Cauchy systems for fredholm integral equations with parameter imbedding

摘要

Consider the family of Fredholm integral equations u(t,γ)=g(t)+γ∫10k(t,y)u(y,γ)dy, where γ is sufficiently small to guarantee a solution, and the Cauchy system uγ(t,γ)=∫10K(t,y,γ)u(y,γ)dy,Kγ(t,y,γ)=∫10K(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.