روش طيفي پتروف-گالركين با پيچيدگي محاسباتي خطي براي حل معادلات ديفرانسيل مرتبه‌ي كسري در دامنه‌ي نيمه‌نامتناهي

In recent years, mathematical models involving fractional-order derivatives have garnered significant attention for addressing complex problems across various disciplines, including the basic sciences, physics, an‎d medicine. Owing to their superior capability in accurately modeling natural phenomena an‎d predicting the behavior of dynamic systems, these models have become increasingly vital. Concurrently, the Sturm–Liouville problem stan‎ds as a cornerstone in scientific an‎d engineering domains, playing a crucial role in the theoretical an‎d numerical development of these fields. Moreover, spectral methods have consistently been of interest for constructing accurate basis functions, especially when dealing with challenging an‎d highdimensional problems. This thesis investigates the fractional-order Sturm–Liouville problem defined on a semi-infinite domain an‎d derives a new family of basis functions that serve as its solutions. To facilitate this, a novel an‎d flexible spectral scheme—referred to as the Laguerre Petrov–Galerkin method—is introduced. This method is specifically designed to solve initial value problems involving constantcoefficient fractional derivatives of order up to one on unbounded (semi-infinite) domains. The proposed scheme results in a structured matrix system that can be solved with a computational complexity of O(N logN), significantly improving the efficiency of fractional differential equation solvers. To derive closed-form expressions for the entries of the mass an‎d stiffness matrices, recurrence relations for generalized Laguerre functions are exploited. The resulting matrices are constructed in a separable form as the product of a diagonal matrix an‎d a lower-triangular Toeplitz matrix, which drastically reduces computational cost an‎d enhances performance in numerical simulations. Additionally, this method is extended to fractional differential equations involving multiple fractional derivatives. In such cases, the equations are reformulated into systems of linear algebraic equations through appropriate polynomial approximations an‎d the application of the Gauss–Legendre quadrature rule, making them amenable to efficient numerical solution.

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