Some applications of collocation method based on legendre polynomials

In this study, approximate solutions of linear differential equations, Fredholm and Volterra integral equations are investigated using the matrix sorting method based on Legendre polynomials. The first section of the study, along with a literature review, highlights the theoretical importance of Legendre polynomials. The second section provides a detailed discussion of the definition, basic properties, orthogonality, series, generating functions, recurrence relations of Legendre polynomials, establishing the mathematical basis of the method. The third section presents a detailed presentation of the Legendre sorting method, demonstrating how differential and integral equations are solved by converting them into matrix systems. The fourth section presents approximate solutions obtained by applying the method to selected example problems. The conclusion section highlights the method's place in the literature and its applicability to various scientific fields, emphasizing the study's contribution to the field.