomials

Mathematics ◽  
2021 ◽  
Vol 9 (9) ◽  
pp. 979
Author(s):  
Sandeep Kumar ◽  
Rajesh K. Pandey ◽  
H. M. Srivastava ◽  
G. N. Singh
Keyword(s):  
Differential Equation ◽  
Fractional Derivative ◽  
Collocation Method ◽  
Caputo Fractional Derivative ◽  
Fractional Derivatives ◽  
Special Cases ◽  
Collocation Approach ◽  
Linear And Nonlinear ◽  
Simulation Results ◽  
Theoretical Results

In this paper, we present a convergent collocation method with which to find the numerical solution of a generalized fractional integro-differential equation (GFIDE). The presented approach is based on the collocation method using Jacobi poly-fractonomials. The GFIDE is defined in terms of the B-operator introduced recently, and it reduces to Caputo fractional derivative and other fractional derivatives in special cases. The convergence and error analysis of the proposed method are also established. Linear and nonlinear cases of the considered GFIDEs are numerically solved and simulation results are presented to validate the theoretical results.

scholarly journals Caputo Fractional Derivative and Quantum-Like Coherence

Entropy ◽  
2021 ◽  
Vol 23 (2) ◽  
pp. 211
Author(s):  
Garland Culbreth ◽  
Mauro Bologna ◽  
Bruce J. West ◽  
Paolo Grigolini
Keyword(s):  
Diffusion Equation ◽  
Fractional Derivative ◽  
Anomalous Diffusion ◽  
Caputo Fractional Derivative ◽  
Quantum Coherence ◽  
Time Derivative ◽  
The Other ◽  
Diffusion Equations ◽  
Self Organization ◽  
Ordinary Time

We study two forms of anomalous diffusion, one equivalent to replacing the ordinary time derivative of the standard diffusion equation with the Caputo fractional derivative, and the other equivalent to replacing the time independent diffusion coefficient of the standard diffusion equation with a monotonic time dependence. We discuss the joint use of these prescriptions, with a phenomenological method and a theoretical projection method, leading to two apparently different diffusion equations. We prove that the two diffusion equations are equivalent and design a time series that corresponds to the anomalous diffusion equation proposed. We discuss these results in the framework of the growing interest in fractional derivatives and the emergence of cognition in nature. We conclude that the Caputo fractional derivative is a signature of the connection between cognition and self-organization, a form of cognition emergence different from the other source of anomalous diffusion, which is closely related to quantum coherence. We propose a criterion to detect the action of self-organization even in the presence of significant quantum coherence. We argue that statistical analysis of data using diffusion entropy should help the analysis of physiological processes hosting both forms of deviation from ordinary scaling.

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