scholarly journals On the influence of the method of approximation of an unknown function on the stability of numerical methods for solving the anomalous diffusion equation

2019 ◽  
pp. 23-29
Author(s):  
Vladimir Andreevich Litvinov
Keyword(s):  
Diffusion Equation ◽  
Numerical Solution ◽  
Anomalous Diffusion ◽  
Caputo Derivative ◽  
Numerical Algorithms ◽  
Difference Approximation ◽  
Liouville Derivative ◽  
The Subject ◽  
The Difference ◽  
The Stability

The subject of the research is numerical algorithms for solving fractional partial differential equations. The object of the study is the stability of several algorithms for the numerical solution of the anomalous diffusion equation. Algorithms based on the difference representation of the fractional Riemann-Liuville derivative and the Caputo derivative for various orders of accuracy are considered. A comparison is made of the results of numerical calculations using the analyzed algorithms for a model problem with the exact solution of the anomalous diffusion equation for various orders of the fractional derivative with respect to the spatial coordinate. The results of the work were obtained on the basis of the analysis of the constructed difference schemes for stability, the conducted numerical experiments and a comparative analysis of the data obtained. The main conclusions of the study are the advantage of using the approximation of the fractional Caputo derivative compared to using the difference scheme for the fractional Riemann-Liouville derivative in the numerical solution of the anomalous diffusion equation. The paper also indicates the importance of choosing the method of difference approximation of the second derivative, which is a derivative of the Caputo.

A new difference scheme for time fractional heat equations based on the Crank-Nicholson method

2013 ◽  
Vol 16 (4) ◽  
Author(s):  
Ibrahim Karatay ◽  
Nurdane Kale ◽  
Serife Bayramoglu
Keyword(s):  
Diffusion Equation ◽  
Heat Equation ◽  
Difference Scheme ◽  
Numerical Solution ◽  
Caputo Derivative ◽  
Numerical Experiments ◽  
Time Derivative ◽  
Heat Equations ◽  
First Order ◽  
Fractional Heat Equation

AbstractIn this paper, we consider the numerical solution of a time-fractional heat equation, which is obtained from the standard diffusion equation by replacing the first-order time derivative with the Caputo derivative of order α, where 0 < α < 1. The main purpose of this work is to extend the idea on the Crank-Nicholson method to the time-fractional heat equations. By the method of the Fourier analysis, we prove that the proposed method is stable and the numerical solution converges to the exact one with the order O(τ 2-α + h 2), conditionally. Numerical experiments are carried out to support the theoretical claims.

scholarly journals II. On the mathematical theory of the stability of earth-work and masonry

1857 ◽  
Vol 8 ◽  
pp. 60-61 ◽  
Keyword(s):  
Compressive Stress ◽  
Royal Society ◽  
Mathematical Theory ◽  
Natural Slope ◽  
The Earth ◽  
The Subject ◽  
The Difference ◽  
The Stability

In the preparation of my course of lectures, I have found it necessary to re-investigate much of the above-named branch of mechanics, and I have now a paper in preparation on the subject, which I propose to offer to the Royal Society when it is ready. In the meanwhile, it appears to me that the two fundamental principles on which my researches are based are of such a nature, that they may very properly be communicated to the Royal Society at once. They are as follows:― I. Principle o f the Stability of Earth . At each point in a mass of earth the directions of greatest and least compressive stress are at right angles to each other; and the condition of stability is, that at each point the ratio of the difference of those stresses to their sum shall not exceed the sine of the angle of natural slope of the earth.

scholarly journals Stability Analysis of Perturbed Linear Non-integer Differential Systems

2021 ◽  
pp. 24-29
Author(s):  
Ubong D. Akpan
Keyword(s):  
Stability Analysis ◽  
Asymptotic Stability ◽  
Differential System ◽  
Caputo Derivative ◽  
Differential Systems ◽  
Fractional Differential System ◽  
Liouville Derivative ◽  
Fractional Differential ◽  
The Stability

In this work, the effect of perturbation on linear fractional differential system is studied. The analysis is done using Riemann-Liouville derivative and the conclusion extended to using Caputo derivative since the result is similar. Conditions for determining the stability and asymptotic stability of perturbed linear fractional differential system are given.

scholarly journals On Numerical Solution Of The Time Fractional Advection-Diffusion Equation Involving Atangana-Baleanu-Caputo Derivative

Open Physics ◽  
2019 ◽  
Vol 17 (1) ◽  
pp. 816-822 ◽  
Author(s):  
Mohammad Partohaghighi ◽  
Mustafa Inc ◽  
Mustafa Bayram ◽  
Dumitru Baleanu
Keyword(s):  
Diffusion Equation ◽  
Fractional Derivative ◽  
Numerical Solution ◽  
Caputo Derivative ◽  
Initial Guess ◽  
Advection Diffusion Equation ◽  
Advection Diffusion ◽  
Time Fractional Derivative ◽  
Group Preserving Scheme ◽  
New Equation

Abstract A powerful algorithm is proposed to get the solutions of the time fractional Advection-Diffusion equation(TFADE): $^{ABC}\mathcal{D}_{0^+,t}^{\beta}u(x,t) =\zeta u_{xx}(x,t)- \kappa u_x(x,t)+$ F(x, t), 0 < β ≤ 1. The time-fractional derivative $^{ABC}\mathcal{D}_{0^+,t}^{\beta}u(x,t)$is described in the Atangana-Baleanu Caputo concept. The basis of our approach is transforming the original equation into a new equation by imposing a transformation involving a fictitious coordinate. Then, a geometric scheme namely the group preserving scheme (GPS) is implemented to solve the new equation by taking an initial guess. Moreover, in order to present the power of the presented approach some examples are solved, successfully.

scholarly journals Fourth-Order Difference Approximation for Time-Fractional Modified Sub-Diffusion Equation

Symmetry ◽  
2020 ◽  
Vol 12 (5) ◽  
pp. 691 ◽  
Author(s):  
Umair Ali ◽  
Muhammad Sohail ◽  
Muhammad Usman ◽  
Farah Aini Abdullah ◽  
Ilyas Khan ◽  
...  
Keyword(s):  
Diffusion Equation ◽  
Difference Scheme ◽  
Fourth Order ◽  
Difference Approximation ◽  
Implicit Difference Scheme ◽  
Order Difference ◽  
Von Neumann ◽  
Symmetry Properties ◽  
Fractional Differential ◽  
The Stability

Fractional differential equations describe nature adequately because of the symmetry properties which describe physical and biological processes. In this article, a fourth-order new implicit difference scheme is formulated and applied to solve the two-dimensional time-fractional modified sub-diffusion equation involving two times Riemann–Liouville fractional derivatives. The stability of the fourth-order implicit difference scheme is investigated using the von Neumann technique. The proposed scheme is shown to be unconditionally stable. Numerical examples are given to illustrate the feasibility of the proposed scheme.

Compact Crank–Nicolson and Du Fort–Frankel Method for the Solution of the Time Fractional Diffusion Equation

2015 ◽  
Vol 12 (06) ◽  
pp. 1550041 ◽  
Author(s):  
Faoziya Al-Shibani ◽  
Ahmad Ismail
Keyword(s):  
Finite Difference ◽  
Diffusion Equation ◽  
Finite Difference Methods ◽  
Fractional Diffusion ◽  
Fractional Diffusion Equation ◽  
Finite Difference Approximation ◽  
Difference Approximation ◽  
Time Fractional Diffusion Equation ◽  
The Stability ◽  
The One

In this paper, two compact implicit finite difference methods are developed and analyzed for solving the one-dimensional time fractional diffusion equation. The temporal derivative is approximated by using Grünwald–Letnikov formula. Compact finite difference approximation is used for the second-order derivative in space. The local truncation errors are discussed. The stability analysis and the convergence of the proposed methods are investigated by means of Fourier series method. A comparison between the results of these methods and the exact solution is made. Numerical tests are given to verify the feasibility and accuracy of the methods.

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