Convergence Analysis of a Block-by-Block Method for Fractional Differential Equations

2012 ◽  
Vol 5 (2) ◽  
pp. 229-241 ◽  
Author(s):  
Jianfei Huang
Keyword(s):  
Differential Equations ◽  
Convergence Analysis ◽  
Fractional Differential Equations ◽  
Block Method ◽  
Fractional Differential

Convergence Analysis of a High Order Schema for the Ordinary Fractional Diffusion Equation

2014 ◽  
Vol 602-605 ◽  
pp. 3088-3091
Author(s):  
Jun Ying Cao ◽  
Zi Qiang Wang
Keyword(s):  
Differential Equations ◽  
Diffusion Equation ◽  
Convergence Analysis ◽  
Fractional Differential Equations ◽  
Fractional Diffusion ◽  
High Order ◽  
Fractional Diffusion Equation ◽  
Block Method ◽  
Fractional Differential ◽  
Block Approach

The block-by-block method extended by Kumar and Agrawal to fractional differential equations. Cao et al. proposed a high order schema which is based on an improved block-by-block approach, which consists in finding 4 unknowns simultaneously at each step block through solving a 4 × 4 system. We rigorously analytically prove that this method is convergent with order for , and order 6 for .

Nonpolynomial collocation approximation of solutions to fractional differential equations

2013 ◽  
Vol 16 (4) ◽  
Author(s):  
Neville Ford ◽  
M. Morgado ◽  
Magda Rebelo
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Error Analysis ◽  
Convergence Analysis ◽  
Fractional Differential Equations ◽  
Order Of Convergence ◽  
Optimal Order ◽  
Optimal Order Of Convergence ◽  
Fractional Differential ◽  
Polynomial Collocation

AbstractWe propose a non-polynomial collocation method for solving fractional differential equations. The construction of such a scheme is based on the classical equivalence between certain fractional differential equations and corresponding Volterra integral equations. Usually, we cannot expect the solution of a fractional differential equation to be smooth and this poses a challenge to the convergence analysis of numerical schemes. In this paper, the approach we present takes into account the potential non-regularity of the solution, and so we are able to obtain a result on optimal order of convergence without the need to impose inconvenient smoothness conditions on the solution. An error analysis is provided for the linear case and several examples are presented and discussed.

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