Orthogonal cubic spline basis and its applications to a partial integro-differential equation with a weakly singular kernel

Abstract

In this paper, a new orthogonal basis for the space of cubic splines has been introduced. This basis is obtained based on B-splines and using the Gram–Schmidt orthogonalization process. A linear combination of these basis is used to estimate functions and numerical solutions of a partial integro-differential equation with a weakly singular kernel. The convergence analysis in the approximate scheme is investigated. The accuracy of the proposed method is demonstrated by three test problems. The results of numerical experiments are compared with analytical solutions by calculating errors $$L_2$$ -norm, $$L_\infty $$ -norm and $$H^1$$ -norm. The numerical results are found to be in good agreement with the exact solutions.