A cubic B-spline collocation method with new approximation for the numerical treatment of the heat equation with classical and non-classical boundary conditions

In this paper, a cubic B-spline collocation method equipped with new approximations for second-order derivatives is used to approximate the solution of the heat equation. This technique depends on the typical finite difference scheme to discretize the time derivative while cubic B-splines are utilized as interpolation functions in the space dimension. The key advantage of using this approach is that the solution is obtained as a piecewise continuous function empowering one to find approximation at any desired location of the domain. The stability and convergence analysis of the presented method are studied rigorously. The capability of the scheme is checked by some test problems. The effectiveness and exactness of the proposed method are confirmed by computing the error norms. Numerical results are contrasted with some existing numerical schemes to exhibit the predominance of our scheme. (paper)