Efficient Collocation Algorithm for High-Order Boundary Value Problems via Novel Exponential-Type Chebyshev Polynomials

Abstract

This paper presents an innovative collocation algorithm designed to effectively handle a specific class of boundary value problems with high-order characteristics. The approach involves utilizing a novel variant of exponential type Chebyshev polynomials that meet all the necessary equation conditions. A key aspect of the algorithm is the transformation of both linear and nonlinear forms of the equations, along with their respective boundary conditions, into systems of algebraic equations. By solving these systems, a unique iterative technique is employed that significantly reduces the computational time required for solving these types of equations. To validate the effectiveness of the algorithm, numerous experiments using various examples with differing orders and types are conducted. The proposed technique is compared against other similar methods. The results obtained demonstrate the exceptional accuracy of the proposed approach and its potential for extension to other models in the future. Additionally, a comprehensive and detailed error analysis of the proposed method is developed, further confirming its robustness and precision in practical applications.