The solution of mixed integral equation of the first kind using toeplitz matrix method

Abstract

In this work, the existence and uniqueness of the solution of mixed integral equation (MIE) of the first kind is considered in the  L₂  *C[0,T], T<1 ,  is the domain of integration with respect to position and T is the time. Then, a numerical method is used to obtain a system of Fredholm integral equations (SFIE). The discontinuous kernel of the SFIE takes the form of Carleman function and logarithmic kernel. The existence and uniqueness of the solution SFIE can be proved. Moreover, Toeplitz matrix method (TMM) is used to obtain a linear algebraic system (LAS). The LAS is solved numerically, to get the eigenvalues and eigenfunctions of SFIE.