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Fast iterative refinement method for mixed systems of integral and fractional integro-differential equations. (English) Zbl 06912520
Comput. Appl. Math. 37, No. 2, 2354-2379 (2018); erratum ibid. 37, No. 2, 2380 (2018).
Summary: The authors’ new method of iterative refinement has been successfully applied to the linear fractional Fredholm integro-differential equation. In this work, we adapt our method to obtain approximate analytical solutions for linear Volterra equations of both the first and the second kind of fractional type. A detailed convergence analysis is presented for each kind. The authors also apply their method to the challenging first kind linear Fredholm integral equation. Three different cases are considered to test the efficacy of our method. These include mixed systems of various forms of linear integral and fractional integro-differential equations. We compare our method with six well-established methods: Picard’s successive approximations, an accelerated form of the latter, the Adomian decomposition, the variational iteration, the homotopy perturbation and the homotopy analysis. The results show the versatility of the new method in solving a wide variety of equations, its high convergence speed and accuracy and its capability of directly dealing with equations of the first kind. We also prove that Picard’s method for linear equations is a special case of our method.
Reviewer: Reviewer (Berlin)
MSC:
65R20 Numerical methods for integral equations
45B05 Fredholm integral equations
45D05 Volterra integral equations
65F10 Iterative numerical methods for linear systems
26A33 Fractional derivatives and integrals
65L03 Numerical methods for functional-differential equations
34A08 Fractional ordinary differential equations and fractional differential inclusions
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References:
[1] Abbasbandy, S, Numerical solutions of the integral equations: homotopy perturbation method and adomian’s decomposition method, Appl Math Comput, 173, 493-500, (2006) · Zbl 1090.65143
[2] Allan, F, Derivation of the Adomian decomposition method using the homotopy analysis method, Appl Math Comput, 190, 6-14, (2007) · Zbl 1125.65063
[3] Arikoglu, A; Ozkol, I, Solution of fractional integro-differential equations by using fractional differential transform method, Chaos Solitons Fractals, 40, 521-529, (2009) · Zbl 1197.45001
[4] Atkinson K (1997) The numerical solution of integral equations of the second kind. Cambridge University Press, UK · Zbl 0899.65077
[5] Biazar, J; Babolian, E; Islam, R, Solution of a system of Volterra integral equations of the first kind by Adomian method, Appl Math Comput, 139, 249-258, (2003) · Zbl 1027.65180
[6] Borwein, D; Borwein, JM; Crandall, RE, Effective Laguerre asymptotics, SIAM J Numer Anal, 46, 3285-3312, (2008) · Zbl 1196.33010
[7] Brunner H (2017) Volterra integral equations: an introduction to theory and applications. Cambridge University Press, UK. doi:10.1017/9781316162491 · Zbl 1376.45002
[8] Deif, SA; Grace, SR, Iterative refinement for a system of linear integro-differential equations of fractional type, J Comput Appl Math, 294, 138-150, (2016) · Zbl 1327.65276
[9] El-Sayed, AMA; Hashem, HHG; Ziada, EAA, Picard and Adomian methods for quadratic integral equation, Comput Appl Math, 29, 447-463, (2010) · Zbl 1208.45007
[10] Golberg, MA, A note on the decomposition method for operator equations, Appl Math Comput, 106, 215-220, (1999) · Zbl 1022.65067
[11] Goldberg S (1966) Unbounded linear operators: theory and applications. McGraw-Hill, New York · Zbl 0148.12501
[12] Golub GH, Van Loan CH (1983) Matrix computations. Johns Hopkins Univ. Press, Baltimore · Zbl 0559.65011
[13] Groetsch, CW, Integral equations of the first kind, inverse problems and regularization: a crash course, J Phys Conf Ser, 73, 1-32, (2007)
[14] Hansen PC (1998) Rank-deficient and discrete ill-posed problems: numerical aspects of linear inversion. SIAM monographs on mathematical modeling and computation. SIAM, Philadelphia · Zbl 1309.26011
[15] Huang, L; Li, X-F; Zhao, Y; Duan, X-Y, Approximate solution of fractional integro-differential equations by Taylor expansion method, Comput Math Appl, 62, 1127-1134, (2011) · Zbl 1228.65133
[16] Kanwal R (2013) Linear integral equations: theory and technique. Modern Birkhauser Classics, New York · Zbl 1259.45001
[17] Kreyszig E (1978) Introductory functional analysis with applications. Wiley, New York · Zbl 0368.46014
[18] Liao S (2004) Beyond perturbation: introduction to the homotopy analysis method. Chapman & Hall/CRC, Boca Raton · Zbl 1051.76001
[19] Liao, S, Comparison between the homotopy analysis method and homotopy perturbation method, Appl Math Comput, 169, 1186-1194, (2005) · Zbl 1082.65534
[20] Machado, JAT; Kiryakova, V; Mainardi, F, Recent history of fractional calculus, Commun Nonlinear Sci Numer Simul, 16, 1140-1153, (2011) · Zbl 1221.26002
[21] Machado, JAT; Mainardi, F; Kiryakova, V, Fractional calculus: quo vadimus? (where are we going?), Fract Calc Appl Anal, 18, 495-526, (2015) · Zbl 1309.26011
[22] Moghaddam BP, Machado JAT, Behforooz H (2017) An integro quadratic spline approach for a class of variable-order fractional initial value problems. Chaos Solitons Fractals. doi:10.1016/j.chaos.2017.03.065 · Zbl 1422.65131
[23] Moghaddam, BP; Machado, JAT, Extended algorithms for approximating variable order fractional derivatives with applications, J Sci Comput, 71, 1351-1374, (2017) · Zbl 1370.26017
[24] Momani, S; Qaralleh, R, An efficient method for solving systems of fractional integro-differential equations, Comput Math Appl, 52, 459-470, (2006) · Zbl 1137.65072
[25] Nawaz, Y, Variational iteration method and homotopy perturbation method for fourth-order fractional integro-differential equations, Comput Math Appl, 61, 2330-2341, (2011) · Zbl 1219.65081
[26] Oldham KB, Myland J, Spanier J (2009) An atlas of functions: with equator, the atlas function calculator. Springer, Heidelberg · Zbl 1167.65001
[27] Podlubny I (1999) Fractional differential equations. Academic press, San Diego · Zbl 0924.34008
[28] Sabatier J, Agrawal OP, Machado JAT (2007) Advances in fractional calculus. Springer, Dordrecht · Zbl 1116.00014
[29] Saberi-Nadjafi, J; Ghorbani, A, He’s homotopy perturbation method: an effective tool for solving nonlinear integral and integro-differential equations, Comput Math Appl, 58, 2379-2390, (2009) · Zbl 1189.65173
[30] Gorder, RA, The variational iteration method is a special case of the homotopy analysis method, Appl Math Lett, 45, 81-85, (2015) · Zbl 1325.65118
[31] Vanani, SK; Aminataei, A, Operational tau approximation for a general class of fractional integro-differential equations, Comput Appl Math, 30, 655-674, (2011) · Zbl 1247.65174
[32] Wazwaz AM (2011) Linear and nonlinear integral equations: methods and applications. Springer, Heidelberg · Zbl 1227.45002
[33] Wilkinson JH (1965) The algebraic eigenvalue problem. Clarendon Press, Oxford · Zbl 0258.65037
[34] Zurigat, M; Momani, S; Alawneh, A, Homotopy analysis method for systems of fractional integro-differential equations, Neural Parallel Sci Comput, 17, 169-186, (2009) · Zbl 1180.65181
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