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We consider a singularly perturbed initial-third boundary value Sobolev problem. Firstly, the asymptotic behaviour of the exact solution is analysed. Then, a second-order finite difference scheme is constructed on the special non-uniform mesh. By using energy estimate, the stability and convergence of the proposed scheme are investigated in the discrete energy norm. Finally, three numerical examples are solved to validate the theory.

A second-order difference scheme for the singularly perturbed Sobolev problems with third type boundary conditions on Bakhvalov mesh