مسائل مقدار مرزي شتورم ليوويل كسري در دامنه‌هاي نامتناهي: نظريه و كاربرد‌ها

Fractional Riemann-Liouville Derivative an‎d Integral , Confluent Hypergeometric Function , Associated Laguerre Polynomials , Generalized Associated Laguerre Functions of the First an‎d Second Kinds , Fractional Sturm–Liouville Problems , Self-Adjoint Operators , Eigen-Functions an‎d Eigenvalues , Orthogonal Projections , Truncation Errors

Fractional Sturm-Liouville Boundary Value Problems in Unbounded Domains: Theory an‎d Applications

رياضي كاربردي و علوم كامپيوتر

In this study, we first examine the fractional Sturm–Liouville operators in confluent hypergeometric differential equations an‎d then investigate two boundary-value problems—referred to as the first an‎d second types of fractional Sturm–Liouville operators—in the semi-infinite domain. Subsequently, we analytically derive the corresponding eigenfunctions an‎d prove that these eigenfunctions, called the generalized associated Laguerre functions of the first an‎d second types, form appropriate orthogonal bases for solving fractional-order differential equations using spectral methods. We then present the fundamental properties of these operators, including the self-adjointness of the Sturm–Liouville problems, the non-polynomial nature of the eigenfunctions, the orthogonality of the eigenfunctions, the orthogonality of their fractional derivatives, the asymptotic behavior of the eigenvalues, the three-term recurrence relations, an‎d the fact that the derivatives of the eigenfunctions admit closed forms within the same family, as well as the reality of all eigenvalues. Furthermore, we establish completeness of the eigenfunctions, orthogonal projections, an‎d upper bounds for projection truncation errors. This study also derives the operational matrices for fractional integration an‎d differentiation an‎d extends the spectral method for solving fractional differential equations. Numerical results are provided to assess the effectiveness an‎d efficiency of the two introduced bases, namely the generalized associated Laguerre functions of the first an‎d second types. These results demonstrate that the proposed basis functions significantly improve accuracy an‎d convergence rates compared to the Laguerre polynomials an‎d their generalized forms, particularly for problems involving fractional differential equations. They show that the generalized associated Laguerre functions of the second type perform remarkably well in modeling problems that involve damping over time, such as clock pendulums, as well as in simulating phenomena observed in nature, such as many diseases. In contrast, classical orthogonal Laguerre polynomials an‎d their stan‎dard dependent forms may fail in this regard.

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