A new multiscale algorithm for solving the heat conduction equation

In this paper, a multiscale algorithm has been initially proposed for a numerical solution to the heat conduction equation. Firstly, the differential form of the heat conduction equation is transformed into an integral form, and then the time variable is discretized by using the compound trapezoidal formula. Secondly, the orthonormal basis with compactly supported property is constructed in the reproducing kernel space W2[0, b]. For the spatial variable, the e-approximation solution of the operator equation is obtained by using the orthonormal basis. Finally, the numerical solution of the heat conduction equation is obtained based on interpolation theory. In the meantime, the feasibility of the algorithm is verified theoretically and numerically. & COPY; 2023 THE AUTHORS. Published by Elsevier BV on behalf of Faculty of Engineering, Alexandria University. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/ licenses/by-nc-nd/4.0/).