On a Mixed Problem for Hilfer Type Fractional Differential Equation with Degeneration

Yuldashev, T. K.; Kadirkulov, B. J.; Bandaliyev, Rovshan A.

doi:10.1134/s1995080222040229

Cited by 15 publications

(4 citation statements)

References 25 publications

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“…These operators play a pivotal role in formulating and solving the fractional partial differential equation under consideration. To establish the groundwork, we present fundamental properties associated with these fractional operators, drawing upon existing literature [9,10,11,12,13,14].…”

Section: Introductionmentioning

confidence: 99%

Initial-Boundary Value Problems to the Time-Nonlocal Diffusion Equation

Mambetov

2024

jour

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This article investigates a fractional diffusion equation involving Caputo fractional derivative and Riemann-Liouville fractional integral. The equation is supplemented by initial and boundary conditions in the domain defined by the interval by space 0<x<1 and interval by time 0<t<T. The fractional operators are defined rigorously, utilizing the Caputo fractional derivative of order β and the Riemann-Liouville fractional integral of order α, where 0<α<β≤1. The main results include the presentation of well-known properties associated with fractional operators and the establishment of the unique solution to the given problem. The key findings are summarized through a theorem that provides the explicit form of the solution. The solution is expressed as a series involving the two-parameter Mittag-Leffler function and orthonormal eigenfunctions of the Sturm-Liouville operator. The uniqueness of the solution is proven, ensuring that the problem has a single, well-defined solution under specific conditions on the initial function. Furthermore, the article introduces and proves estimates related to the Mittag-Leffler function, providing bounds crucial for the convergence analysis. The convergence of the series is investigated, and conditions for the solution to belong to a specific function space are established. The uniqueness of the solution is demonstrated, emphasizing its singularity within the given problem. Finally, the continuity of the solution in the specified domain is confirmed through the uniform convergence of the series.

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Section: Introductionmentioning

confidence: 99%

Initial-Boundary Value Problems to the Time-Nonlocal Diffusion Equation

Mambetov

2024

jour

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“…Recently, in [14] the initial-boundary value problems of Dirichlet and Neumann for the Caputo type time-fractional diffusion equation are considered. Also, the regular solution of a mixed problem for the Hilfer type nonlinear partial differential equation in three-dimensional domain is studied in [15].…”

Section: Introductionmentioning

confidence: 99%

Cauchy problems for the time-fractional degenerate diffusion equations

Smadiyeva²

2023

JMMCS

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This paper is devoted to the Cauchy problems for the one-dimensional linear time-fractional diffusion equations with ∂ α t the Caputo fractional derivative of order α ∈ (0, 1) in the variable t and time-degenerate diffusive coefficients t β with β > −α. The solutions of Cauchy problems for the one-dimensional time-fractional degenerate diffusion equations with the time-fractional derivative ∂ α t of order α ∈ (0, 1) in the variable t, are shown. In the "Problem statement and main results"section of the paper, the solution of the time-fractional degenerate diffusion equation in a variable coefficient with two different initial conditions are considered. In this work, a solution is found by using the Kilbas-Saigo function E α,m,l (z) and applying the Fourier transform F and inverse Fourier transform F −1 . Convergence of solution of problem 1 and problem 2 are proven using Plancherel theorem. The existence and uniqueness of the solution of the problem are confirmed.

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“…В работе [8] для линейного обыкновенного дифференциального уравнения дробного порядка вида (3) с производными Римана-Лиувилля была сформулирована и решена начальная задача. Краевые и начальные задачи для вырождающихся уравнений с дробным производным Хилфера исследовались в работах [9][10][11][12], а с дробными производными Римана-Лиувилля и Капуто в работах [2], [13][14]. В данной работе в терминах функции Килбаса-Сайго строится явное представление фундаментальной системы решений уравнения (1).…”

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“…,γm} 0y u (y) = λy s u, y > 0, λ ∈ C, s ≥ 0. limy→0 D α 0 0y u (y) = A 0 , lim y→0 D α 1 0y u (y) = A 1 , ... lim y→0 D α m−1 0y u (y) u (y) = A m−1 . (11) здесь A i = const, i = 0, 1, ..., m − 1, D α 0 0y = D γ 0 −1 0y , D α k 0y = D γ k −1 ≤ k ≤ s − 1, Γ (1 + α k ) , k = s, Γ(1+α k ) Γ(1+α k −αs) y α k −αs , s < k ≤ m.Подставив представление (9) в начальные условия(11) , получимlim y→0 D α 0 0y u (y) = d 0 Γ (1 + α 0 ) = A 0 , lim y→0 D α 1 0y u (y) = d 1 Γ (1 + α 1 ) = A 1 , ... lim y→0 D α m−1 0y u (y) = d m−1 Γ (1 + α m−1 ) = A m−1 .Значит решение задачи Коши будет иметь вид u (y) =…”

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Построение Частных Решений С Помощью Специальных Функций Для Уравнений С Дробной Производной

Irgashev¹

2022

Matematicheskie Zametki

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On a Mixed Problem for Hilfer Type Fractional Differential Equation with Degeneration

Cited by 15 publications

References 25 publications

Initial-Boundary Value Problems to the Time-Nonlocal Diffusion Equation

Initial-Boundary Value Problems to the Time-Nonlocal Diffusion Equation

Cauchy problems for the time-fractional degenerate diffusion equations

Построение Частных Решений С Помощью Специальных Функций Для Уравнений С Дробной Производной

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