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

The asymptotic behavior of the solution of the singularly perturbed boundary value problem is examined. The derivations prove that a unique pair exists, in which components and satisfy the equation and boundary value conditions . 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 problem is 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.