Some Estimates For The Carleman Function

In this article we consider Carlеman’s functions, to find integral representation for the polygarmonious functions defined in unbounded domain of Euclidean space which Satisfies

An asymptotic model for solving mixed integral equation in some domains

In this paper, we discuss the solution of mixed integral equation with generalized potential function in position and the kernel of Volterra integral term in time. The solution will be discussed in the space $$L_{2} (\Omega ) \times C[0,T],$$ L 2 ( Ω ) × C [ 0 , T ] , $$0 \le t \le T < 1$$ 0 ≤ t ≤ T < 1 , where $$\Omega$$ Ω is the domain of position and $$t$$ t is the time. The mixed integral equation is established from the axisymmetric problems in the theory of elasticity. Many special cases when kernel takes the potential function, Carleman function, the elliptic function and logarithmic function will be established.

New Model for Solving Mixed Integral Equation of the First Kind with Generalized Potential Kernel

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.