Commenced in January 2007
Frequency: Monthly
Edition: International
Paper Count: 30127
Research of a Multistep Method Applied to Numerical Solution of Volterra Integro-Differential Equation

Authors: M.Imanova, G.Mehdiyeva, V.Ibrahimov

Abstract:

Solution of some practical problems is reduced to the solution of the integro-differential equations. But for the numerical solution of such equations basically quadrature methods or its combination with multistep or one-step methods are used. The quadrature methods basically is applied to calculation of the integral participating in right hand side of integro-differential equations. As this integral is of Volterra type, it is obvious that at replacement with its integrated sum the upper limit of the sum depends on a current point in which values of the integral are defined. Thus we receive the integrated sum with variable boundary, to work with is hardly. Therefore multistep method with the constant coefficients, which is free from noted lack and gives the way for finding it-s coefficients is present.

Keywords: Volterra integro-differential equations, multistepmethods, finite-difference methods, initial value problem

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1076702

Procedia APA BibTeX Chicago EndNote Harvard JSON MLA RIS XML ISO 690 PDF Downloads 1084

References:


[1] V.Volterra, "Theorie of functionals and of integral and integrodifferential equations" London, 1931.
[2] Ye. M. Polishuk, Vito Volterra. Leningrad, nauka, 1977, 114 p.
[3] M.N. Imanova, "Numerical solution of Volterra integro-differential equation" Proceeding of institute of mathematics and mechanics, XXII, Baku, 2005.
[4] H.Brunner,"Implicit Runqe-Kutta methods of optimal order for Volterra inteqro-differential equations" Vol 442, Ôäû106, January 1984, pp.95- 109.
[5] Ch.Lubich, "Runge-Kutta theory for Volterra and Abel integral equations of the second kind" Mathematics of computation volume 41, number 163, July 1983, p. 87-102.
[6] V.R. Ibrahimov, "On the maximum degree of the k-step Obrechkoff-s method" Bulletin of Iranian Mathematical Society, Vol. 20, No. 1, pp.1-28 (2002).
[7] V.R. Ibrahimov, M.N. Imanova, "On the new method for solving Volterra integral equation" Transactions issue mathematics and mechanics series of physical-technical and mathematical science, XXVI, Ôäû1, 2007.
[8] V.R. Ibrahimov, G.Yu.Mehdiyeva, "On way of generalizing quadrature method", Baku State University, series of physicomath. sciences, 1, 2008, p.92-88.
[9] A.F. Verlan, V.S. Sizikov, "Integral equations: methods, algorithms, programs" (In Russian) Naukova Dumka, Kiev (1986), 544 p.
[10] G. Dahlquist, "Convergence and stability in the numerical integration of ordinary differential equations" Math. Scand. - 1956, Ôäû4, p.33-53.