The purpose of this study is to develop an efficient and accurate numerical technique for solving nonlinear boundary value problems, with particular focus on the Darcy–Brinkman–Forchheimer equation (DBFE) and the quartic strongly nonlinear heat transfer equation (QSNHTE).
The approach is developed by constructing a matrix-based method that employs Pell-Lucas polynomials (PLPs) and utilizes the evenly spaced collocation points. Initially, the solution is formulated in a matrix representation, allowing all terms within the problems to be expressed accordingly. The Pell-Lucas collocation method (PLCM) is established by using the matrix formulation with the evenly spaced collocation points. Through this procedure, the original nonlinear equations are transformed into a system of linear algebraic equations, and solving this system yields the coefficient matrix corresponding to the PLP-based solutions. An error analysis is conducted for both problems. Subsequently, numerical implementations are carried out using MATLAB. Additionally, the L∞ and root mean square error norms are computed for various polynomial degrees and parameter values, providing quantitative validation of the method's accuracy.
The numerical results demonstrate that the proposed PLCM provides highly accurate and stable solutions for both DBFE and QSNHTE. The computed error norms decrease significantly with increasing polynomial degree, confirming the convergence and reliability of the method. Comparisons with existing numerical approaches reported in the literature show that the proposed technique achieves competitive accuracy, which is further illustrated through tabulated data and graphical representations.
The proposed PLCM has the limitation that it is currently formulated and tested for one-dimensional nonlinear problems, and its performance has been demonstrated specifically for the DBFE and QSNHTE models; therefore, further studies are needed to evaluate its applicability and efficiency for higher-dimensional and more complex nonlinear systems.
The proposed PLCM provides a reliable and computationally efficient tool for solving nonlinear boundary value problems arising in porous media flow and nonlinear heat transfer. Its matrix-based structure and straightforward MATLAB implementation make it suitable for practical engineering applications requiring high accuracy and numerical stability.
By improving the accuracy and stability of numerical simulations for porous media flow and nonlinear heat transfer models, the proposed method may contribute indirectly to more efficient energy systems and environmentally sustainable engineering designs.
The originality of this work lies in the matrix representation of nonlinear differential operators based on PLPs, enabling the numerical treatment of strongly nonlinear boundary value problems arising from the DBFE and QSNHTE.
