A preconditioned fast finite difference method for space-time fractional partial differential equations

2017 ◽  
Vol 20 (1) ◽  
Author(s):  
Hongfei fu ◽  
Hong Wang
Keyword(s):  
Finite Difference ◽  
Finite Difference Method ◽  
Difference Method ◽  
Numerical Scheme ◽  
Mathematical Structure ◽  
Space Time ◽  
Optimal Order ◽  
Conservation Property ◽  
Long Time ◽  
The Stability

AbstractWe develop a fast space-time finite difference method for space-time fractional diffusion equations by fully utilizing the mathematical structure of the scheme. A circulant block preconditioner is proposed to further reduce the computational costs. The method has optimal-order memory requirement and approximately linear computational complexity. The method is not lossy, as no compression of the underlying numerical scheme has been employed. Consequently, the method retains the stability, accuracy, and, in particular, the conservation property of the underlying numerical scheme. Numerical experiments are presented to show the efficiency and capacity of long time modelling of the new method.

scholarly journals Construction of a Class of Copula Using the Finite Difference Method

2021 ◽  
Vol 2021 ◽  
pp. 1-8
Author(s):  
Remi Guillaume Bagré ◽  
Frédéric Béré ◽  
Vini Yves Bernadin Loyara
Keyword(s):  
Finite Difference ◽  
Finite Difference Method ◽  
Difference Method ◽  
Numerical Scheme ◽  
Approximate Solutions ◽  
Copula Function ◽  
Common Technique ◽  
The Finite Difference Method ◽  
Definition Of ◽  
General Method

The definition of a copula function and the study of its properties are at the same time not obvious tasks, as there is no general method for constructing them. In this paper, we present a method that allows us to obtain a class of copula as a solution to a boundary value problem. For this, we use the finite difference method which is a common technique for finding approximate solutions of partial differential equations which consists in solving a system of relations (numerical scheme) linking the values of the unknown functions at certain points sufficiently close to each other.

scholarly journals Computational Analysis of the Stability of 2D Heat Equation on Elliptical Domain Using Finite Difference Method

2020 ◽  
pp. 8-19
Author(s):  
Mehwish Naz Rajput ◽  
Asif Ali Shaikh ◽  
Shakeel Ahmed Kamboh
Keyword(s):  
Finite Difference ◽  
Finite Difference Method ◽  
Heat Equation ◽  
Stability Condition ◽  
Difference Method ◽  
Stable Solution ◽  
Finite Difference Schemes ◽  
Step Size ◽  
Stability Range ◽  
The Stability

Aims: The aim and objective of the study to derive and analyze the stability of the finite difference schemes in relation to the irregularity of domain. Study Design: First of all, an elliptical domain has been constructed with the governing two dimensional (2D) heat equation that is discretized using the Finite Difference Method (FDM). Then the stability condition has been defined and the numerical solution by writing MATLAB codes has been obtained with the stable values of time domain. Place and Duration of Study: The work has been jointly conducted at the MUET, Jamshoro and QUEST, Nawabshah Pakistan from January 2019 to December 2019.  Methodology: The stability condition over an elliptical domain with the non-uniform step size depending upon the boundary tracing function is derived by using Von Neumann method. Results: From the results it was revealed that stability region for the small number of mesh points remains larger and gets smaller as the number of mesh nodes is increased. Moreover, the ranges for the time steps are defined for varied spatial step sizes that help to find the stable solution. Conclusion: The corresponding stability range for number of nodes N=10, 20, 30, 40, 50, and 60 was found respectively. Within this range the solution remains smooth as time increases. The results of this study attempt to provide the stable solution of partial differential equations on irregular domains.

scholarly journals Modified Chebyshev Collocation Method for Solving Differential Equations

CAUCHY ◽  
2015 ◽  
Vol 3 (4) ◽  
pp. 208
Author(s):  
M Ziaul Arif ◽  
Ahmad Kamsyakawuni ◽  
Ikhsanul Halikin
Keyword(s):  
Exact Solution ◽  
Finite Difference ◽  
Finite Difference Method ◽  
Differential Equations ◽  
Collocation Method ◽  
Difference Method ◽  
Numerical Scheme ◽  
Chebyshev Collocation Method ◽  
Chebyshev Collocation ◽  
Polynomial Collocation

This paper presents derivation of alternative numerical scheme for solving differential equations, which is modified Chebyshev (Vieta-Lucas Polynomial) collocation differentiation matrices. The Scheme of modified Chebyshev (Vieta-Lucas Polynomial) collocation method is applied to both Ordinary Differential Equations (ODEs) and Partial Differential Equations (PDEs) cases. Finally, the performance of the proposed method is compared with finite difference method and the exact solution of the example. It is shown that modified Chebyshev collocation method more effective and accurate than FDM for some example given.

Sign in / Sign up

Export Citation Format

Share Document