On the Canonical Solution of the Sturm–Liouville Problem with Singularity and Turning Point of Even Order

2011 ◽  
Vol 54 (3) ◽  
pp. 506-518 ◽  
Author(s):  
A. Neamaty ◽  
S. Mosazadeh
Keyword(s):  
Liouville Equation ◽  
Turning Point ◽  
Infinite Product ◽  
Liouville Problem ◽  
Asymptotic Estimates ◽  
Product Representation ◽  
Representation Of Solutions ◽  
Even Order ◽  
Sturm Liouville ◽  
Infinite Product Representation

AbstractIn this paper, we are going to investigate the canonical property of solutions of systems of differential equations having a singularity and turning point of even order. First, by a replacement, we transform the system to the Sturm–Liouville equation with turning point. Using of the asymptotic estimates provided by Eberhard, Freiling, and Schneider for a special fundamental system of solutions of the Sturm–Liouville equation, we study the infinite product representation of solutions of the systems. Then we transform the Sturm–Liouville equation with turning point to the equation with singularity, then we study the asymptotic behavior of its solutions. Such representations are relevant to the inverse spectral problem.

scholarly journals DUAL EQUATION AND INVERSE PROBLEM FOR AN INDEFINITE STURM–LIOUVILLE PROBLEM WITH M TURNING POINTS OF EVEN ORDER

2012 ◽  
Vol 17 (5) ◽  
pp. 618-629
Author(s):  
Hamidreza Marasi ◽  
Aliasghar Jodayree Akbarfam
Keyword(s):  
Inverse Problem ◽  
Infinite Product ◽  
Turning Points ◽  
Liouville Problem ◽  
Product Representation ◽  
Dual Equation ◽  
Number Of Zeros ◽  
Even Order ◽  
Sturm Liouville ◽  
Infinite Product Representation

In this paper the differential equation y″ + (ρ 2 φ 2 (x) –q(x))y = 0 is considered on a finite interval I, say I = [0, 1], where q is a positive sufficiently smooth function and ρ 2 is a real parameter. Also, [0, 1] contains a finite number of zeros of φ 2 , the so called turning points, 0 < x 1 < x 2 < … < x m < 1. First we obtain the infinite product representation of the solution. The product representation, satisfies in the original equation. As a result the associated dual equation is derived and then we proceed with the solution of the inverse problem.

scholarly journals A New Type of Sturm-Liouville Equation in the Non-Newtonian Calculus

2021 ◽  
Vol 2021 ◽  
pp. 1-8
Author(s):  
Sertac Goktas
Keyword(s):  
Liouville Equation ◽  
Spectral Properties ◽  
Liouville Problem ◽  
Mathematical Physics ◽  
Asymptotic Estimates ◽  
One Dimensional ◽  
Multiplicative Calculus ◽  
New Type ◽  
Sturm Liouville ◽  
The One

In mathematical physics (such as the one-dimensional time-independent Schrödinger equation), Sturm-Liouville problems occur very frequently. We construct, with a different perspective, a Sturm-Liouville problem in multiplicative calculus by some algebraic structures. Then, some asymptotic estimates for eigenfunctions of the multiplicative Sturm-Liouville problem are obtained by some techniques. Finally, some basic spectral properties of this multiplicative problem are examined in detail.

scholarly journals On partial fractional Sturm–Liouville equation and inclusion

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Zohreh Zeinalabedini Charandabi ◽  
Hakimeh Mohammadi ◽  
Shahram Rezapour ◽  
Hashem Parvaneh Masiha
Keyword(s):  
Differential Equation ◽  
Liouville Equation ◽  
Existence Of Solutions ◽  
Liouville Problem ◽  
Existence Of A Solution ◽  
Contractive Mappings ◽  
Sturm Liouville

AbstractThe Sturm–Liouville differential equation is one of interesting problems which has been studied by researchers during recent decades. We study the existence of a solution for partial fractional Sturm–Liouville equation by using the α-ψ-contractive mappings. Also, we give an illustrative example. By using the α-ψ-multifunctions, we prove the existence of solutions for inclusion version of the partial fractional Sturm–Liouville problem. Finally by providing another example and some figures, we try to illustrate the related inclusion result.

scholarly journals Solutions of Direct and Inverse Even-Order Sturm-Liouville Problems Using Magnus Expansion

Mathematics ◽  
2019 ◽  
Vol 7 (6) ◽  
pp. 544 ◽  
Author(s):  
Upeksha Perera ◽  
Christine Böckmann
Keyword(s):  
Boundary Conditions ◽  
Liouville Problem ◽  
Group Method ◽  
Universal Method ◽  
Magnus Expansion ◽  
Arbitrary Boundary ◽  
Lie Group Method ◽  
Present Technique ◽  
Even Order ◽  
Sturm Liouville

In this paper Lie group method in combination with Magnus expansion is utilized to develop a universal method applicable to solving a Sturm–Liouville problem (SLP) of any order with arbitrary boundary conditions. It is shown that the method has ability to solve direct regular (and some singular) SLPs of even orders (tested for up to eight), with a mix of (including non-separable and finite singular endpoints) boundary conditions, accurately and efficiently. The present technique is successfully applied to overcome the difficulties in finding suitable sets of eigenvalues so that the inverse SLP problem can be effectively solved. The inverse SLP algorithm proposed by Barcilon (1974) is utilized in combination with the Magnus method so that a direct SLP of any (even) order and an inverse SLP of order two can be solved effectively.

scholarly journals Sturm–Liouville problems with eigenparameter dependent boundary conditions

1994 ◽  
Vol 37 (1) ◽  
pp. 57-72 ◽  
Author(s):  
P. A. Binding ◽  
P. J. Browne ◽  
K. Seddighi
Keyword(s):  
Boundary Conditions ◽  
Liouville Problem ◽  
Asymptotic Estimates ◽  
Comparison Results ◽  
Sturm Liouville ◽  
Image Position ◽  
Sturm Theory

Sturm theory is extended to the equationfor 1/p, q, r∈L1 [0, 1] with p, r > 0, subject to boundary conditionsandOscillation and comparison results are given, and asymptotic estimates are developed. Interlacing of eigenvalues with those of a standard Sturm–Liouville problem where the boundary conditions are ajy(j) = cj(py′)(j), j=0, 1, forms a key tool.

scholarly journals An application of comparison criteria to fractional spectral problem having Coloumb potential

2018 ◽  
Vol 22 (Suppl. 1) ◽  
pp. 79-85 ◽  
Author(s):  
Erdal Bas ◽  
Turk Metin
Keyword(s):  
Boundary Condition ◽  
Spectral Problem ◽  
Spectral Theory ◽  
Liouville Equation ◽  
Coulomb Potential ◽  
Liouville Problem ◽  
Comparison Theorems ◽  
Liouville Theory ◽  
Sturm Liouville ◽  
Comparison Criteria

In this study, the zeros of eigen functions of spectral theory are considered in fractional Sturm-Liouville problem. The 1st and 2nd comparison theorems for fractional Sturm-Liouville equation with boundary condition and their proofs are given. In this way, our new approximation will contribute to construct fractional Sturm-Liouville theory. Also, its an application is given in case of Coulomb potential and the results are presented by a symbolic graph.

Consequences of the connection formulae for Sturm-Liouville spectral functions

2002 ◽  
Vol 132 (2) ◽  
pp. 387-393 ◽  
Author(s):  
S. M. Riehl
Keyword(s):  
Explicit Formula ◽  
Liouville Equation ◽  
Liouville Problem ◽  
Initial Condition ◽  
Spectral Functions ◽  
Sturm Liouville ◽  
Image Position ◽  
Connection Formulae ◽  
Special Case

For a special case of the Sturm-Liouville equation, −(py′)′ + qy = λwy on [0, ∞) with the initial condition y(0) cos α + p(0)y′(0) sin α = 0, α ∈ [0, π), it is shown that, given the spectral derivative for two values of α ∈ [0, π) at a fixed μ = Re{λ} ≥ Λ0, it is possible to uniquely determine . An explicit formula is derived to accomplish this. Further, in a more general case of the Sturm-Liouville problem for μ with both finite and positive, then the following inequality holds

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