# American Institute of Mathematical Sciences

September  2018, 23(7): 2679-2694. doi: 10.3934/dcdsb.2017188

## 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

Received  October 2016 Revised  May 2017 Published  July 2017

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

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.

Citation: Roberto Garrappa, Eleonora Messina, Antonia Vecchio. Effect of perturbation in the numerical solution of fractional differential equations. Discrete & Continuous Dynamical Systems - B, 2018, 23 (7) : 2679-2694. doi: 10.3934/dcdsb.2017188
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##### References:
Plot of $t^{\alpha} E_{1, \alpha+1}(-\lambda t)$ when $\lambda=1$ (left plot) and $\lambda=10$ (right plot)
Values of $\eta$ as function of $\lambda$ in logarithmic scale
Solution of the problem test and bound (6) for $\alpha=0.8$
Comparison of the difference $\delta y(t)$ between the exact and perturbed solutions and the bound (9)
Relationship between global and truncation errors with respect to Theorem 4.2 for $\alpha=0.4$ (here $\eta\approx0.320$, $K=1.0$, $A\approx1.578$)
 $h$ $\eta + h^{\alpha}A$ $\displaystyle\sup_{n=0, N}|e_n|$ $\displaystyle\frac{\sup_{n=0, N} |T_n(h)|}{1 - (\eta + h^{\alpha}A)}$ $2^{-4}$ $0.840$ $2.5692(-2)$ $1.2938(-1)$ $2^{-5}$ $0.714$ $1.3885(-2)$ $4.0188(-2)$ $2^{-6}$ $0.618$ $7.4352(-3)$ $1.6738(-2)$ $2^{-7}$ $0.546$ $4.0198(-3)$ $7.8711(-3)$ $2^{-8}$ $0.491$ $2.1891(-3)$ $3.9330(-3)$ $2^{-9}$ $0.450$ $1.1840(-3)$ $2.0099(-3)$ $2^{-10}$ $0.418$ $6.1270(-4)$ $1.0002(-3)$
 $h$ $\eta + h^{\alpha}A$ $\displaystyle\sup_{n=0, N}|e_n|$ $\displaystyle\frac{\sup_{n=0, N} |T_n(h)|}{1 - (\eta + h^{\alpha}A)}$ $2^{-4}$ $0.840$ $2.5692(-2)$ $1.2938(-1)$ $2^{-5}$ $0.714$ $1.3885(-2)$ $4.0188(-2)$ $2^{-6}$ $0.618$ $7.4352(-3)$ $1.6738(-2)$ $2^{-7}$ $0.546$ $4.0198(-3)$ $7.8711(-3)$ $2^{-8}$ $0.491$ $2.1891(-3)$ $3.9330(-3)$ $2^{-9}$ $0.450$ $1.1840(-3)$ $2.0099(-3)$ $2^{-10}$ $0.418$ $6.1270(-4)$ $1.0002(-3)$
Relationship between global and truncation errors with respect to Theorem 4.2 for $\alpha=0.6$ (here $\eta\approx0.240$, $K=1.0$, $A\approx1.791$)
 $h$ $\eta + h^{\alpha}A$ $\displaystyle\sup_{n=0, N}|e_n|$ $\displaystyle\frac{\sup_{n=0, N} |T_n(h)|}{1 - (\eta + h^{\alpha}A)}$ $2^{-3}$ $0.755$ $8.5380(-3)$ $3.1044(-2)$ $2^{-4}$ $0.580$ $4.5485(-3)$ $9.7699(-3)$ $2^{-5}$ $0.464$ $2.0622(-3)$ $3.5605(-3)$ $2^{-6}$ $0.388$ $8.9133(-4)$ $1.3787(-3)$ $2^{-7}$ $0.338$ $3.8127(-4)$ $5.5464(-4)$ $2^{-8}$ $0.305$ $1.6256(-4)$ $2.2801(-4)$ $2^{-9}$ $0.283$ $6.8446(-5)$ $9.3858(-5)$ $2^{-10}$ $0.268$ $2.7444(-5)$ $3.7099(-5)$
 $h$ $\eta + h^{\alpha}A$ $\displaystyle\sup_{n=0, N}|e_n|$ $\displaystyle\frac{\sup_{n=0, N} |T_n(h)|}{1 - (\eta + h^{\alpha}A)}$ $2^{-3}$ $0.755$ $8.5380(-3)$ $3.1044(-2)$ $2^{-4}$ $0.580$ $4.5485(-3)$ $9.7699(-3)$ $2^{-5}$ $0.464$ $2.0622(-3)$ $3.5605(-3)$ $2^{-6}$ $0.388$ $8.9133(-4)$ $1.3787(-3)$ $2^{-7}$ $0.338$ $3.8127(-4)$ $5.5464(-4)$ $2^{-8}$ $0.305$ $1.6256(-4)$ $2.2801(-4)$ $2^{-9}$ $0.283$ $6.8446(-5)$ $9.3858(-5)$ $2^{-10}$ $0.268$ $2.7444(-5)$ $3.7099(-5)$
Relationship between global and truncation errors with respect to Theorem 4.2 for $\alpha=0.8$ (here $\eta\approx0.203$, $K=1.0$, $A\approx1.933$)
 $h$ $\eta + h^{\alpha}A$ $\displaystyle\sup_{n=0, N}|e_n|$ $\displaystyle\frac{\sup_{n=0, N} |T_n(h)|}{1 - (\eta + h^{\alpha}A)}$ $2^{-2}$ $0.840$ $1.6121(-2)$ $9.4144(-2)$ $2^{-3}$ $0.569$ $3.6570(-3)$ $7.9159(-3)$ $2^{-4}$ $0.413$ $7.7545(-4)$ $1.2719(-3)$ $2^{-5}$ $0.324$ $1.7330(-4)$ $2.4809(-4)$ $2^{-6}$ $0.272$ $6.8838(-5)$ $9.2726(-5)$ $2^{-7}$ $0.243$ $2.4179(-5)$ $3.1554(-5)$ $2^{-8}$ $0.226$ $8.1096(-6)$ $1.0403(-5)$ $2^{-9}$ $0.216$ $2.6392(-6)$ $3.3532(-6)$ $2^{-10}$ $0.210$ $8.1173(-7)$ $1.0258(-6)$
 $h$ $\eta + h^{\alpha}A$ $\displaystyle\sup_{n=0, N}|e_n|$ $\displaystyle\frac{\sup_{n=0, N} |T_n(h)|}{1 - (\eta + h^{\alpha}A)}$ $2^{-2}$ $0.840$ $1.6121(-2)$ $9.4144(-2)$ $2^{-3}$ $0.569$ $3.6570(-3)$ $7.9159(-3)$ $2^{-4}$ $0.413$ $7.7545(-4)$ $1.2719(-3)$ $2^{-5}$ $0.324$ $1.7330(-4)$ $2.4809(-4)$ $2^{-6}$ $0.272$ $6.8838(-5)$ $9.2726(-5)$ $2^{-7}$ $0.243$ $2.4179(-5)$ $3.1554(-5)$ $2^{-8}$ $0.226$ $8.1096(-6)$ $1.0403(-5)$ $2^{-9}$ $0.216$ $2.6392(-6)$ $3.3532(-6)$ $2^{-10}$ $0.210$ $8.1173(-7)$ $1.0258(-6)$
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)
 $h$ $\tilde{\eta} + h^{\alpha}\tilde{A}$ $\displaystyle\sup_{n=0, N}|\tilde{e}_n|$ $\textrm{Bound (23)}$ $2^{-4}$ 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)
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)
 $h$ $\tilde{\eta} + h^{\alpha}\tilde{A}$ $\displaystyle\sup_{n=0, N}|\tilde{e}_n|$ $\textrm{Bound (23)}$ $2^{-3}$ 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)
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)
 $h$ $\tilde{\eta} + h^{\alpha}\tilde{A}$ $\displaystyle\sup_{n=0, N}|\tilde{e}_n|$ $\textrm{Bound (23)}$ $2^{-2}$ 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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