Orthogonal cubic splines for the numerical solution of nonlinear parabolic partial differential equations

In this paper, a new orthogonal basis for the space of cubic splines has been introduced. A linear combination of cubic orthogonal splines is considered to approximate the functions in which the coefficients are calculated with numerically stable formulae. Applications to the numerical solutions of some parabolic partial differential equations are given, in which the approximations are obtained using the first and second integral of orthogonal splines which leads to an efficient solution procedure. The convergence analysis in the approximate scheme is investigated. A comparison of the obtained numerical solutions with some other papers indicates that the presented method is reliable and yields result with good accuracy. The main parts of our study are as follows:•We propose a robust approach based on the orthogonal cubic splines procedure in conjunction with the operational matrix.•The convergence in the approximate scheme is analyzed.•Numerical examples show that the proposed method is very accurate.