Abstract

New technique model is used to solve the mixed integral equation (\textbf{MIE}) of the first kind, with a position kernel contains a generalized potential function multiplying by a continuous function and continuous kernel in time, in the space $L_{2} (\Omega )\times C[0,T],\, 0\leq T<1$, $\Omega$ is the domain of integration and $T$ is the time. The integral equation arises in the treatment of various semi-symmetric contact problems, in one, two, and  three dimensions, with mixed boundary conditions in the mechanics of continuous media. The solution of the \textbf{MIE }when the kernel of position takes the potential function form, elliptic function form, Carleman function and logarithmic kernel are discussed and obtain in  this work. Moreover, many special cases are derived.

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