Successive Approximation Method example2

Example 1:- https://youtu.be/y7VOb3LyqLs The method of successive approximations, that was used earlier in Chapter 2 Fredlohm integral equation for handling Fredholm integral equation will be implemented here to solve Volterra integral equations as well. There is a variety on analytic and numerical techniques, traditional and new, that are usually used in studying Volterra integral equations. Accordingly we will first start with the recent methods. Volterra integral equation playlist :- https://www.youtube.com/playlist?list=PL8HByMi5awLpRyGT5vqse9xWYdYwyh0go 1) Adomain Decomposition Method for VIE :- https://youtu.be/lFpN3kDQFZs 2)Noise term Phenomenon:- https://youtu.be/t17Vt59uBvE 3) Series solution method:- https://youtu.be/Guk6CPzlnMc 4) Successive approximation method:- https://youtu.be/_M-JlJRtqBk ...…...................................................................... Like. Share. Comment Subscribe Follow me on Instagram :- @kulsankekumar Integral equation playlist:- https://www.youtube.com/playlist?list=PL8HByMi5awLqLOOFFjOHuGxMlrP_Hd8NA .....…...................................................................... What is Integral Equation:- https://youtu.be/P_BayV54k7o What is Fredholm integral equation and its kinds:- https://youtu.be/SpdcPZe1w3g Homogeneous and non homogeneous Integral equation:- https://youtu.be/Wpgd5LKlGU8 ...…...................................................................... In this method, we replace the unknown function u(x) under the integral sign of the Volterra equation. By any selective real valued continuous function u0(x), called the zeroth approximation.This substitution will give the first approximation u1(x). It is obvious that u1(x) is continuous whenever f(x) , K(x, t) and u0(x) are continuous. The second approximation u2(x) of u(x) can be obtained similarly by replacing u0(x) by u1(x) obtained above, hence we find U2 This process can be continued in the same manner to obtain the nth approximation. In other words, the various approximations of the solution u(x) of can be obtained in a recursive scheme. This process can be continued in the same manner to obtain the nth approximation. In other words, the various approximations of the solution u(x) of (134) can be obtained in a recursive scheme given by The most commonly selected functions for u0(x) are 0, 1 or x. At the limit, the solution u(x) of the equation (134) is obtained by so that the resulting solution u(x) is independent of the choice of the zeroth approximation u0(x). It is useful, for comparison reasons, to distinguish between the recursive schemes used in the decomposition method and in the successive approximations method. In the decomposition method, we decompose the solution u(x) into components u0, u1, u2, ... where each component is evaluated subsequently, and in this case the solution is given in a series form