Explicit and implicit numerical investigations of one-dimensional heat equation based on spline collocation and Thomas algorithm

This study uses the cubic spline method to solve the one-dimensional (1D) (one spatial and one temporal dimension) heat problem (a parametric linear partial differential equation) numerically using both explicit and implicit strategies. The set of simultaneous equations acquired in both the explicit and implicit method may be solved using the Thomas algorithm from the tridiagonal dominating matrix, and the spline offers a continuous solution. The results are implemented with very fine meshes and with relatively small-time steps. Using mesh refinement, it was possible to find better temperature distribution in the thin bar. Five numerical examples are used to support the efficiency and accuracy of the current scheme. The findings are also compared with analytical results and other results in terms of error and error norms \({L}_{2}\) $L_2$ and \({L}_{\infty }\) $L_{\infty }$ . The Von-Neuman technique is used to analyse stability. The truncation error of both systems is calculated and determined to have a convergence of order \(O\left( {h + \Delta t^{2} } \right).\) $O\left(h+\mathrm{\Delta }{t}^2\right).$