Liouville Operator Research Articles

On <i>q</i> Sturm Liouville Operator with Periodic Boundary Conditions

In this study, we consider q-Sturm Liouville operator with periodic boundary conditions. An asymptotic expression of the solution is obtained. With the help of this asymptotic representation, an asymptotic solution of the characteristic equation is presented. An application of the Rouche theorem, asymptotic expressions of eigenvalues are obtained.

On proportional hybrid operators in the discrete setting

In this article, we introduce a new nonlocal operator defined as a linear combination of the discrete fractional Caputo operator and the fractional sum operator. A new dual operator is also introduced by replacing the discrete fractional Caputo operator with the discrete fractional Riemann–Liouville operator. It is shown that it corresponds to a natural discretization of a proportional hybrid operator defined by the Riemann–Liouville operator instead of Caputo hybrid operator. We then analyze the most important properties of these operators, such as their inverse operator and the ‐transform, among others. As an application, we solve difference equations equipped with these operators and obtain explicit solutions for them in terms of trivariate Mittag‐Leffler sequences.

Heat equation for Sturm–Liouville operator with singular propagation and potential

Abstract This article considers the initial boundary value problem for the heat equation with the time-dependent Sturm–Liouville operator with singular potentials. To obtain a solution by the method of separation of variables, the problem is reduced to the problem of eigenvalues of the Sturm–Liouville operator. Further on, the solution to the initial boundary value problem is constructed in the form of a Fourier series expansion. A heterogeneous case is also considered. Finally, we establish the well-posedness of the equation in the case when the potential and initial data are distributions, also for singular time-dependent coefficients.

Theoretical Results on Positive Solutions in Delta Riemann–Liouville Setting

This article primarily focuses on examining the existence and uniqueness analysis of boundary fractional difference equations in a class of Riemann–Liouville operators. To this end, we firstly recall the general solution of the homogeneous fractional operator problem. Then, the Green function to the corresponding fractional boundary value problems will be reconstructed, and homogeneous boundary conditions are used to find the unknown constants. Next, the existence of solutions will be studied depending on the fixed-point theorems on the constructed Green’s function. The uniqueness of the problem is also derived via Lipschitz constant conditions.

Solutions of Second-Order Nonlinear Implicit ψ-Conformable Fractional Integro-Differential Equations with Nonlocal Fractional Integral Boundary Conditions in Banach Algebra

In this paper, we introduce and thoroughly examine new generalized ψ-conformable fractional integral and derivative operators associated with the auxiliary function ψ(t). We rigorously analyze and confirm the essential properties of these operators, including their semigroup behavior, linearity, boundedness, and specific symmetry characteristics, particularly their invariance under time reversal. These operators not only encompass the well-established Riemann–Liouville and Hadamard operators but also extend their applicability. Our primary focus is on addressing complex fractional boundary value problems, specifically second-order nonlinear implicit ψ-conformable fractional integro-differential equations with nonlocal fractional integral boundary conditions within Banach algebra. We assess the effectiveness of these operators in solving such problems and investigate the existence, uniqueness, and Ulam–Hyers stability of their solutions. A numerical example is presented to demonstrate the theoretical advancements and practical implications of our approach. Through this work, we aim to contribute to the development of fractional calculus methodologies and their applications.

On the spaces introduced by N. K. Karapetyants and B. S. Rubin, and their connection with Lorentz–Zygmund spaces

We consider spaces introduced by N. K. Karapetyants and B. S. Rubin in 1982, to characterize, in particular, the image of the fractional integral Riemann–Liouville operator. These spaces lie near L∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${L}^{\\infty }$$\\end{document}. We show that they coincide with well-known Lorentz–Zygmund spaces. This allows us to reformulate one result from N. K. Karapetyants and B. S. Rubin dealing with Riemann–Liouville fractional integral operator J0+α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${J}_{0+}^{\\alpha }$$\\end{document} defined on Lp0,1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${L}^{p}\\left(\ ext{0,1}\\right)$$\\end{document} (1<p<∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$1<p<\\infty$$\\end{document}) in the borderline case α=1/p\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha =1/p$$\\end{document}. Using of the well-developed theory of Lorentz–Zygmund spaces leads to new results on the fractional integral Riemann–Liouville operator.

The resolvent of impulsive dynamic singular Sturm–Liouville operators

The resolvent of impulsive dynamic singular Sturm–Liouville operators

On the approximation to fractional calculus operators with multivariate Mittag-Leffler function in the kernel

Several numerical techniques have been developed to approximate Riemann–Liouville (R - L) and Caputo fractional calculus operators. Recently linear positive operators have been started to use to approximate fractional calculus operators such as R - L, Caputo, Prabhakar and operators containing bivariate Mittag-Leffler functions. In the present paper, we first define and investigate the fractional calculus properties of Caputo derivative operator containing the multivariate Mittag-Leffler function in the kernel. Then we introduce approximating operators by using the modified Kantorovich operators for the approximation to fractional integral and Caputo derivative operators with multivariate Mittag-Leffler function in the kernel. We study the convergence properties of the operators and compute the degree of approximation by means of modulus of continuity and Hölder continuous functions. The obtained results corresponds to a large family of fractional calculus operators including R- L, Caputo and Prabhakar models.

Bicomplex Caputo Derivative: A Comparative Study with Bicomplex Riemann–Liouville Operators and Applications

Bicomplex Caputo Derivative: A Comparative Study with Bicomplex Riemann–Liouville Operators and Applications

Distribution of the Spectrum of a Sturm–Liouville Operator Perturbed by a Delta Interaction

Distribution of the Spectrum of a Sturm–Liouville Operator Perturbed by a Delta Interaction

Spectral properties of Sturm–Liouville operators on infinite metric graphs

Spectral properties of Sturm–Liouville operators on infinite metric graphs

Lipschitz and Dini–Lipschitz functions for the Sturm–Liouville transform

In this work, we introduce Lipschitz and Dini–Lipschitz classes of functions associated with the Sturm–Liouville operator L := d 2 d x 2 + A ′ ( x ) A ( x ) d d x , where A is a nonnegative function satisfying certain conditions; and we prove two versions of Titchmarsh's theorems for the Sturm–Liouville transform F . The harmonic analysis associated with the operator L plays an important role in establishing the results of this paper.

Fractional calculus of modified special functions involving the generalized M-series in their kernels and illustrative examples

In this paper we apply the Riemann–Liouville, Erdelyi–Kober and Caputo fractional operators to the modified beta, modified Gauss hypergeometric and modified confluent hypergeometric functions in which the generalized M-series are included in their kernels. Furthermore, as examples, we obtain solutions of some fractional differential equations involving the above modified special functions.

Recovering of the Stiffness Coefficients of the Sturm–Liouville Operator on a Star Graph from a Finite Set of Its Eigenvalues

Recovering of the Stiffness Coefficients of the Sturm–Liouville Operator on a Star Graph from a Finite Set of Its Eigenvalues

Integration of the Korteweg-de Vries equation with time-dependent coefficients in the case of moving eigenvalues of the Sturm–Liouville operator

The inverse scattering method is used to integrate the Korteweg-de Vries equation with time-dependent coefficients. We derive the evolution of the scattering data of the Sturm–Liouville operator whose coefficient is a solution of the Korteweg-de Vries equation with time-dependent coefficients. An algorithm for constructing exact solutions of the Korteweg-de Vries equation with time-dependent coefficients is also proposed; we reduce it to the inverse problem of scattering theory for the Sturm–Liouville operator. Examples illustrating the stated algorithm are given.

On recovering Sturm–Liouville operators with two delays

Abstract We study the inverse spectral problems of recovering Sturm–Liouville differential operator with two constant delays a 1 {a_{1}} and a 2 {a_{2}} greater than one third of the interval. It has been proved that the operator can be recovered uniquely from four spectra under the condition 2 ⁢ a 1 + a 2 2 ≥ π {2a_{1}+\frac{a_{2}}{2}\geq\pi} , while it is not possible otherwise.

Inverse problems for discrete Sturm–Liouville operator having Bessel-type potential

In this paper, inverse problems are studied for Sturm–Liouville difference operators having p 2 − 1 / 4 m 2 type singularity. First, we define the Green's function and the corresponding Weyl function, some of their properties are investigated, and we give a method to find the potential function. Then we give the solution of the inverse nodal problem and present a constructive procedure for the solution of the inverse problem by using nodes.

A note on Kirchhoff type boundary value problem involving Riemann-Liouville fractional derivative

In this paper, we study some nonlinear Kirchhoff boundary value problems of fractional differential equations involving Riemann Liouville operator. Under appropriate assumptions on the functions in the given problem, we establish the existence of solutions using variational methods combined with the mountain pass theorem. Moreover, an illustrative example is presented to prove the validity of the main result.

Schrödinger equation for Sturm–Liouville operator with singular propagation and potential

In this paper, we consider an initial/boundary value problem for the Schrödinger equation with the Hamiltonian involving the fractional Sturm–Liouville operator with singular propagation and potential. To construct a solution, first considering the coefficients in a regular sense, the method of separation of variables is used, which leads the solution of the equation to the eigenvalue and eigenfunction problem of the Sturm–Liouville operator. Next, using the Fourier series expansion in eigenfunctions, a solution to the Schrödinger equation is constructed. Important estimates related to the Sobolev space are also obtained. In addition, the equation is studied in the case where the initial data, propagation, and potential are strongly singular. For this case, the concept of “very weak solutions” is used. The existence, uniqueness, negligibility, and consistency of very weak solution of the Schrödinger equation are established.

Donoghue 𝑚-functions for Singular Sturm–Liouville operators

Let A ˙ \dot {A} be a densely defined, closed, symmetric operator in the complex, separable Hilbert space H \mathcal {H} with equal deficiency indices and denote by N i = ker ⁡ ( ( A ˙ ) ∗ − i I H ) \mathcal {N}_i = \ker ((\dot {A})^* - i I_{\mathcal {H}}) , dim ⁡ ( N i ) = k ∈ N ∪ { ∞ } \dim (\mathcal {N}_i)=k\in \mathbb {N} \cup \{\infty \} , the associated deficiency subspace of A ˙ \dot {A} . If A A denotes a self-adjoint extension of A ˙ \dot {A} in H \mathcal {H} , the Donoghue m m -operator M A , N i D o ( ⋅ ) M_{A,\mathcal {N}_i}^{Do} (\,\cdot \,) in N i \mathcal {N}_i associated with the pair ( A , N i ) (A,\mathcal {N}_i) is given by M A , N i D o ( z ) = z I N i + ( z 2 + 1 ) P N i ( A − z I H ) − 1 P N i | N i M_{A,\mathcal {N}_i}^{Do}(z)=zI_{\mathcal {N}_i} + (z^2+1) P_{\mathcal {N}_i} (A - z I_{\mathcal {H}})^{-1} P_{\mathcal {N}_i} \vert _{\mathcal {N}_i} , z ∈ C ∖ R , z\in \mathbb {C}\setminus \mathbb {R}, with I N i I_{\mathcal {N}_i} the identity operator in N i \mathcal {N}_i , and P N i P_{\mathcal {N}_i} the orthogonal projection in H \mathcal {H} onto N i \mathcal {N}_i . Assuming the standard local integrability hypotheses on the coefficients p , q , r p, q,r , we study all self-adjoint realizations corresponding to the differential expression τ = 1 r ( x ) [ − d d x p ( x ) d d x + q ( x ) ] \tau =\frac {1}{r(x)}[-\frac {d}{dx}p(x)\frac {d}{dx} + q(x)] for a.e. x ∈ ( a , b ) ⊆ R x\in (a,b) \subseteq \mathbb {R} , in L 2 ( ( a , b ) ; r d x ) L^2((a,b); rdx) , and, as our principal aim in this paper, systematically construct the associated Donoghue m m -functions (respectively, ( 2 × 2 ) (2 \times 2) matrices) in all cases where τ \tau is in the limit circle case at least at one interval endpoint a a or b b .