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This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License
María B. Pintarelli
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DOI:10.17265/2159-5291/2015.01.004
Grupo de Aplicaciones Matematicas y Estadisticas de la Facultad de Ingenieria (GAMEFI), UNLP, and Departamento de Matematica, Facultad de Ciencias Exactas, Universidad Nacional de La Plata.
In this paper we will see that, under certain conditions, the techniques
of generalized moment problem will apply to numerically solve an Volterra integral
equation of first kind or second kind. Volterra integral equation is transformed into a one-dimensional generalized
moment problem, and shall apply the moment problem techniques to find a numerical
approximation of the solution. Specifically you will see that solving the Volterra
integral equation of first kindft=atKt,sxsdsa≤t≤b
or solve the Volterra
integral equation of the second kind xt=ft+atKt,sxsdsa≤t≤b
is equivalent to solving
a generalized moment problem of the form μn=abgn(s)xsdsn=0,1,2,….
This shall apply for to
find the solution of an integrodifferential equation of the form x't=ft+atKt,sxsds
fora≤t≤b
andxa=a0
Also considering the nonlinear
integral equation:fx=axyx-tytdt
This integral equation
is transformed a two-dimensional generalized moment problem. In all cases, we will
find an approximated solution and bounds for the error of the estimated solution
using the techniques ofgeneralized moment problem.
Generalized moment problems, solution stability, Volterra integral equations, nonlinear integral equations.
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