Numerical Algorithm of the Fractional Diffusion Equation Based on Sinc-Chebyshev Collocation

2014 ◽  
Vol 926-930 ◽  
pp. 3105-3108
Author(s):  
Zhi Mao ◽  
Ting Ting Wang
Keyword(s):  
Diffusion Equation ◽  
Numerical Algorithm ◽  
Variable Coefficient ◽  
Fractional Diffusion ◽  
Diffusion Equations ◽  
Algebraic Equations ◽  
Fractional Diffusion Equations ◽  
Linear Algebraic Equations ◽  
Collocation Technique ◽  
Sinc Functions

Fractional diffusion equations have recently been applied in various area of engineering. In this paper, a new numerical algorithm for solving the fractional diffusion equations with a variable coefficient is proposed. Based on the collocation technique where the shifted Chebyshev polynomials in time and the sinc functions in space are utilized respectively, the problem is reduced to the solution of a system of linear algebraic equations. The procedure is tested and the efficiency of the proposed algorithm is confirmed through the numerical example.

scholarly journals Sinc-Chebyshev Collocation Method for a Class of Fractional Diffusion-Wave Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Zhi Mao ◽  
Aiguo Xiao ◽  
Zuguo Yu ◽  
Long Shi
Keyword(s):  
Variable Coefficient ◽  
Fractional Derivatives ◽  
Wave Equations ◽  
Fractional Diffusion ◽  
Algebraic Equations ◽  
Chebyshev Collocation Method ◽  
Diffusion Wave ◽  
Linear Algebraic Equations ◽  
Collocation Technique ◽  
Sinc Functions

This paper is devoted to investigating the numerical solution for a class of fractional diffusion-wave equations with a variable coefficient where the fractional derivatives are described in the Caputo sense. The approach is based on the collocation technique where the shifted Chebyshev polynomials in time and the sinc functions in space are utilized, respectively. The problem is reduced to the solution of a system of linear algebraic equations. Through the numerical example, the procedure is tested and the efficiency of the proposed method is confirmed.

scholarly journals Stochastic model for multi-term time-fractional diffusion equations with noise

2021 ◽  
Vol 25 (Spec. issue 2) ◽  
pp. 287-293
Author(s):  
Vahid Hosseini ◽  
Mohamad Remazani ◽  
Wennan Zou ◽  
Seddigheh Banihashemii
Keyword(s):  
Numerical Solution ◽  
Numerical Approach ◽  
Fractional Diffusion ◽  
Diffusion Equations ◽  
Data Driven ◽  
Algebraic Equations ◽  
Fractional Diffusion Equations ◽  
Collocation Points ◽  
Linear Algebraic Equations ◽  
Collocation Approach

This paper studies a spectral collocation approach for evaluating the numerical solution of the stochastic multi-term time-fractional diffusion equations associated with noisy data driven by Brownian motion. This model describes the symmetry breaking in molecular vibrations. The numerical solution of the stochastic multi-term time-fractional diffusion equations is proposed by means of collocation points method based on sixth-kind Chebyshev polynomial approach. For this purpose, the problem under consideration is reduced to a system of linear algebraic equations. Two examples highlight the robustness and accuracy of the proposed numerical approach.

An Efficient Operational Matrix Technique for Multidimensional Variable-Order Time Fractional Diffusion Equations

2016 ◽  
Vol 11 (6) ◽  
Author(s):  
M. A. Zaky ◽  
S. S. Ezz-Eldien ◽  
E. H. Doha ◽  
J. A. Tenreiro Machado ◽  
A. H. Bhrawy
Keyword(s):  
Present Method ◽  
Operational Matrix ◽  
Solution Process ◽  
Fractional Diffusion ◽  
Diffusion Equations ◽  
Variable Order ◽  
Algebraic Equations ◽  
Global Approximation ◽  
Fractional Diffusion Equations ◽  
Accurate Numerical Algorithm

This paper derives a new operational matrix of the variable-order (VO) time fractional partial derivative involved in anomalous diffusion for shifted Chebyshev polynomials. We then develop an accurate numerical algorithm to solve the 1 + 1 and 2 + 1 VO and constant-order fractional diffusion equation with Dirichlet conditions. The contraction of the present method is based on shifted Chebyshev collocation procedure in combination with the derived shifted Chebyshev operational matrix. The main advantage of the proposed method is to investigate a global approximation for spatial and temporal discretizations, and it reduces such problems to those of solving a system of algebraic equations, which greatly simplifies the solution process. In addition, we analyze the convergence of the present method graphically. Finally, comparisons between the algorithm derived in this paper and the existing algorithms are given, which show that our numerical schemes exhibit better performances than the existing ones.

Numerical approximations for fractional diffusion equations via a Chebyshev spectral-tau method

Open Physics ◽  
2013 ◽  
Vol 11 (10) ◽  
Author(s):  
Eid Doha ◽  
Ali Bhrawy ◽  
Samer Ezz-Eldien
Keyword(s):  
Numerical Technique ◽  
Operational Matrix ◽  
Fractional Diffusion ◽  
High Accuracy ◽  
Variable Coefficients ◽  
Diffusion Equations ◽  
Algebraic Equations ◽  
Fractional Differentiation ◽  
Fractional Diffusion Equations ◽  
General Method

AbstractIn this paper, a class of fractional diffusion equations with variable coefficients is considered. An accurate and efficient spectral tau technique for solving the fractional diffusion equations numerically is proposed. This method is based upon Chebyshev tau approximation together with Chebyshev operational matrix of Caputo fractional differentiation. Such approach has the advantage of reducing the problem to the solution of a system of algebraic equations, which may then be solved by any standard numerical technique. We apply this general method to solve four specific examples. In each of the examples considered, the numerical results show that the proposed method is of high accuracy and is efficient for solving the time-dependent fractional diffusion equations.

scholarly journals Finite Difference and Sinc-Collocation Approximations to a Class of Fractional Diffusion-Wave Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-11 ◽  
Author(s):  
Zhi Mao ◽  
Aiguo Xiao ◽  
Zuguo Yu ◽  
Long Shi
Keyword(s):  
Finite Difference ◽  
Wave Equations ◽  
Fractional Diffusion ◽  
Stability And Convergence ◽  
Algebraic Equations ◽  
Time Step ◽  
Diffusion Wave ◽  
Linear Algebraic Equations ◽  
Sinc Functions ◽  
Exponential Rate Of Convergence

We propose an efficient numerical method for a class of fractional diffusion-wave equations with the Caputo fractional derivative of orderα. This approach is based on the finite difference in time and the global sinc collocation in space. By utilizing the collocation technique and some properties of the sinc functions, the problem is reduced to the solution of a system of linear algebraic equations at each time step. Stability and convergence of the proposed method are rigorously analyzed. The numerical solution is of3-αorder accuracy in time and exponential rate of convergence in space. Numerical experiments demonstrate the validity of the obtained method and support the obtained theoretical results.

scholarly journals Identification of points sources via time fractional diffusion equation

Filomat ◽  
2018 ◽  
Vol 32 (18) ◽  
pp. 6189-6201 ◽  
Author(s):  
A. Ghanmi ◽  
R. Mdimagh ◽  
I.B. Saad
Keyword(s):  
Inverse Problem ◽  
Diffusion Equation ◽  
Stability Result ◽  
Fractional Diffusion ◽  
Cauchy Data ◽  
Diffusion Equations ◽  
Lipschitz Stability ◽  
Single Measurement ◽  
Fractional Diffusion Equations ◽  
Time Fractional Diffusion Equation

This article investigates the source identification in the fractional diffusion equations, by performing a single measurement of the Cauchy data on the accessible boundary. The main results of this work consist in giving an identifiability result and establishing a local Lipschitz stability result. To solve the inverse problem of identifying fractional sources from such observations, a non-iterative algebraical method based on the Reciprocity Gap functional is proposed.

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