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

2021 ◽  
Author(s):  
Alireza Afarideh ◽  
Farhad Dastmalchi Saei ◽  
Mehrdad Lakestani ◽  
Behzad Nemati Saray
Keyword(s):  
Algebraic System ◽  
Eigenvalue Problems ◽  
Fractional Integration ◽  
Pseudospectral Method ◽  
Liouville Problem ◽  
Derivative Operator ◽  
Weakly Singular ◽  
Fractional Integration Operator ◽  
Sturm Liouville ◽  
Cardinal Functions

Abstract 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.

scholarly journals The dual eigenvalue problems of the conformable fractional Sturm–Liouville problems

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Yan-Hsiou Cheng
Keyword(s):  
Eigenvalue Problems ◽  
Liouville Problem ◽  
First Eigenvalue ◽  
Liouville Property ◽  
The First Eigenvalue ◽  
Eigenvalue Gap ◽  
Single Well ◽  
Eigenvalue Ratio ◽  
Sturm Liouville ◽  
Prüfer Substitution

AbstractIn this paper, we are concerned with the eigenvalue gap and eigenvalue ratio of the Dirichlet conformable fractional Sturm–Liouville problems. We show that this kind of differential equation satisfies the Sturm–Liouville property by the Prüfer substitution. That is, the nth eigenfunction has $n-1$ n − 1 zero in $( 0,\pi ) $ ( 0 , π ) for $n\in \mathbb{N}$ n ∈ N . Then, using the homotopy argument, we find the minimum of the first eigenvalue gap under the class of single-well potential functions and the first eigenvalue ratio under the class of single-barrier density functions. The result of the eigenvalue gap is different from the classical Sturm–Liouville problem.

Fluxon Centering in Josephson Junctions with Exponentially Varying Width

2012 ◽  
Vol 2 (3) ◽  
pp. 204-213
Author(s):  
E. G. Semerdjieva ◽  
M. D. Todorov
Keyword(s):  
Josephson Junctions ◽  
Eigenvalue Problems ◽  
Liouville Problem ◽  
Static Solution ◽  
Physical Parameters ◽  
Nonlinear Eigenvalue Problems ◽  
Continuous Analogue ◽  
Sturm Liouville ◽  
The Stability ◽  
The Impact

AbstractNonlinear eigenvalue problems for fluxons in long Josephson junctions with exponentially varying width are treated. Appropriate algorithms are created and realized numerically. The results obtained concern the stability of the fluxons, the centering both magnetic field and current for the magnetic flux quanta in the Josephson junction as well as the ascertaining of the impact of the geometric and physical parameters on these quantities. Each static solution of the nonlinear boundary-value problem is identified as stable or unstable in dependence on the eigenvalues of associated Sturm-Liouville problem. The above compound problem is linearized and solved by using of the reliable Continuous analogue of Newton method.

scholarly journals A New Approach for Linear Eigenvalue Problems and Nonlinear Euler Buckling Problem

2012 ◽  
Vol 2012 ◽  
pp. 1-21 ◽  
Author(s):  
Meltem Evrenosoglu Adiyaman ◽  
Sennur Somali
Keyword(s):  
Eigenvalue Problems ◽  
Liouville Problem ◽  
Numerical Results ◽  
High Order Accuracy ◽  
Order Accuracy ◽  
Eigenvalues And Eigenfunctions ◽  
New Approach ◽  
Euler Buckling ◽  
Large Step ◽  
Sturm Liouville

We propose a numerical Taylor's Decomposition method to compute approximate eigenvalues and eigenfunctions for regular Sturm-Liouville eigenvalue problem and nonlinear Euler buckling problem very accurately for relatively large step sizes. For regular Sturm-Liouville problem, the technique is illustrated with three examples and the numerical results show that the approximate eigenvalues are obtained with high-order accuracy without using any correction, and they are compared with the results of other methods. The numerical results of Euler Buckling problem are compared with theoretical aspects, and it is seen that they agree with each other.

scholarly journals Fundamental Results of Conformable Sturm-Liouville Eigenvalue Problems

Complexity ◽  
2017 ◽  
Vol 2017 ◽  
pp. 1-7 ◽  
Author(s):  
Mohammed Al-Refai ◽  
Thabet Abdeljawad
Keyword(s):  
Boundary Conditions ◽  
Eigenvalue Problems ◽  
Liouville Problem ◽  
Rayleigh Quotient ◽  
First Eigenvalue ◽  
Suggested Model ◽  
Conformable Derivative ◽  
Arbitrary Boundary ◽  
Sturm Liouville ◽  
Natural Boundary Conditions

We suggest a regular fractional generalization of the well-known Sturm-Liouville eigenvalue problems. The suggested model consists of a fractional generalization of the Sturm-Liouville operator using conformable derivative and with natural boundary conditions on bounded domains. We establish fundamental results of the suggested model. We prove that the eigenvalues are real and simple and the eigenfunctions corresponding to distinct eigenvalues are orthogonal and we establish a fractional Rayleigh Quotient result that can be used to estimate the first eigenvalue. Despite the fact that the properties of the fractional Sturm-Liouville problem with conformable derivative are very similar to the ones with the classical derivative, we find that the fractional problem does not display an infinite number of eigenfunctions for arbitrary boundary conditions. This interesting result will lead to studying the problem of completeness of eigenfunctions for fractional systems.

Multiparameter Variational Eigenvalue Problems With Indefinite Nonlinearity

1997 ◽  
Vol 49 (5) ◽  
pp. 1066-1088 ◽  
Author(s):  
Tetsutaro Shibata
Keyword(s):  
Asymptotic Behavior ◽  
Asymptotic Formula ◽  
Optimal Condition ◽  
Level Set ◽  
Eigenvalue Problems ◽  
Liouville Problem ◽  
General Level ◽  
Sturm Liouville ◽  
Image Position

AbstractWe consider the multiparameter nonlinear Sturm-Liouville problemwhere are parameters. We assume that1 ≤ q ≤ p1 < p2 < ... ≤ pn < 2q + 3.We shall establish an asymptotic formula of variational eigenvalue λ = λ(μ, α) obtained by using Ljusternik-Schnirelman theory on general level set Nμ, α(α < 0 : parameter of level set). Furthermore,we shall give the optimal condition of {(μ, α)}, under which μi(m + 1 ≤ i ≤ n : fixed) dominates the asymptotic behavior of λ(μ, α).

scholarly journals Asymptotic spectrum of multiparameter eigenvalue problems

1996 ◽  
Vol 39 (1) ◽  
pp. 119-132 ◽  
Author(s):  
Hans Volkmer
Keyword(s):  
Hilbert Space ◽  
Maximum Principle ◽  
Eigenvalue Problem ◽  
Eigenvalue Problems ◽  
Liouville Problem ◽  
Asymptotic Spectrum ◽  
Multiparameter Eigenvalue Problem ◽  
Sturm Liouville

Results are given for the asymptotic spectrum of a multiparameter eigenvalue problem in Hilbert space. They are based on estimates for eigenvalues derived from the minim un-maximum principle. As an application, a multiparameter Sturm-Liouville problem is considered.

scholarly journals ON THE UPPER ESTIMATES FOR THE FIRST EIGENVALUE OF A STURM - LIOUVILLE PROBLEM WITH A WEIGHTED INTEGRAL CONDITION

2017 ◽  
Vol 21 (6) ◽  
pp. 124-129
Author(s):  
M.Yu. Telnova
Keyword(s):  
Weight Function ◽  
Eigenvalue Problems ◽  
Liouville Problem ◽  
Integral Condition ◽  
First Eigenvalue ◽  
Second Order Differential Equations ◽  
The First Eigenvalue ◽  
Lagrange Problem ◽  
Sturm Liouville ◽  
The Given

In this paper a problem for which the origin problem was a problem known as the Lagrange problem or the problem on finding the form of the firmest column of the given volume is viewed. The Lagrange problem was the source for different extremal eigenvalue problems, among them for eigenvalue problems for the second-order differential equations, with an integral condition on the potential. In this paper the problem of that kind is considered under the con- dition that the integral condition contains a weight function. The method of finding the sharp upper estimates for the first eigenvalue of a Sturm - Liouville problem with Dirichlet conditions for some values of parameters in the integral condition was found and attainability of those estimates was proved. In this paper a problem for which the origin problem was a problem known as the Lagrange problem or the problem on finding the form of the firmest column of the given volume is viewed. The Lagrange problem was the source for different extremal eigenvalue problems, among them for eigenvalue problems for the second-order differential equations, with an integral condition on the potential. In this paper the problem of that kind is considered under the con- dition that the integral condition contains a weight function. The method of finding the sharp upper estimates for the first eigenvalue of a Sturm - Liouville problem with Dirichlet conditions for some values of parameters in the integral condition was found and attainability of those estimates was proved.

On a method for solving the inverse Sturm–Liouville problem

2019 ◽  
Vol 27 (3) ◽  
pp. 401-407 ◽  
Author(s):  
Vladislav V. Kravchenko
Keyword(s):  
Algebraic System ◽  
Finite Interval ◽  
Series Representation ◽  
Liouville Problem ◽  
Simple Algorithm ◽  
System Of Equations ◽  
Linear Algebraic System ◽  
Legendre Series ◽  
Sturm Liouville

Abstract A method for solving the inverse Sturm–Liouville problem on a finite interval is proposed. It is based on a Fourier–Legendre series representation of the integral transmutation kernel. Substitution of the representation into the Gel’fand–Levitan equation leads to a linear algebraic system of equations and consequently to a simple algorithm for recovering the potential. Numerical illustrations are presented.

Homogeneous robin boundary conditions and discrete spectrum of fractional eigenvalue problem

2019 ◽  
Vol 22 (1) ◽  
pp. 78-94 ◽  
Author(s):  
Malgorzata Klimek
Keyword(s):  
Eigenvalue Problem ◽  
Boundary Conditions ◽  
Eigenvalue Problems ◽  
Liouville Problem ◽  
Continuous Functions ◽  
Real Spectrum ◽  
Robin Boundary Conditions ◽  
Schmidt Operator ◽  
Sturm Liouville ◽  
Robin Boundary

Abstract We discuss a fractional eigenvalue problem with the fractional Sturm-Liouville operator mixing the left and right derivatives of order in the range (1/2, 1], subject to a variant of Robin boundary conditions. The considered differential fractional Sturm-Liouville problem (FSLP) is equivalent to an integral eigenvalue problem on the respective subspace of continuous functions. By applying the properties of the explicitly calculated integral Hilbert-Schmidt operator, we prove the existence of a purely atomic real spectrum for both eigenvalue problems. The orthogonal eigenfunctions’ systems coincide and constitute a basis in the corresponding weighted Hilbert space. An analogous result is obtained for the reflected fractional Sturm-Liouville problem.

Spectrum of Fractional and Fractional Prabhakar Sturm–Liouville Problems with Homogeneous Dirichlet Boundary Conditions

Symmetry ◽  
2021 ◽  
Vol 13 (12) ◽  
pp. 2265
Author(s):  
Malgorzata Klimek
Keyword(s):  
Boundary Conditions ◽  
Fractional Derivatives ◽  
Dirichlet Boundary ◽  
Eigenvalue Problems ◽  
Liouville Problem ◽  
Dirichlet Boundary Conditions ◽  
Sturm Liouville ◽  
Left And Right ◽  
Homogeneous Dirichlet Boundary Conditions ◽  
Liouville Operators

In this study, we consider regular eigenvalue problems formulated by using the left and right standard fractional derivatives and extend the notion of a fractional Sturm–Liouville problem to the regular Prabhakar eigenvalue problem, which includes the left and right Prabhakar derivatives. In both cases, we study the spectral properties of Sturm–Liouville operators on function space restricted by homogeneous Dirichlet boundary conditions. Fractional and fractional Prabhakar Sturm–Liouville problems are converted into the equivalent integral ones. Afterwards, the integral Sturm–Liouville operators are rewritten as Hilbert–Schmidt operators determined by kernels, which are continuous under the corresponding assumptions. In particular, the range of fractional order is here restricted to interval (1/2,1]. Applying the spectral Hilbert–Schmidt theorem, we prove that the spectrum of integral Sturm–Liouville operators is discrete and the system of eigenfunctions forms a basis in the corresponding Hilbert space. Then, equivalence results for integral and differential versions of respective eigenvalue problems lead to the main theorems on the discrete spectrum of differential fractional and fractional Prabhakar Sturm–Liouville operators.

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