Article Contents
Article Contents

Effect of perturbation in the numerical solution of fractional differential equations

• * Corresponding author: E. Messina

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.

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

 Citation:

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

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

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

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

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

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

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

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

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

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

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)

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)

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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