On an Explicit Difference Method for Fractional Diffusion and Diffusion-Wave Equations

2009 ◽  
Author(s):  
Joaqui´n Quintana Murillo ◽  
Santos Bravo Yuste
Keyword(s):  
Wave Equations ◽  
Its Region ◽  
Fractional Diffusion ◽  
Diffusion Wave ◽  
Von Neumann ◽  
Stability Bound ◽  
Riemann Zeta ◽  
The Stability ◽  
And Diffusion ◽  
Explicit Difference Schemes

An explicit difference scheme for solving fractional diffusion and fractional diffusion-wave equations, in which the fractional derivative is in the Caputo form, is considered. The two equations are studied separately: for the fractional diffusion equation, the L1 discretization formula is employed, whereas the L2 discretization formula is used for the fractional diffusion-wave equation. Its accuracy is similar to other well-known explicit difference schemes, but its region of stability is larger. The stability analysis is carried out by means of a procedure similar to the standard von Neumann method. The stability bound, which is given in terms of the the Riemann Zeta function, is checked numerically.

An Explicit Difference Method for Solving Fractional Diffusion and Diffusion-Wave Equations in the Caputo Form

2010 ◽  
Vol 6 (2) ◽  
Author(s):  
Joaquín Quintana Murillo ◽  
Santos Bravo Yuste
Keyword(s):  
Fractional Derivative ◽  
Difference Method ◽  
Wave Equations ◽  
Time Derivative ◽  
Fourier Method ◽  
Its Region ◽  
Fractional Diffusion ◽  
Diffusion Wave ◽  
Von Neumann ◽  
The Stability

An explicit difference method is considered for solving fractional diffusion and fractional diffusion-wave equations where the time derivative is a fractional derivative in the Caputo form. For the fractional diffusion equation, the L1 discretization formula of the fractional derivative is employed, whereas the L2 discretization formula is used for the fractional diffusion-wave equation. In both equations, the spatial derivative is approximated by means of the three-point centered formula. The accuracy of the present method is similar to other well-known explicit difference schemes, but its region of stability is larger. The stability analysis is carried out by means of a kind of fractional von Neumann (or Fourier) method. The stability bound so obtained, which is given in terms of the Riemann zeta function, is checked numerically.

scholarly journals Numerical method for fractional diffusion-wave equations with functional delay

2021 ◽  
Vol 57 ◽  
pp. 156-169
Author(s):  
V.G. Pimenov ◽  
E.E. Tashirova
Keyword(s):  
Numerical Method ◽  
Wave Equations ◽  
Order Of Convergence ◽  
Fractional Diffusion ◽  
Explicit Method ◽  
Diffusion Wave ◽  
Functional Delay ◽  
The Stability ◽  
The Given ◽  
Implicit Numerical Method

For a fractional diffusion-wave equation with a nonlinear effect of functional delay, an implicit numerical method is constructed. The scheme is based on the L2-method of approximation of the fractional derivative of the order from 1 to 2, interpolation and extrapolation with the given properties of discrete prehistory and an analogue of the Crank-Nicolson method. The order of convergence of the method is investigated using the ideas of the general theory of difference schemes with heredity. The order of convergence of the method is more significant than in previously known methods, depending on the order of the starting values. The main point of the proof is the use of the stability of the L2-method. The results of comparing numerical experiments with other schemes are presented: a purely implicit method and a purely explicit method, these results showed, in general, the advantages of the proposed scheme.

scholarly journals Approximate Solutions of Time Fractional Diffusion Wave Models

Mathematics ◽  
2019 ◽  
Vol 7 (10) ◽  
pp. 923 ◽  
Author(s):  
Abdul Ghafoor ◽  
Sirajul Haq ◽  
Manzoor Hussain ◽  
Poom Kumam ◽  
Muhammad Asif Jan
Keyword(s):  
Finite Difference ◽  
Wave Equations ◽  
Quadrature Rule ◽  
Approximate Solutions ◽  
Fractional Diffusion ◽  
Haar Wavelets ◽  
Test Problems ◽  
Algebraic Equations ◽  
Diffusion Wave ◽  
Wave Models

In this paper, a wavelet based collocation method is formulated for an approximate solution of (1 + 1)- and (1 + 2)-dimensional time fractional diffusion wave equations. The main objective of this study is to combine the finite difference method with Haar wavelets. One and two dimensional Haar wavelets are used for the discretization of a spatial operator while time fractional derivative is approximated using second order finite difference and quadrature rule. The scheme has an excellent feature that converts a time fractional partial differential equation to a system of algebraic equations which can be solved easily. The suggested technique is applied to solve some test problems. The obtained results have been compared with existing results in the literature. Also, the accuracy of the scheme has been checked by computing L 2 and L ∞ error norms. Computations validate that the proposed method produces good results, which are comparable with exact solutions and those presented before.

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