Journal of Prime Research in Mathematics

A cubic trigonometric B-spline collocation method based on Hermite formula for the numerical solution of the heat equation with classical and non-classical boundary conditions

Aatika Yousaf\(*{1,*}\), Muhammad Yaseen\(^a\)
\(^a\)Department of Mathematics, University of Sargodha,40100, Sargodha, Pakistan.
Correspondence should be addressed to: Muhammad Yaseen at yaseen.yaqoob@uos.edu.pk

Abstract

In this article, a new trigonometric cubic B-spline collocation method based on the Hermite formula is presented for the numerical solution of the heat equation with classical and non-classical boundary conditions. This scheme depends on the standard finite difference scheme to discretize the time derivative while cubic trigonometric B-splines are utilized to discretize the derivatives in space. The scheme is further refined utilizing the Hermite formula. The stability analysis of the scheme is established by standard Von-Neumann method. The numerical solution is obtained as a piecewise smooth function empowering us to find approximations at any location in the domain. The relevance of the method is checked by some test problems. The suitability and exactness of the proposed method are shown by computing the error norms. Numerical results are compared with some current numerical procedures to show the effectiveness of the proposed scheme.

Keywords:

Heat equation, cubic B-splines collocation method, cubic trigonometric basis function, Hermite formula, stability.