scholarly journals Fractional-compact numerical algorithms for Riesz spatial fractional reaction-dispersion equations

2017 ◽  
Vol 20 (3) ◽  
Author(s):  
Hengfei Ding ◽  
Changpin Li
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Numerical Approximation ◽  
Numerical Algorithms ◽  
High Order ◽  
Stability And Convergence ◽  
Dispersion Equations ◽  
Approximation Formulas ◽  
Fractional Differential ◽  
Fractional Reaction

AbstractIt is well known that using high-order numerical algorithms to solve fractional differential equations leads to almost the same computational cost with low-order ones but the accuracy (or convergence order) is greatly improved due to the nonlocal properties of fractional operators. Therefore, developing some high-order numerical approximation formulas for fractional derivatives plays a more important role in numerically solving fractional differential equations. This paper focuses on constructing (generalized) high-order fractional-compact numerical approximation formulas for Riesz derivatives. Then we apply the developed formulas to the one- and two-dimension Riesz spatial fractional reaction-dispersion equations. The stability and convergence of the derived numerical algorithms are strictly studied by using the energy analysis method. Finally, numerical simulations are given to demonstrate the efficiency and convergence orders of the presented numerical algorithms.

scholarly journals High order algorithms for numerical solution of fractional differential equations

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Mohammad Shahbazi Asl ◽  
Mohammad Javidi ◽  
Yubin Yan
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Numerical Algorithms ◽  
Interpolation Polynomial ◽  
High Order ◽  
The Novel ◽  
Fractional Differential ◽  
Novel Algorithms ◽  
The Stability ◽  
Theoretical Results

AbstractIn this paper, two novel high order numerical algorithms are proposed for solving fractional differential equations where the fractional derivative is considered in the Caputo sense. The total domain is discretized into a set of small subdomains and then the unknown functions are approximated using the piecewise Lagrange interpolation polynomial of degree three and degree four. The detailed error analysis is presented, and it is analytically proven that the proposed algorithms are of orders 4 and 5. The stability of the algorithms is rigorously established and the stability region is also achieved. Numerical examples are provided to check the theoretical results and illustrate the efficiency and applicability of the novel algorithms.

ψ-Haar wavelets method for numerically solving fractional differential equations

2020 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Amjid Ali ◽  
Teruya Minamoto ◽  
Umer Saeed ◽  
Mujeeb Ur Rehman
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Numerical Approximation ◽  
Numerical Solutions ◽  
Operational Matrix ◽  
Nonlinear Problems ◽  
Mathematical Tool ◽  
Content Type ◽  
Fractional Models ◽  
Fractional Differential

Purpose The purpose of this paper is to obtain a numerical scheme for finding numerical solutions of linear and nonlinear fractional differential equations involving ψ-Caputo derivative. Design/methodology/approach An operational matrix to find numerical approximation of ψ-fractional differential equations (FDEs) is derived. This study extends the method to nonlinear FDEs by using quasi linearization technique to linearize the nonlinear problems. Findings The error analysis of the proposed method is discussed in-depth. Accuracy and efficiency of the method are verified through numerical examples. Research limitations/implications The method is simple and a good mathematical tool for finding solutions of nonlinear ψ-FDEs. The operational matrix approach offers less computational complexity. Originality/value Engineers and applied scientists may use the present method for solving fractional models appearing in applications.

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 .

scholarly journals Sinc Based Inverse Laplace Transforms, Mittag-Leffler Functions and Their Approximation for Fractional Calculus

2021 ◽  
Vol 5 (2) ◽  
pp. 43
Author(s):  
Gerd Baumann
Keyword(s):  
Differential Equations ◽  
Fractional Calculus ◽  
Laplace Transform ◽  
Fractional Differential Equations ◽  
Numerical Approximation ◽  
Laplace Transforms ◽  
Inverse Laplace Transform ◽  
Exact Methods ◽  
Sinc Methods ◽  
Fractional Differential

We shall discuss three methods of inverse Laplace transforms. A Sinc-Thiele approximation, a pure Sinc, and a Sinc-Gaussian based method. The two last Sinc related methods are exact methods of inverse Laplace transforms which allow us a numerical approximation using Sinc methods. The inverse Laplace transform converges exponentially and does not use Bromwich contours for computations. We apply the three methods to Mittag-Leffler functions incorporating one, two, and three parameters. The three parameter Mittag-Leffler function represents Prabhakar’s function. The exact Sinc methods are used to solve fractional differential equations of constant and variable differentiation order.

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