Asymptotic Estimates of the Solution of a Restoration Problem with an Initial Jump

Abstract

The asymptotic behavior of the solution of the singularly perturbed boundary value problemLεy=htλ,Liy+σiλ=ai,i=1,n+1̅is examined. The derivations prove that a unique pair(ỹt,λ̃ε,ε,λ̃ε)exists, in which componentsy(t,λ̃ε,ε)andλ̃(ε)satisfy the equationLεy=h(t)λand boundary value conditionsLiy+σiλ=ai,i=1,n+1̅. The issues of limit transfer of the perturbed problem solution to the unperturbed problem solution as a small parameter approaches zero and the existence of the initial jump phenomenon are studied. This research is conducted in two stages. In the first stage, the Cauchy function and boundary functions are introduced. Then, on the basis of the introduced Cauchy function and boundary functions, the solution of the restoration problemLεy=htλ,Liy+σiλ=ai,i=1,n+1̅is obtained from the position of the singularly perturbed problem with the initial jump. Through this process, the formula of the initial jump and the asymptotic estimates of the solution of the considered boundary value problem are identified.