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.
Highlights
The field of integral and integro-differential equations is a very important subject in applied mathematics, because mathematical formulation of many physical phenomena contains integral and integro-differential equations
New technique model is used to solve the mixed integral equation (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 L2(Ω) × C[0, T ], 0 ≤ T < 1, Ω 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
Summary
The field of integral and integro-differential equations is a very important subject in applied mathematics, because mathematical formulation of many physical phenomena contains integral and integro-differential equations H., 2013).Integral equations are very important branch of mathematics, which come in application in many physical problems. & Lima, P., 2008).different analytic and numeric methods have been established to obtain the solutions of the integral equations. In (Ordokhani, Y., 2013), a collocation method based on the Bessel polynomials was used for solution of the nonlinear Fredholm-Volterra Hammerstein integro-differential equations, under mixed conditions. & Armand, A., 2016), the approximate solution for the nonlinear Volterra-Fredholm-Hammerste in integral equations was obtained by using the Tau-collocation method. The integro differential equation with Cauchy kernel is derived with its special cases. This equation has appeared in both combined infrared gaseous radiation and molecular condition, and elastic contact studies
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