Discontinuous fractional Sturm–Liouville problems with transmission conditions

Akdoğan, Z.; Yakar, Ali; Demirci, Mustafa

doi:10.1016/j.amc.2018.12.049

Cited by 11 publications

(3 citation statements)

References 21 publications

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“…Recently, there has been an increasing interest in Sturm-Liouville boundary value problems defined on two or more disjoint segments with common ends, the so-called many-interval SLPs (see, for example, [9][10][11][12][13][14][15][16][17][18][19][20][21] and references cited therein). To deal with such multi-interval boundary value problems, naturally, additional conditions (the so-called transmission conditions, jump conditions, interface conditions, and impulsive conditions) are imposed at these common endpoints.…”

Section: Introductionmentioning

confidence: 99%

Non-classical periodic boundary value problems with impulsive conditions

ÖZTÜRK

Mukhtarov

Aydemir

2023

Journal of New Results in Science

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This study investigates some spectral properties of a new type of periodic Sturm-Liouville problem. The problem under consideration differs from the classical ones in that the differential equation is given on two disjoint segments that have a common end, and two additional interaction conditions are imposed on this common end (such interaction conditions are called various names, including transmission conditions, jump conditions, interface conditions, impulsive conditions, etc.). At first, we proved that all eigenvalues are real and there is a corresponding real-valued eigenfunction for each eigenvalue. Then we showed that two eigenfunctions corresponding to different eigenvalues are orthogonal. We also defined some left and right-hand solutions, in terms of which we constructed a new transfer characteristic function. Finally, we have defined asymptotic formulas for the transfer characteristic functions and also for the eigenvalues. The results obtained are a generalization of similar results of the classical Sturm-Liouville theory.

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Section: Introductionmentioning

confidence: 99%

Non-classical periodic boundary value problems with impulsive conditions

ÖZTÜRK

Mukhtarov

Aydemir

2023

Journal of New Results in Science

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“…See also [14][15][16][17][18] and the references therein. Nonetheless, some recent publications discuss the existence, uniqueness, and stability features of variable fractional order differential equations [19][20][21][22][23][24][25][26][27][28][29][30][31][32].…”

Section: Introductionmentioning

confidence: 99%

Existence, Uniqueness, and Stability of Solutions to Variable Fractional Order Boundary Value Problems

Souid

Bouazza

Yakar

2022

Journal of New Theory

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This paper investigates the sufficient conditions for the existence and uniqueness of a class of Riemann-Liouville fractional differential equations of variable order with fractional boundary conditions. The problem is converted into differential equations of constant orders by combining the concepts of generalized intervals and piecewise constant functions. We derive the required conditions for ensuring the uniqueness of the problem in order to utilize the Banach fixed point theorem. The stability of the obtained solution in the Ulam-Hyers-Rassias (UHR) sense is also investigated, and we finally provide an illustrative example.

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“…Among these methods, we can mention sinc collocation method and differential quadrature method [3], finite-element method [4], Sinc-Galerkin and differential transform methods [5], boundary Value Methods [6], differential quadrature method [7], Haar wavelet method [8] and homotopy analysis method [9]. For further studies, we can refer the readers to [10][11][12]13].…”

Section: Introductionmentioning

confidence: 99%

Pseudospectral method for solving fractional Sturm-Liouville problem using Chebyshev cardinal functions

et al. 2021

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This work deals with the pseudospectral method to solve the Sturm–Liouville eigenvalue problems with Caputo fractional derivative using Chebyshev cardinal functions. The method is based on reducing the problem to a weakly singular Volterra integro-diﬀerential equation. Then, using the matrices obtained from the representation of the fractional integration operator and derivative operator based on Chebyshev cardinal functions, the equation becomes an algebraic system. To get the eigenvalues, we ﬁnd the roots of the characteristics polynomial of the coeﬃcients matrix. We have proved the convergence of the proposed method. To illustrate the ability and accuracy of the method, some numerical examples are presented.

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Discontinuous fractional Sturm–Liouville problems with transmission conditions

Cited by 11 publications

References 21 publications

Non-classical periodic boundary value problems with impulsive conditions

Non-classical periodic boundary value problems with impulsive conditions

Existence, Uniqueness, and Stability of Solutions to Variable Fractional Order Boundary Value Problems

Pseudospectral method for solving fractional Sturm-Liouville problem using Chebyshev cardinal functions

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