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.

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 A solution to the inverse problem for the Sturm-Liouville-type equation with a delay

Filomat ◽  
2013 ◽  
Vol 27 (7) ◽  
pp. 1237-1245 ◽  
Author(s):  
Milenko Pikula ◽  
Vladimir Vladicic ◽  
Olivera Markovic
Keyword(s):  
Inverse Problem ◽  
Boundary Conditions ◽  
Fourier Coefficients ◽  
Liouville Problem ◽  
Type Equation ◽  
Liouville Type ◽  
Separated Boundary Conditions ◽  
Sturm Liouville ◽  
Delay Factor ◽  
Spectral Assignment

The paper is devoted to study of the inverse problem of the boundary spectral assignment of the Sturm-Liouville with a delay. -y'(x) + q(x)y(? ? x) = ?y(x), q ? AS[0, ?], ? ? (0,1] (1) with separated boundary conditions: y(0) = y(?) = 0 (2) y(0) = y'(?) = 0 (3) It is argued that if the sequence of eigenvalues is given ?n(1) and ?n(2) tasks (1-2) and (1-3) respectively, then the delay factor ? ? (0,1) and the potential q ? AS[0, ?] are unambiguous. The potential q is composed by means of trigonometric Fourier coefficients. The method can be easily transferred to the case of ? = 1 i.e. to the classical Sturm-Liouville problem.

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.

A special case of the Sturm–Liouville inverse problem by three spectra: uniqueness results

2006 ◽  
Vol 136 (1) ◽  
pp. 181-187 ◽  
Author(s):  
Vyacheslav Pivovarchik
Keyword(s):  
Inverse Problem ◽  
Liouville Problem ◽  
Uniqueness Results ◽  
Special Cases ◽  
Sturm Liouville ◽  
Special Case

A three spectral inverse Sturm–Liouville problem is considered arising in the theory of vibrating strings. It is shown that in some special cases the solution of this problem is unique.

Unbounded solution components for nonlinear Hill's equations

1995 ◽  
Vol 125 (6) ◽  
pp. 1131-1167 ◽  
Author(s):  
Thomas Mrziglod
Keyword(s):  
Original Problem ◽  
Liouville Problem ◽  
Unbounded Solution ◽  
Complicated Structure ◽  
Number Of Zeros ◽  
Extended Problem ◽  
Sturm Liouville ◽  
Hill’S Equations ◽  
New Phenomena ◽  
Parameter Dependent

A class of nonlinear Hill's equations on ℝ is considered, where the nonlinearity is concentrated on a compact interval [−N, N]. For values of the parameter λ not in the spectrum of the linearised equation (which is purely continuous) an equivalent nonlinear Sturm–Liouville problem on [−N, N] with parameter-dependent boundary conditions at x = ± N is given. Extending this problem to all real values of the parameter in a suitable way makes it possible to prove the existence of unbounded solution components for both the extended Sturm–Liouville problem and the original problem. The complicated structure of the extended problem results in new phenomena. For example, the number of zeros of different functions in the same solution component may be different.

Sign in / Sign up

Export Citation Format

Share Document