Effect of perturbation in the numerical solution of fractional differential equations

Roberto Garrappa; Eleonora Messina; Antonia Vecchio

doi:10.3934/dcdsb.2017188

Article Contents

2018, Volume 23, Issue 7: 2679-2694. Doi: 10.3934/dcdsb.2017188

This issue Previous Article Positive symplectic integrators for predator-prey dynamics Next Article On the stability of

$\vartheta$

-methods for stochastic Volterra integral equations

Effect of perturbation in the numerical solution of fractional differential equations

1.
Dipartimento di Matematica Università degli Studi di Bari Via E. Orabona 4,70125 Bari, Italy
2.
Dipartimento di Matematica e Applicazioni Università degli Studi di Napoli "Federico Ⅱ" Via Cintia, I-80126 Napoli, Italy
3.
C.N.R. National Research Council of Italy Institute for Computational Application "Mauro Picone" Via P. Castellino, 111 -80131 Napoli -Italy

* Corresponding author: E. Messina
* Corresponding author: E. Messina

Received: September 30, 2016

Revised: April 30, 2017

Early access: June 2017

Published: July 2018

This work is supported under the INdAM-GNCS project 2016 "Metodi numerici per operatori non-locali nella simulazione di fenomeni complessi"

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Abstract

The equations describing engineering and real-life models are usually derived in an approximated way. Thus, in most cases it is necessary to deal with equations containing some kind of perturbation. In this paper we consider fractional differential equations and study the effects on the continuous and numerical solution, of perturbations on the given function, over long-time intervals. Some bounds on the global error are also determined.

Keywords:

Mathematics Subject Classification: Primary: 34A08, 65L07; Secondary: 65R20.

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Figure 1. Plot of $t^{\alpha} E_{1, \alpha+1}(-\lambda t)$ when $\lambda=1$ (left plot) and $\lambda=10$ (right plot)

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Figure 2. Values of $\eta$ as function of $\lambda$ in logarithmic scale

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Figure 3. Solution of the problem test and bound (6) for $\alpha=0.8$

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Figure 4. Comparison of the difference $\delta y(t)$ between the exact and perturbed solutions and the bound (9)

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Table 1. Relationship between global and truncation errors with respect to Theorem 4.2 for $\alpha=0.4$ (here $\eta\approx0.320$, $K=1.0$, $A\approx1.578$)

$ h $	$ \eta + h^{\alpha}A $	$ \displaystyle\sup_{n=0, N}\|e_n\| $	$ \displaystyle\frac{\sup_{n=0, N} \|T_n(h)\|}{1 - (\eta + h^{\alpha}A)} $
$ 2^{-4} $	$ 0.840 $	$ 2.5692(-2) $	$ 1.2938(-1) $
$ 2^{-5} $	$ 0.714 $	$ 1.3885(-2) $	$ 4.0188(-2) $
$ 2^{-6} $	$ 0.618 $	$ 7.4352(-3) $	$ 1.6738(-2) $
$ 2^{-7} $	$ 0.546 $	$ 4.0198(-3) $	$ 7.8711(-3) $
$ 2^{-8} $	$ 0.491 $	$ 2.1891(-3) $	$ 3.9330(-3) $
$ 2^{-9} $	$ 0.450 $	$ 1.1840(-3) $	$ 2.0099(-3) $
$ 2^{-10} $	$ 0.418 $	$ 6.1270(-4) $	$ 1.0002(-3) $

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Table 2. Relationship between global and truncation errors with respect to Theorem 4.2 for $\alpha=0.6$ (here $\eta\approx0.240$, $K=1.0$, $A\approx1.791$)

$ h $	$ \eta + h^{\alpha}A $	$ \displaystyle\sup_{n=0, N}\|e_n\| $	$ \displaystyle\frac{\sup_{n=0, N} \|T_n(h)\|}{1 - (\eta + h^{\alpha}A)} $
$ 2^{-3} $	$ 0.755 $	$ 8.5380(-3) $	$ 3.1044(-2) $
$ 2^{-4} $	$ 0.580 $	$ 4.5485(-3) $	$ 9.7699(-3) $
$ 2^{-5} $	$ 0.464 $	$ 2.0622(-3) $	$ 3.5605(-3) $
$ 2^{-6} $	$ 0.388 $	$ 8.9133(-4) $	$ 1.3787(-3) $
$ 2^{-7} $	$ 0.338 $	$ 3.8127(-4) $	$ 5.5464(-4) $
$ 2^{-8} $	$ 0.305 $	$ 1.6256(-4) $	$ 2.2801(-4) $
$ 2^{-9} $	$ 0.283 $	$ 6.8446(-5) $	$ 9.3858(-5) $
$ 2^{-10} $	$ 0.268 $	$ 2.7444(-5) $	$ 3.7099(-5) $

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Table 3. Relationship between global and truncation errors with respect to Theorem 4.2 for $\alpha=0.8$ (here $\eta\approx0.203$, $K=1.0$, $A\approx1.933$)

$ h $	$ \eta + h^{\alpha}A $	$ \displaystyle\sup_{n=0, N}\|e_n\| $	$ \displaystyle\frac{\sup_{n=0, N} \|T_n(h)\|}{1 - (\eta + h^{\alpha}A)} $
$ 2^{-2} $	$ 0.840 $	$ 1.6121(-2) $	$ 9.4144(-2) $
$ 2^{-3} $	$ 0.569 $	$ 3.6570(-3) $	$ 7.9159(-3) $
$ 2^{-4} $	$ 0.413 $	$ 7.7545(-4) $	$ 1.2719(-3) $
$ 2^{-5} $	$ 0.324 $	$ 1.7330(-4) $	$ 2.4809(-4) $
$ 2^{-6} $	$ 0.272 $	$ 6.8838(-5) $	$ 9.2726(-5) $
$ 2^{-7} $	$ 0.243 $	$ 2.4179(-5) $	$ 3.1554(-5) $
$ 2^{-8} $	$ 0.226 $	$ 8.1096(-6) $	$ 1.0403(-5) $
$ 2^{-9} $	$ 0.216 $	$ 2.6392(-6) $	$ 3.3532(-6) $
$ 2^{-10} $	$ 0.210 $	$ 8.1173(-7) $	$ 1.0258(-6) $

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Table 4. Perturbed problem: relationship between global error and bounds from Theorem 4.5 for $\alpha=0.4$ (here $\tilde{\eta}\approx0.318$, $\tilde{K}=1.0$, $\tilde{A}\approx1.578$)

$h$	$\tilde{\eta} + h^{\alpha}\tilde{A}$	$\displaystyle\sup_{n=0, N}\|\tilde{e}_n\|$	$\textrm{Bound (23)}$
$2^{-4}$	0.838	9.8267(-2)	2.5169(-1)
$2^{-5}$	0.712	9.4799(-2)	1.6371(-1)
$2^{-6}$	0.617	9.3258(-2)	1.4043(-1)
$2^{-7}$	0.544	9.2595(-2)	1.3162(-1)
$2^{-8}$	0.489	9.2311(-2)	1.2770(-1)
$2^{-9}$	0.448	9.2198(-2)	1.2578(-1)
$2^{-10}$	0.416	9.2154(-2)	1.2477(-1)

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Table 5. Perturbed problem: relationship between global error and bounds from Theorem 4.5 for $\alpha=0.6$ (here $\tilde{\eta}\approx0.238$, $\tilde{K}=1.0$, $\tilde{A}\approx1.791$)

$h$	$\tilde{\eta} + h^{\alpha}\tilde{A}$	$\displaystyle\sup_{n=0, N}\|\tilde{e}_n\| $	$\textrm{Bound (23)}$
$ 2^{-3}$	0.753	6.1846(-2)	1.1423(-1)
$2^{-4}$	0.578	6.3482(-2)	9.3146(-2)
$2^{-5}$	0.462	6.3552(-2)	8.6966(-2)
$2^{-6}$	0.386	6.3352(-2)	8.4792(-2)
$2^{-7}$	0.336	6.3249(-2)	8.3971(-2)
$2^{-8}$	0.303	6.3206(-2)	8.3645(-2)
$2^{-9}$	0.281	6.3189(-2)	8.3511(-2)
$2^{-10}$	0.266	6.3183(-2)	8.3455(-2)

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Table 6. Perturbed problem: relationship between global error and bounds from Theorem 4.5 for $\alpha=0.8$ (here $\tilde{\eta}\approx0.201$, $\tilde{K}=1.0$, $\tilde{A}\approx1.933$)

$h$	$\tilde{\eta} + h^{\alpha}\tilde{A}$	$\displaystyle\sup_{n=0, N}\|\tilde{e}_n\|$	$\textrm{Bound (23)}$
$2^{-2}$	0.839	2.7785(-2)	1.5982(-1)
$2^{-3}$	0.568	4.4303(-2)	7.4435(-2)
$2^{-4}$	0.412	4.8468(-2)	6.7814(-2)
$2^{-5}$	0.322	4.9372(-2)	6.6793(-2)
$2^{-6}$	0.271	4.9568(-2)	6.6638(-2)
$2^{-7}$	0.241	4.9612(-2)	6.6577(-2)
$2^{-8}$	0.224	4.9620(-2)	6.6556(-2)
$2^{-9}$	0.215	4.9622(-2)	6.6549(-2)
$2^{-10}$	0.209	4.9622(-2)	6.6547(-2)

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