توابع پايه‌اي لاگرانژ غير كلاسيك و كاربردهاي آن در معادلات ديفرانسيل كسري

This thesis investigates the development and application of a novel pseudo-spectral method based on Müntz-functions for solving fractional differential equations. The study builds upon the Jacobi-Müntz functions, proposing two new non-classical Lagrange basis functions derived from these functions. These basis functions offer more general forms compared to traditional Jacobi polynomials and their generalized forms and are used to create two non-classical interpolants. The research includes the derivation of pseudo-spectral fractional differentiation matrices. Numerical results are presented to eva‎luate the effectiveness and computational efficiency of the proposed methods. The results demonstrate that these non-classical Lagrange basis functions offer significant improvements in accuracy and convergence rates compared to classical methods, especially for problems involving fractional differential equations. The findings indicate that the Müntz pseudo-spectral method can be a powerful tool for solving complex problems in applied mathematics and engineering, where traditional methods may fall short.

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