The first way is to establish a mixed integral equations from the thermoelasticity problem, using the general fundamental equations, in the general form, with the aid of potential theory method. The third way, we used two different famous numerical methods: Toeplitz matrix method and product Nystrom method to discuss, numerically the solution of the mixed integral equation? The fourth way is using the integral operator transforms to discuss some different kinds of problems in some mathematical physic problems especially in elastic and... The second way is to discuss the existence and uniqueness of this new equation in view of pure mathematics using Picard method and Banach fixed point theorem when the Picard method fails! Beside this some lemmas and theorems for the local and the total error, for each method, are discussed and proved. Integral equations play an important role for solving many different kinds of the mathematical physics and mechanics. In this book, we used four ways.