Numerical Solution of Nonlinear Advection Equation Using Reproducing Kernel Method

In this study, an iterative approximation is proposed by using the reproducing kernel method (RKM) for the nonlinear advection equation. To apply the iterative RKM, specific reproducing kernel spaces are defined and their kernel functions are presented. The proposed method requires homogenising the initial or boundary conditions of the problem under consideration. After homogenising the initial condition of the advection equation, a linear operator selection is made, and then the approximate solution is constructed using orthonormal basis functions in serial form. Convergence analysis of the approximate solution is demonstrated through the lemma and theorem. Numerical outcomes are provided in the form of graphics and tables to show the efficiency and accuracy of the presented method.