ERROR ESTIMATES IN PICARD’S METHOD OF SUCCESSIVE APPROXIMATIONS FOR A PARTICULAR SECOND ORDER INITIAL VALUE PROBLEM


By

Jervin Zen Lobo

Department of Mathematics, St. Xavier’s College, Mapusa - 403507, Goa, India

Email:zenlobo1990@gmail.com

(Received : August 02,2021; In format : August 14,2021; Revised in final form : March 27,2022)


Abstract

In this paper, we extend the method of successive approximations to second order Initial Value Problems (IVPs) of the type y'' = f(x, y), y(x0) = y0, y' (x0) = y1, without converting it to a system of first order differential equations. We obtain an upper bound in the closed form for the difference between two successive iterates. Further, we calculate an error bound for the solution and see that we get a tighter bound for the second order IVP as compared to its first order counterpart.


2020 Mathematical Sciences Classification: 34A12, 34A40.

Keywords and Phrases: Gronwall Inequality, Initial Value Problem, Integral equations, Lipschitz condition, Weierstrass M-test.



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