Fredholm‐Volterra integral equation with potential kernel

M A Abdou,A A El-Bary

doi:10.1155/s0161171201005981

Abstract

A method is used to solve the Fredholm‐Volterra integral equation of the first kind in the space L2(Ω) × C(0, T), , z = 0, and T < ∞. The kernel of the Fredholm integral term considered in the generalized potential form belongs to the class C([Ω] × [Ω]), while the kernel of Volterra integral term is a positive and continuous function that belongs to the class C[0, T]. Also in this work the solution of Fredholm integral equation of the second and first kind with a potential kernel is discussed. Many interesting cases are derived and established in the paper.

Highlights

In [14, 15] Mkhitaryan and Abdou obtained the general formulas, even and odd, of the potential analytic function, using Krein’s method [13], for the Fredholm integral equation of the first kind with Carleman kernel [15] and logarithmic kernel [14]
Many problems of mathematical physics, theory of elasticity, and mixed problems of mechanics of continuous media reduce to an integral equation with a kernel that has one of the following forms: Knα,m γ (x, y ) = y xα +γ−1 Wnα,m (x, y ), ∞Wnα,m(x, y) = λαJn(xλ)Jm(yλ) dλ, (1.1) (1.2)where Jn(x) is a Bessel function of the first kind of order n
The Fredholm integral equations of the first and second kind with a generalized potential kernel are established and their solutions are discussed, the kernel is represented in the Weber-Sonin integral formula

Summary

Introduction

In [14, 15] Mkhitaryan and Abdou obtained the general formulas, even and odd, of the potential analytic function, using Krein’s method [13], for the Fredholm integral equation of the first kind with Carleman kernel [15] and logarithmic kernel [14] Abdou in [1] obtained the solution of Fredholm integral equation of the second kind with potential function kernel, K(x − ξ, y − η) = 1/ (x − ξ)2 + (y − η)2 , In [3], the structure resolvent for the Fredholm integral equation of the second kind with potential function kernel is obtained by Abdou.

Results

Conclusion