ASYMPTOTIC EXPANSION OF SOLUTION OF BVP WITH INITIAL JUMPS FOR SINGULARLY PERTURBED INTEGRO-DIFFERENTIAL EQUATION

Abstract

We consider the undivided boundary value problem for singularly perturbed linear integro-differential equations of th order with Fredholm’s integral terms have an initial jump of the phenomenon th order on the left end of the segment. Defined regular and boundary layer members of the asymptotic expansions of the solutions. Regular members of the asymptotics constructed in the form of integro-differential equations, which different from the usual unperturbed equations by the presence of additional term called initial jump of integral terms. The values of the initial jumps are defined. Boundary conditions for the regular members of the asymptotics also contain additional term, called an initial jump of the derivatives th order. Thus, for the determination of the regular members of the asymptotics obtained boundary value problems for linear integro-differential with additional parameter. To determine the boundary layer of the asymptotic expansion of solution we obtained initial problems for homogeneous an inhomogeneous ordinary differential equations with constant coefficients. Exponential estimates for boundary layer terms of the asymptotics are obtained. A theorem of existence and uniqueness and the asymptotic representations of solution with an estimate of the remainder term of the asymptotics. It is found that the constructed asymptotics approximation to the solution of the original singularly perturbed integro-differential boundary value problem is uniformly throughout the considered interval.