Compact fourth-order cubic spline approximation for nonlinear two-point Robin boundary value problems

In this paper, we aim to propose a new fourth-order compact numerical algorithm based on cubic spline approximations for the solution of nonlinear two-point boundary value problems with mixed boundary conditions. We require only two half-step points and a central point to discretize the nonlinear differential equation at each internal grid point, and two internal grid points near the boundary to discretize the mixed boundary conditions at the boundary. We do not require any fictitious points for computation and the method is directly applicable to solve problems irrespective of coordinates, which is the main significance of the proposed study. The error analysis of the proposed algorithm in the form of a nonlinear tri-diagonal system is described. In order to demonstrate the utility of the proposed method, we solved some benchmark problems, including singular problems and compared the results with the results obtained by other researchers. The numerical results confirm the superiority of the suggested method over other existing methods. Furthermore, the method is also applied to typically singular local fractional problem to validate its reliability.