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 Weak solvability of the unconditionally stable difference scheme for the coupled sine-Gordon system

2020 ◽  
Vol 25 (6) ◽  
pp. 997-1014
Author(s):  
Ozgur Yildirim ◽  
Meltem Uzun
Keyword(s):  
Difference Scheme ◽  
Numerical Method ◽  
Existence And Uniqueness ◽  
Numerical Experiments ◽  
Finite Difference Schemes ◽  
Order Of Accuracy ◽  
Unconditionally Stable ◽  
First Order ◽  
The Difference ◽  
Weak Solvability

In this paper, we study the existence and uniqueness of weak solution for the system of finite difference schemes for coupled sine-Gordon equations. A novel first order of accuracy unconditionally stable difference scheme is considered. The variational method also known as the energy method is applied to prove unique weak solvability.We also present a new unified numerical method for the approximate solution of this problem by combining the difference scheme and the fixed point iteration. A test problem is considered, and results of numerical experiments are presented with error analysis to verify the accuracy of the proposed numerical method.

scholarly journals Determination of source term for fractional heat equation on the sphere

2020 ◽  
Vol 24 (Suppl. 1) ◽  
pp. 361-370
Author(s):  
Nguyen Phuong ◽  
Tran Binh ◽  
Nguyen Luc ◽  
Nguyen Can
Keyword(s):  
Diffusion Equation ◽  
Heat Equation ◽  
Exact Solutions ◽  
Source Term ◽  
Fractional Diffusion ◽  
Inverse Source Problem ◽  
Fractional Heat Equation ◽  
Ill Posed ◽  
Time Fractional Diffusion Equation

In this work, we study a truncation method to solve a time fractional diffusion equation on the sphere of an inverse source problem which is ill-posed in the sense of Hadamard. Through some priori assumption, we present the error estimates between the regularized and exact solutions.

scholarly journals No local L1 solutions for semilinear fractional heat equations

2017 ◽  
Vol 20 (6) ◽  
Author(s):  
Kexue Li
Keyword(s):  
Cauchy Problem ◽  
Heat Equation ◽  
Heat Equations ◽  
Fractional Heat Equation ◽  
The Cauchy Problem

AbstractWe study the Cauchy problem for the semilinear fractional heat equation

scholarly journals Phase-Fitted and Amplification-Fitted Higher Order Two-Derivative Runge-Kutta Method for the Numerical Solution of Orbital and Related Periodical IVPs

2017 ◽  
Vol 2017 ◽  
pp. 1-11 ◽  
Author(s):  
N. A. Ahmad ◽  
N. Senu ◽  
F. Ismail
Keyword(s):  
Numerical Solution ◽  
Numerical Experiments ◽  
Initial Value ◽  
Oscillatory Solutions ◽  
First Order ◽  
Runge Kutta Method ◽  
Algebraic Order ◽  
Runge Kutta ◽  
The Stability ◽  
Order Four

A phase-fitted and amplification-fitted two-derivative Runge-Kutta (PFAFTDRK) method of high algebraic order for the numerical solution of first-order Initial Value Problems (IVPs) which possesses oscillatory solutions is derived. We present a sixth-order four-stage two-derivative Runge-Kutta (TDRK) method designed using the phase-fitted and amplification-fitted property. The stability of the new method is analyzed. The numerical experiments are carried out to show the efficiency of the derived methods in comparison with other existing Runge-Kutta (RK) methods.

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.

scholarly journals Analytical Solutions of a Class of Fluids Models with the Caputo Fractional Derivative

2022 ◽  
Vol 6 (1) ◽  
pp. 35
Author(s):  
Ndolane Sene
Keyword(s):  
Diffusion Equation ◽  
Prandtl Number ◽  
Fractional Derivative ◽  
Analytical Solutions ◽  
Caputo Fractional Derivative ◽  
Caputo Derivative ◽  
Fractional Diffusion ◽  
Fluid Models ◽  
Fractional Operators ◽  
Fractional Heat Equation

This paper studies the analytical solutions of the fractional fluid models described by the Caputo derivative. We combine the Fourier sine and the Laplace transforms. We analyze the influence of the order of the Caputo derivative the Prandtl number, the Grashof numbers, and the Casson parameter on the dynamics of the fractional diffusion equation with reaction term and the fractional heat equation. In this paper, we notice that the order of the Caputo fractional derivative plays the retardation effect or the acceleration. The physical interpretations of the influence of the parameters of the model have been proposed. The graphical representations illustrate the main findings of the present paper. This paper contributes to answering the open problem of finding analytical solutions to the fluid models described by the fractional operators.

scholarly journals Meshless Technique for the Solution of Time-Fractional Partial Differential Equations Having Real-World Applications

2020 ◽  
Vol 2020 ◽  
pp. 1-17 ◽  
Author(s):  
Mehnaz Shakeel ◽  
Iltaf Hussain ◽  
Hijaz Ahmad ◽  
Imtiaz Ahmad ◽  
Phatiphat Thounthong ◽  
...  
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Difference Scheme ◽  
Caputo Derivative ◽  
Time Derivative ◽  
Order Derivative ◽  
Fractional Partial Differential Equations ◽  
Partial Differential ◽  
Central Difference ◽  
Definition Of

In this article, radial basis function collocation scheme is adopted for the numerical solution of fractional partial differential equations. This method is highly demanding because of its meshless nature and ease of implementation in high dimensions and complex geometries. Time derivative is approximated by Caputo derivative for the values of α ∈ 0 , 1 and α ∈ 1 , 2 . Forward difference scheme is applied to approximate the 1st order derivative appearing in the definition of Caputo derivative for α ∈ 0 , 1 , whereas central difference scheme is used for the 2nd order derivative in the definition of Caputo derivative for α ∈ 1 , 2 . Numerical problems are given to judge the behaviour of the proposed method for both the cases of α . Error norms are used to asses the accuracy of the method. Both uniform and nonuniform nodes are considered. Numerical simulation is carried out for irregular domain as well. Results are also compared with the existing methods in the literature.

Time-fractional diffusion equation in the fractional Sobolev spaces

2015 ◽  
Vol 18 (3) ◽  
Author(s):  
Rudolf Gorenflo ◽  
Yuri Luchko ◽  
Masahiro Yamamoto
Keyword(s):  
Diffusion Equation ◽  
Sobolev Spaces ◽  
Caputo Derivative ◽  
Fractional Integration ◽  
Time Derivative ◽  
Fractional Diffusion ◽  
Fractional Diffusion Equation ◽  
Fractional Sobolev Spaces ◽  
Smooth Functions ◽  
Time Fractional Diffusion Equation

AbstractThe Caputo time-derivative is usually defined pointwise for well-behaved functions, say, for the continuously differentiable functions. Accordingly, in the publications devoted to the theory of the partial fractional differential equations with the Caputo derivatives, the functional spaces where the solutions are looked for are often the spaces of smooth functions that appear to be too narrow for several important applications. In this paper, we propose a definition of the Caputo derivative on a finite interval in the fractional Sobolev spaces and investigate it from the operator theoretic viewpoint. In particular, some important equivalences of the norms related to the fractional integration and differentiation operators in the fractional Sobolev spaces are given. These results are then applied for proving the maximal regularity of the solutions to some initial-boundary-value problems for the time-fractional diffusion equation with the Caputo derivative in the fractional Sobolev spaces.

Modeling Extreme-Event Precursors with the Fractional Diffusion Equation

2015 ◽  
Vol 18 (1) ◽  
Author(s):  
Michele Caputo ◽  
José M. Carcione ◽  
Marco A. B. Botelho
Keyword(s):  
Diffusion Equation ◽  
Difference Operator ◽  
Extreme Event ◽  
Inverse Fourier Transform ◽  
Time Derivative ◽  
First Order ◽  
The Past ◽  
Moderate Earthquake ◽  
The Green Function ◽  
Derivatives Of

AbstractExtreme catastrophic events such as earthquakes, terrorism and economic collapses are difficult to predict. We propose a tentative mathematical model for the precursors of these events based on a memory formalism and apply it to earthquakes suggesting a physical interpretation. In this case, a precursor can be the anomalous increasing rate of events (aftershocks) following a moderate earthquake, contrary to Omori's law. This trend constitute foreshocks of the main event and can be modelled with fractional time derivatives. A fractional derivative of order 0 < v < 2 replaces the first-order time derivative in the classical diffusion equation.We obtain the frequency-domain Green's function and the corresponding time-domain solution by performing an inverse Fourier transform. Alternatively, we propose a numerical algorithm, where the time derivative is computed with the Grünwald-Letnikov expansion, which is a finitedifference generalization of the standard finite-difference operator to derivatives of fractional order. The results match the analytical solution obtained from the Green function. The calculation requires to store the whole field in the computer memory since anomalous diffusion “remembers the past”.

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