A higher-order compact finite difference method for third-order boundary value problems with a novel treatment of Robin boundary conditions

This paper presents a higher-order compact finite difference method for solving third-order boundary value problems subject to Robin boundary conditions. The proposed scheme achieves sixth-order accuracy in the interior domain and incorporates a new boundary closure technique for Robin conditions to maintain this accuracy at the boundaries. A detailed error analysis is performed to establish the theoretical convergence properties of the method. Numerical experiments on several benchmark problems are conducted to validate the accuracy and efficiency of the scheme. The computed results confirm that the method attains the expected sixth-order rate of convergence and demonstrate its superiority over existing lower-order methods in terms of accuracy and computational efficiency. The proposed approach provides a reliable and effective tool for solving third-order boundary value problems subject to Robin boundary conditions.