Numerical Methods for Fractional Percolation Equation with Riesz Space Fractional Derivative

2020 ◽  
Vol 13 (6) ◽  
pp. 438
Author(s):  
Iman Isho Gorial
Keyword(s):  
Numerical Methods ◽  
Fractional Derivative ◽  
Riesz Space

scholarly journals Novel Second-Order Accurate Implicit Numerical Methods for the Riesz Space Distributed-Order Advection-Dispersion Equations

2015 ◽  
Vol 2015 ◽  
pp. 1-14 ◽  
Author(s):  
X. Wang ◽  
F. Liu ◽  
X. Chen
Keyword(s):  
Numerical Methods ◽  
Quadrature Rule ◽  
Second Order ◽  
Riesz Space ◽  
Stability And Convergence ◽  
Dispersion Equations ◽  
Advection Dispersion ◽  
Alternating Direction ◽  
The Stability ◽  
Distributed Order

We derive and analyze second-order accurate implicit numerical methods for the Riesz space distributed-order advection-dispersion equations (RSDO-ADE) in one-dimensional (1D) and two-dimensional (2D) cases, respectively. Firstly, we discretize the Riesz space distributed-order advection-dispersion equations into multiterm Riesz space fractional advection-dispersion equations (MT-RSDO-ADE) by using the midpoint quadrature rule. Secondly, we propose a second-order accurate implicit numerical method for the MT-RSDO-ADE. Thirdly, stability and convergence are discussed. We investigate the numerical solution and analysis of the RSDO-ADE in 1D case. Then we discuss the RSDO-ADE in 2D case. For 2D case, we propose a new second-order accurate implicit alternating direction method, and the stability and convergence of this method are proved. Finally, numerical results are presented to support our theoretical analysis.

Soliton Solutions of Nonlinear and Nonlocal Sine-Gordon Equation Involving Riesz Space Fractional Derivative

2015 ◽  
Vol 70 (8) ◽  
pp. 659-667 ◽  
Author(s):  
Santanu Saha Ray
Keyword(s):  
Fourier Transform ◽  
Approximate Solution ◽  
Fractional Derivative ◽  
Homotopy Analysis Method ◽  
Gordon Equation ◽  
Soliton Solutions ◽  
Riesz Space ◽  
Analysis Method ◽  
Sine Gordon Equation ◽  
Homotopy Analysis

AbstractIn this article, a novel approach comprising modified homotopy analysis method with Fourier transform has been implemented for the approximate solution of fractional sine-Gordon equation (SGE) ${u_{tt}}\; - \;{}^RD_x^\alpha u\; + \;{\rm{sin}}u\; = \;0,$ where $^RD_x^\alpha $ is the Riesz space fractional derivative, 1 ≤ α ≤ 2. For α=2, it becomes classical SGE utt−uxx + sinu=0, and corresponding to α=1, it becomes nonlocal SGE utt−Hu + sinu=0, which arises in the Josephson junction theory, where H is the Hilbert transform. The fractional SGE is considered as an interpolation between the classical SGE (corresponding to α=2) and nonlocal SGE (corresponding to α=1). Here, the approximate solution of fractional SGE is derived by using modified homotopy analysis method with the Fourier transform. Then, we analyse the results by numerical simulations, which demonstrate the simplicity and effectiveness of the present method.

Fractional nonlinear stochastic heat equation with variable thermal conductivity

2020 ◽  
Vol 23 (6) ◽  
pp. 1762-1782
Author(s):  
Miloš Japundžić ◽  
Danijela Rajter-Ćirić
Keyword(s):  
Thermal Conductivity ◽  
Heat Equation ◽  
Fractional Derivative ◽  
Variable Thermal Conductivity ◽  
Riesz Space ◽  
Stochastic Heat Equation ◽  
Semigroups Of Operators ◽  
Infinite Domain ◽  
Approximate Problem ◽  
Uniformly Continuous

Abstract We consider a nonlinear stochastic heat equation with Riesz space-fractional derivative and variable thermal conductivity, on infinite domain. First we approximate the original problem by regularizing the Riesz space-fractional derivative. Then we prove that the approximate problem has almost surely a unique solution within a Colombeau generalized stochastic process space. In our solving procedure we use the theory of Colombeau generalized uniformly continuous semigroups of operators. At the end, we study the relation of the original and the approximate problem and prove that, under certain conditions, the derivative operators appearing in these two problems are associated. Even more, we prove that under some additional conditions, solutions of the original and the approximate problem are almost certainly associated as well (assuming that the first one almost surely exists).

scholarly journals Applications of Optimal Spline Approximations for the Solution of Nonlinear Time-Fractional Initial Value Problems

Axioms ◽  
2021 ◽  
Vol 10 (4) ◽  
pp. 249
Author(s):  
Enza Pellegrino ◽  
Francesca Pitolli
Keyword(s):  
Numerical Methods ◽  
Fractional Derivative ◽  
Initial Value Problems ◽  
Real Life ◽  
Collocation Methods ◽  
Approximation Properties ◽  
Initial Value ◽  
Numerical Tests ◽  
Fractional Differential ◽  
Life Phenomena

Nonlinear fractional differential equations are widely used to model real-life phenomena. For this reason, there is a need for efficient numerical methods to solve such problems. In this respect, collocation methods are particularly attractive for their ability to deal with the nonlocal behavior of the fractional derivative. Among the variety of collocation methods, methods based on spline approximations are preferable since the approximations can be represented by local bases, thereby reducing the computational load. In this paper, we use a collocation method based on spline quasi-interpolant operators to solve nonlinear time-fractional initial value problems. The numerical tests we performed show that the method has good approximation properties.

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