DEVELOPING THE LIMIT-RESIDUAL FUNCTION METHOD FOR PROVIDING APPROXIMATE SOLUTIONS TO INITIAL AND BOUNDARY-VALUE PROBLEMS

The limit residual function (RF) method was recently introduced in 2024 to provide series solutions for a specific class of differential equations that involve only initial conditions. This paper expands this approach to encompass a broader range of differential equations, incorporating initial and boundary conditions. Two new parameters are introduced in the enhanced method: the embedding parameter and the auxiliary parameter. The auxiliary parameter controls the convergence intervals of the series solution. Additionally, we analyze the error of the approximate solution. A so-called [Formula: see text]-curve is introduced to help determine the optimal value of the [Formula: see text] parameter. To demonstrate the efficiency and effectiveness of the method, we discuss two examples: one involving initial conditions and the other involving boundary conditions.