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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))+ t 0 t g(s,u(s))ds,t 0 <t<,

subject to u (j) (t 0 )=α j ,0<j(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 A. B. Vasilieva and 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 α, and the each following summand is an integral iteration of one or two predecessors. Convergence of the sum to the required solution is proved.

MSC:
65R20Integral equations (numerical methods)
45J05Integro-ordinary differential equations
References:
[1]Lovitt, W. V.: Linear integral equations, (1950)
[2]Delves, L. M.; Mohamed, J. L.: Computational methods for integral equations, (1985)
[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)
[8]Agarwal, R. P.: Boundary value problems for high ordinary differential equations, (1986)
[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)
[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)
[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)
[19]Adomian, G.: Nonlinear stochastic operator equations, (1986)
[20]Adomian, G.: Solving frontier problems of physics: the decomposition method, (1994)
[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)
[26]Stakgold, I.: Green’s functions and boundary value problems, (1998) · Zbl 0897.35001
[27]Widder, D. V.: Advanced calculus, (1989)
[28]Halmos, P. R.: Finite-dimensional vector spaces, (1993)
[29]Lax, P. D.: Functional analysis, (2002)
[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)
[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)
[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)
[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)