A successive numerical scheme for some classes of Volterra-Fredholm integral equations (Q2800491)

The following nonlinear Volterra-Fredholm integral equation is considered NEWLINE\[NEWLINE x(s)=f(s)+\int_{a}^{s}g(s,t,x(t))dt+\int_{a}^{b}h(s,t,x(t))dt, \quad s\in[a,b]. \tag{1} NEWLINE\]NEWLINE It is supposed that (1) has at least one solution. The authors present a numerical scheme for this equation by transforming it into a discretized form NEWLINE\[NEWLINE x^*(s_i)=f(s_i)+\int_{a}^{s_i}g(s_i,t,x^*(t))dt+\int_{a}^{b}h(s_i,t,x^*(t))dt, \quad i=0,1,\dots,n, \tag{2} NEWLINE\]NEWLINE where \(\{a=s_0,s_1,\dots,s_{n-1},s_n=b\}\) is an equidistant partition of \([a,b]\) and \(x^*(t)\) is an analytical solution of (1).NEWLINENEWLINEThen the solution of (2) is approximated by an iterative scheme under some smoothness conditions on the kernels \(g\) and \(h\). The approach suggested in this paper is also illustrated by several numerical examples.