An Asymptotic Model for Solving Mixed Integral Equation in Position and Time

Abstract

In this paper, we considered a mixed integral equation (MIE) of the second kind in the space $L_2\left[-b,b\right]\times C\left[0,T\right],T<1.$ The kernel of position has a singularity and takes some different famous forms, while the kernels of time are positive and continuous. Using an asymptotic method of separating the variables, we have a Fredholm integral equation (FIE) in position with variable parameters in time. Then, using the Toeplitz matrix method (TMM), we obtain a linear algebraic system (LAS) that can be solved numerically. Some applications with the aid of the maple 18 program are discussed when the kernel takes Coleman function, Cauchy kernel, Hilbert kernel, and a generalized logarithmic function. Also the error estimate, in each case, is computed.