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)
•  T. M. Atanacković, S. Pilipović, B. Stanković and D. Zorica, Fractional Calculus with Applications in Mechanics Mechanical Engineering and Solid Mechanics Series. ISTE, London; John Wiley & Sons, Inc. , Hoboken, NJ, 2014. Wave propagation, impact and variational principles. L. C. Becker , Resolvents and solutions of weakly singular linear Volterra integral equations, Nonlinear Anal., 74 (2011) , 1892-1912.  doi: 10.1016/j.na.2010.10.060. H. Brunner, Collocation Methods for Volterra Integral and Related Functional Differential Equations volume 15 of Cambridge Monographs on Applied and Computational Mathematics, Cambridge University Press, Cambridge, 2004. doi: 10.1017/CBO9780511543234. H. Brunner and P. J. van der Houwen, The Numerical Solution of Volterra Equations volume 3 of CWI Monographs, North-Holland Publishing Co. , Amsterdam, 1986. E. Capelas de Oliveira and J. A. T. Machado, A review of definitions for fractional derivatives and integral Math. Probl. Eng. , 2014 (2014), Art. ID 238459, 6 pp. doi: 10.1155/2014/238459. R. Caponetto, G. Dongola, L. Fortuna and I. Petráš, Fractional Order Systems: Modeling and Control Applications volume 72 of Series on Nonlinear Science, Series A, World Scientific, Singapore, 2010. M. Concezzi , R. Garra  and  R. Spigler , Fractional relaxation and fractional oscillation models involving Erdélyi-Kober integrals, Fract. Calc. Appl. Anal., 18 (2015) , 1212-1231.  doi: 10.1515/fca-2015-0070. F. R. de Hoog  and  R. S. Anderssen , Kernel perturbations for a class of second-kind convolution Volterra equations with non-negative kernels, Appl. Math. Lett., 25 (2012) , 1222-1225.  doi: 10.1016/j.aml.2012.02.058. K. Diethelm, The Analysis of Fractional Differential Equations Springer-Verlag, Berlin, 2010. doi: 10.1007/978-3-642-14574-2. K. Diethelm , N. J. Ford  and  A. D. Freed , A predictor-corrector approach for the numerical solution of fractional differential equations, Nonlinear Dynam., 29 (2002) , 3-22.  doi: 10.1023/A:1016592219341. K. Diethelm , N. J. Ford  and  A. D. Freed , Detailed error analysis for a fractional Adams method, Numer. Algorithms, 36 (2004) , 31-52.  doi: 10.1023/B:NUMA.0000027736.85078.be. J. Dixon , On the order of the error in discretization methods for weakly singular second kind Volterra integral equations with nonsmooth solutions, BIT, 25 (1985) , 624-634.  doi: 10.1007/BF01936141. R. Garra , R. Gorenflo , F. Polito  and  Ž. Tomovski , Hilfer-Prabhakar derivatives and some applications, Appl. Math. Comput., 242 (2014) , 576-589.  doi: 10.1016/j.amc.2014.05.129. R. Garrappa , Trapezoidal methods for fractional differential equations: Theoretical and computational aspects, Math. Comput. Simulation, 110 (2015) , 96-112.  doi: 10.1016/j.matcom.2013.09.012. R. Garrappa , Grünwald-Letnikov operators for fractional relaxation in Havriliak-Negami models, Commun. Nonlinear Sci. Numer. Simul., 38 (2016) , 178-191.  doi: 10.1016/j.cnsns.2016.02.015. R. Gorenflo, A. A. Kilbas, F. Mainardi and S. V. Rogosin, Mittag–Leffler Functions, Related Topics and Applications, Springer Monographs in Mathematics. Springer, New York, 2014. A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations volume 204 of {North-Holland Mathematics Studies}, Elsevier Science B. V. , Amsterdam, 2006. F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity Imperial College Press, London, 2010. An introduction to mathematical models. doi: 10.1142/9781848163300. G. Pagnini , Erdélyi-Kober fractional diffusion, Fract. Calc. Appl. Anal., 15 (2012) , 117-127.  doi: 10.2478/s13540-012-0008-1. I. Podlubny, Fractional Differential Equations volume 198 of Mathematics in Science and Engineering, Academic Press, Inc. , San Diego, CA, 1999. An introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications. V. E. Tarasov, Fractional Dynamics Nonlinear Physical Science. Springer, Heidelberg; Higher Education Press, Beijing, 2010. Applications of fractional calculus to dynamics of particles, fields and media. doi: 10.1007/978-3-642-14003-7. G. Teschl, Ordinary Differential Equations and Dynamical Systems volume 140 of Graduate Studies in Mathematics, American Mathematical Society, Providence, 2012. doi: 10.1090/gsm/140. V. V. Uchaikin, Fractional Derivatives for Physicists and Engineers, Volume Ⅱ Nonlinear Physical Science. Higher Education Press, Beijing; Springer, Heidelberg, 2013. Applications. doi: 10.1007/978-3-642-33911-0. A. Young , Approximate product-integration, Proc. Roy. Soc. London Ser. A., 224 (1954) , 552-561.  doi: 10.1098/rspa.1954.0179.

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