Reproducing kernel method for solving singularly perturbed differential-difference equations with boundary layer behavior in Hilbert space

Abstract We consider singularly perturbed differential-difference equation, which contains negative shift in the differentiated term with boundary layer behavior. RKHSM (Reproducing Kernel Hilbert Space Method) without Gram–Schmidt orthogonalization process, is considered in the present paper. We decompose the domain of the problem into two subintervals. One of them has not the boundary layer and the other one has. The side of the interval in which the boundary layer exists is important. If the boundary layer of this problem exists on the left side of interval, the RKHSM will provide a proper approximation of solution, otherwise for the implement of RKHSM, we need to change the variable of the singularly perturbed problem to shift the boundary layer region.

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