Multi-order fractional differential equations and their numerical solution
type: Item
instance of: scholarly article
title: Multi-order fractional differential equations and their numerical solution
zbMATH Open document ID: 1060.65070
DOI: 10.1016/S0096-3003(03)00739-2
author: Kai Diethelm
author: Neville J. Ford
published in: Applied Mathematics and Computation
publication date: 2004-10-01T00:00:00Z
review text: The paper presents the numerical solution of multi-order fractional differential equations of the general (possibly nonlinear) form \[ y^{(\alpha )}(t)=f(t, y(t), y^{(\beta_1)}(t), y^{(\beta_2)}(t),\dots, y^{(\beta_n)}(t)) \] with \(\alpha>\beta_n>\beta_{n-1}>...>\beta_1\) and \(\alpha -\beta_n\leq1\), \(\beta_j -\beta_{j-1}\leq 1\) for all \(j\) and \(0<\beta_1\leq 1\). Its linear case is \(y^{(\alpha )}(t)=\lambda_oy(t)+ \sum_{j=1}^n\lambda_jy^{(\beta_j)}(t)+f(t)\). The initial conditions have the form \(y^{k}(t)=y_o^{(k)}\), \(k=0,1,\dots,\lceil\alpha\rceil -1\). The derivatives are understood in the Caputo sense. A generalization of an approach employed in the solution of ordinary differential equations of order two or higher converting such equation to a system of equations of order one is used. It uses the fact that any real number can be approximated arbitrarily closely by a rational number. Thereby, the assumption on the commensuracy for fractional order equations can be ensured by an appropriate order approximation. A simple generalization of the theorem on the equivalence of a nonlinear system and the linear systems theory is presented first. Then, the nonlinear problem includes two Gronwall-type results for a two-term equation, the general existence-uniqueness as well as the structural stability results. A convergent and stable Adams-type numerical method is proposed including a specific numerical example.
Mathematics Subject Classification ID: 65L05
Mathematics Subject Classification ID: 65L20
Mathematics Subject Classification ID: 34A34
Mathematics Subject Classification ID: 26A33
Mathematics Subject Classification ID: 65L06
zbMATH DE Number: 2105538
zbMATH Keywords: multi-term fractional differential equation
zbMATH Keywords: Caputo derivative
zbMATH Keywords: structural stability
zbMATH Keywords: Adams method
zbMATH Keywords: convergence
zbMATH Keywords: numerical example
reviewed by: Lubomír Bakule
describes a project that uses: FracPECE
MaRDI profile type: Publication
Wikidata QID: Q59411831
cites work: Q3798129
cites work: An algorithm for the numerical solution of differential equations of fractional order
cites work: Fractional Linear Multistep Methods for Abel-Volterra Integral Equations of the Second Kind
cites work: Q3750978
cites work: Numerical solution of the Bagley-Torvik equation.
cites work: Q3365609
cites work: Q4233638
cites work: The fractional calculus. Theory and applications of differentiation and integration to arbitrary order
cites work: Q3141895
cites work: Fractals and fractional calculus in continuum mechanics
cites work: The numerical solution of linear multi-term fractional differential equations: Systems of equations
cites work: Analysis of fractional differential equations
cites work: A predictor-corrector approach for the numerical solution of fractional differential equations
cites work: Detailed error analysis for a fractional Adams method
cites work: The numerical solution of fractional differential equations: speed versus accuracy
full work available at URL: s0096-3003(03)00739-2
OpenAlex ID: W2050533316
DBLP publication ID: journals/amc/DiethelmF04
description: scientific article; zbMATH DE number 2105538