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Iterative and non-iterative methods for non-linear Volterra integro-differential equations. (English) Zbl 1202.65179
The author considers the problem of numerical solving the initial problem for the equation $$ A(t)u^{(n)}(t)=f(t,u(t)) + \int_{t_0}^{t}g(s,u(s))ds, \; t_0<t< \infty, $$ subject to $u^{(j)}(t_0)=\alpha_j, \; 0<j\leq(n-1)$, where $A(t)$ are invertible square matrices of the order $N$ and $u^{(j)}$ denotes the $j$-th order derivative of the unknown $N$-dimensional function $u(t)$. This problem is deep theoretically investigated, and the local existence theorem 1 presented and proved in the paper is presented (with non-essential simplification $A(t)=I$) in the text-book by {\it A. B. Vasilieva} and {\it A. N. Tikhonov} [Integralnye Uravnenia (Russian). Izdat. Moskovskogo Universiteta, Moscow (1986)]. The author presents and discusses a few variants of iterative algorithms of Picard kind, and a series method of solving the initial problem. The latter is the presentation of an approximation of the required solution as a finite functional sum. The first summand is the given initial value $\alpha$, and the each following summand is an integral iteration of one or two predecessors. Convergence of the sum to the required solution is proved.

65R20Integral equations (numerical methods)
45J05Integro-ordinary differential equations
Full Text: DOI
[1] Lovitt, W. V.: Linear integral equations, (1950)
[2] Delves, L. M.; Mohamed, J. L.: Computational methods for integral equations, (1985) · Zbl 0592.65093
[3] Davis, H. T.: Introduction to nonlinear differential and integral equations, (1962) · Zbl 0106.28904
[4] Tricomi, F. G.: Integral equations, (1985)
[5] Petrovski, I.: Lectures on the theory of integral equations, (1971) · Zbl 0233.45001
[6] Krasnov, M.; Kiseliov, A.; Makarenko, G.: Integral equations, (1982)
[7] Porter, D.; Stirling, D. S. G.: Integral equations, (1990) · Zbl 0714.45001
[8] Agarwal, R. P.: Boundary value problems for high ordinary differential equations, (1986) · Zbl 0619.34019
[9] Ramos, J. I.: Piecewise-quasilinearization techniques for singularly perturbed Volterra integro-differential equations, Applied mathematics and computation 188, 1221-1233 (2007) · Zbl 1118.65130 · doi:10.1016/j.amc.2006.10.076
[10] Wazwaz, A. -M.: A comparison study between the modified decomposition method and the traditional methods for solving nonlinear integral equations, Applied mathematics and computation 181, 1703-1712 (2006) · Zbl 1105.65128 · doi:10.1016/j.amc.2006.03.023
[11] Hashim, I.: Adomian decomposition method for solving BVPs for fourth-order integro-differential equations, Journal of computational and applied mathematics 193, 658-664 (2006) · Zbl 1093.65122 · doi:10.1016/j.cam.2005.05.034
[12] He, J. -H.: Variational iteration method -- a kind of non-linear analytical technique: some examples, International journal of nonlinear mechanics 34, 699-708 (1999) · Zbl 05137891
[13] He, J. -H.: Some asymptotic methods for strongly nonlinear equations, International journal of modern physics 20, 1141-1199 (2006) · Zbl 1102.34039 · doi:10.1142/S0217979206033796
[14] Sweilam, N. H.: Fourth-order integro-differential equations using variational iteration method, Computer and mathematics with applications 54, 1086-1091 (2007) · Zbl 1141.65399 · doi:10.1016/j.camwa.2006.12.055
[15] Wang, S. -Q.; He, J. -H.: Variational iteration method for solving integro-differential equations, Physics letters A 367, 188-191 (2007) · Zbl 1209.65152
[16] Shou, D. -H.; He, J. -H.: Beyond Adomian: the variational iteration method for solving heat-like and wave-like equations with variable coefficients, Physics letters A 372, 233-237 (2008) · Zbl 1217.35091 · doi:10.1016/j.physleta.2007.07.011
[17] Saberi-Nadjafi, J.; Tamamgar, M.: The variational iteration method: a highly promising method for solving the system of integro-differential equations, Computer and mathematics with applications 56, 346-351 (2008) · Zbl 1155.65399 · doi:10.1016/j.camwa.2007.12.014
[18] Adomian, G.: Stochastic systems, (1983) · Zbl 0523.60056
[19] Adomian, G.: Nonlinear stochastic operator equations, (1986) · Zbl 0609.60072
[20] Adomian, G.: Solving frontier problems of physics: the decomposition method, (1994) · Zbl 0802.65122
[21] Ramos, J. I.: On the variational iteration method and other iterative techniques for nonlinear differential equations, Applied mathematics and computation 199, 39-69 (2008) · Zbl 1142.65082 · doi:10.1016/j.amc.2007.09.024
[22] Ramos, J. I.: On the Picard Lindelöf method for nonlinear second-order differential equations, Applied mathematics and computation 203, 238-242 (2008) · Zbl 1195.65095 · doi:10.1016/j.amc.2008.04.029
[23] Ramos, J. I.: A non-iterative derivative-free method for nonlinear ordinary differential equations, Applied mathematics and computation 203, 672-678 (2008) · Zbl 1157.65414 · doi:10.1016/j.amc.2008.05.015
[24] Keller, H. B.: Numerical methods for two-point boundary-value problems, (1992)
[25] Stakgold, I.: Boundary value problems of mathematical physics, Boundary value problems of mathematical physics &amp; II (1967) · Zbl 0158.04801
[26] Stakgold, I.: Green’s functions and boundary value problems, (1998) · Zbl 0897.35001
[27] Widder, D. V.: Advanced calculus, (1989) · Zbl 0728.26002
[28] Halmos, P. R.: Finite-dimensional vector spaces, (1993) · Zbl 0107.01404
[29] Lax, P. D.: Functional analysis, (2002) · Zbl 1009.47001
[30] Ortega, J.; Rheinbolt, W.: Iterative solution of nonlinear equations in several variables, (1970) · Zbl 0241.65046
[31] W. Rheinbolt, Methods for solving systems of nonlinear equations, Society for Industrial and Applied Mathematics, Philadelphia, Pennsylvania, 1974.
[32] , The numerical solution of nonlinear problems (1981) · Zbl 0484.65002
[33] Atkinson, K. E.: An introduction to numerical analysis, (1989) · Zbl 0718.65001
[34] Evans, D. J.: Preconditioning methods: theory and applications, (1983)
[35] Y. Saad, Iterative Methods for Sparse Linear Systems, second edition, Society for Industrial and Applied Mathematics, Philadelphia, Pennsylvania, 2003. · Zbl 1031.65046
[36] Axelsson, O.: Iterative solution methods, (1994) · Zbl 0795.65014
[37] Turkel, E.: Preconditioning techniques in computational fluid dynamics, Annual review of fluid dynamics 31, 385-416 (1999)
[38] Yu, Z. -H.: Variational iteration method for solving the multi-pantograph delay equation, Physics letters A 372, 6475-6479 (2008) · Zbl 1225.34024 · doi:10.1016/j.physleta.2008.09.013
[39] J.I. Ramos, Piecewise-adaptive decomposition methods, Chaos, Solitons &amp; Fractals, in press. · Zbl 1198.65149 · doi:10.1016/j.chaos.2007.09.043
[40] Ramos, J. I.: Piecewise homotopy methods for nonlinear ordinary differential equations, Applied mathematics and computation 198, 92-116 (2008) · Zbl 1137.65048 · doi:10.1016/j.amc.2007.08.030
[41] He, Ji-H.: Homotopy perturbation technique, Computer methods in applied mechanics and engineering 178, 257-262 (1999) · Zbl 0956.70017
[42] He, Ji-H.: Homotopy perturbation method: a new nonlinear analytical technique, Applied mathematical computation 135, 73-79 (2003) · Zbl 1030.34013 · doi:10.1016/S0096-3003(01)00312-5
[43] He, Ji-H.: Addendum: new interpretation of homotopy perturbation method, International journal of modern physics 20, 2561-2568 (2006)
[44] J.I. Ramos, Series approach to the Lane -- Emden equation and comparison with the homotopy perturbation method, Chaos, Solitons &amp; Fractals 38 (2008) 400 -- 408. · Zbl 1146.34300 · doi:10.1016/j.chaos.2006.11.018
[45] Ramos, J. I.: An artificial parameter-decomposition method for nonlinear oscillators: applications to oscillators with odd nonlinearities, Journal of sound and vibration 307, 312-329 (2007)