Cauchy systems for fredholm integral equations with parameter imbedding

Abstract

Consider the family of Fredholm integral equations u( t, γ)= g( t)+ γ∫ 1 0 k( t, y) u( y, γ)d y, where γ is sufficiently small to guarantee a solution, and the Cauchy system u γ(t,γ)=∫ 1 0K(t,y,γ)u(y,γ) dy, K γ(t,y,γ)=∫ 1 0K(t,y ′,γ)K(y ′,y,γ) dy ′, 0⩽t, y⩽1, 0⩽γ, u(t,0)=g(t), K(t,γ,0)=k(t,y), 0⩽t, y⩽1 . The equivalence between the family of Fredholm integral equations and the Cauchy system is demonstrated. The numerical method is illustrated with an example.