M.A. Abdou

Fredholm-Volterra integral equation of the first kind and contact problem

Regular and singular asymptotic methods are applied to one- and two-dimensional Fredholm-Volterra integral equation (F-VIE) of the first kind that arise in the treatment of various two-dimensional axisymmetric and three-dimensional problems with mixed boundary conditions in the mechanics of continuous media. The solution of the integral equation is obtained in the space L2(@W)xC(0,T),0=

On a symptotic methods for Fredholm--Volterra integral equation of the second kind in contact problems

A method is used to obtain the general solution of Fredholm-Volterra integral equation of the second kind in the space L2(Ω) × C(0, T), 0 ≤ t ≤ T < ∞ Ω is the domain of integrations.The kernel of the Fredholm integral term belong to C([Ω] × [Ω]) and has a singular term and a smooth term. The kernel of Volterra integral term is a positive continuous in the class C(0,T), while Ω is the domain of integration with respect to the Fredholm integral term.Besides the separation method, the method of orthogonal polynomials has been used to obtain the solution of the Fredholm integral equation. The principal (singular) part of the kernel which corresponds to the selected domain of parameter variation is isolated. The unknown and known functions are expanded in a Chebyshev polynomial and an infinite algebraic system is obtained.

Fredholm-Volterra integral equation with singular kernel

In this paper, under certain conditions, a series in the Legendre polynomials form is used to obtain the solution of Fredholm-Volterra integral equation of the second kind in L2(-1,1)xC(0,T),T<~. The uniqueness of the solution is considered. The kernel of Fredholm integral term is established in a logarithmic form in position, while the kernel of Volterra integral term is a positive continuous function in time and belongs to the class C(0,T),T<~. The solution by series leads us to obtain an infinite system of linear algebraic equations, where the convergence of this system is studied. In the end of this paper, the Fredholm-Volterra integral equation of the first kind is established and its solution is discussed.

On the numerical treatment of the singular integral equation of the second kind

Here, a numerical treatment for solving the integral equation of the second kind with Cauchy kernel is presented. The singular term has been removed and the solution in the Legendre polynomial form has been used to obtain a system of linear algebraic equation. This system is solved numerically.

On the solution of linear and nonlinear integral equation

The purpose of this paper is to establish the solution of Fredholm-Volterra integral equation of the second kind in the space L2(@W)xC[0,T] considering the following when, Fredholm integral term in Li2(@W) and Volterra integral term, in [0,T], are linear. Also, each of Fredholm or Volterra integral term is linear while the other term is nonlinear. Here, @W is defined as the domain of integration with respect to position, while t is consider as the time, [emailxa0protected]?[0,T], T<~. Also the Fredholm-Volterra integral equation of the first kind is established as special case of these work. Many special in one, two and three dimensional problems of integral equation are considered. Representing the kernel of Fredholm integral term, in the form of logarithmic function, Carleman function, potential and generalized potential function and Macdonald function are considered.

On a method for solving a two-dimensional nonlinear integral equation of the second kind

In this article, the existence of at least one solution of a nonlinear integral equation of the second kind is proved. The degenerate method is used to obtain a nonlinear algebraic system, where the existence of at least one solution of this system is discussed. Finally, computational results with error estimates are obtained using Maple software.

Fredholm integral equation of the second kind with potential kernel

A method is used to solve the Fredholm integral equation of the second kind, which is investigated from the semi-symmetric Hertz problem for two different elastic materials in three dimensions. Also the kernel is represented in the nonhomogeneous wave equation form.

On the numerical solutions of Fredholm-Volterra integral equation

Toeplitz matrix method and the product Nystrom method are described for mixed Fredholm-Volterra singular integral equation of the second kind. The results are compared with the exact solution of the integral equation. The error of each method is calculated.

On a method for solving an integral equation in the displacement contact problem

In this paper, the solution, in a series form, of the integral equation of the mixed type in the space L(@W)xC[0,T] is obtained, where @W={(x,y,z):-~

Fredholm-Volterra integral equation and generalized potential kernel

A method is used to solve the Fredholm-Volterra integral equation of the first kind in the space L2(@W)xC(0,T),@W=(x,y)@[emailxa0protected]:x^2+y^2=