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.

在本文中，使用配备了新的二阶导数近似的三次B样条搭配方法来近似热方程的解。该技术依赖于典型的有限差分方案来离散时间导数，而三次B样条被用作空间维的插值函数。使用这种方法的主要优点是，解决方案是作为一个分段的连续函数获得的，该函数使一个函数可以在域的任何所需位置处找到近似值。严格研究了所提方法的稳定性和收敛性。该方案的能力通过一些测试问题来检验。通过计算误差准则，验证了所提方法的有效性和正确性。