Pell-lucas matrix ordering method for solution of linear and nonlinear differential equations and its applications to engineering problems

In this study, a matrix ordering method based on Pell-Lucas polynomials has been developed as an alternative approach to solving linear and nonlinear differential equations under given mixed conditions and numerical solutions have been investigated. The method, which has been developed by utilizing the standard ordering points and the matrix properties of Taylor and Pell–Lucas polynomials, reduces the considered differential equations to a system of algebraic equations whose unknowns are the coefficients of the Pell–Lucas polynomials. By solving this algebraic system, approximate solutions are obtained in terms of a finite number of Pell–Lucas polynomials. The study consists of five sections. In the first section, the aim, scope and motivation of the study are presented; the limitations of classical solution methods and the potential of alternative polynomial-based approaches are discussed. In the second section, Pell and Pell-Lucas polynomials, which form the basis of the thesis, are introduced, and the definition relations, generating functions and some basic properties of these polynomials are examined in detail. In the third section, the theoretical infrastructure of the matrix sorting method is established and the sorting approach based on Pell-Lucas polynomials is defined, and the building blocks of the algorithm are mathematically based. In the fourth section, how the developed method is applied to the boundary layer problem is explained step by step, and the algorithm is run on test matrices of different sizes. In addition, the obtained results are evaluated numerically, the method is compared with other solution methods based on special polynomials in the literature, and its advantages in terms of calculation accuracy are analyzed. In the fifth and last section, a general evaluation is made, the prominent contributions of the study are summarized and suggestions for future studies are presented.