Spectral parameter power series for fourth-order Sturm–Liouville problems

Khmelnytskaya, Kira V.; Kravchenko, Vladislav V.; Baldenebro-Obeso, Jesús A.

doi:10.1016/j.amc.2012.09.055

Cited by 23 publications

(23 citation statements)

References 22 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Indeed, Equation can be written as

{(p u^{'})}^{'} + (q - λ^{*} r) u = (λ - λ^{*}) ru,

then the same arguments that were used to prove Proposition 5 and Theorem 7 can be applied. The procedure for constructing solutions of Equation based on a particular solution f * corresponding to λ = λ * is known as the spectral shift and is of great practical importance especially in numerical applications .…”

Section: Spps Representation For Solutions Of the Sturm–liouville Equmentioning

confidence: 99%

Spectral parameter power series method for discontinuous coefficients

Blancarte

Campos

Khmelnytskaya

2014

Math. Meth. Appl. Sci.

Self Cite

View full text Add to dashboard Buy / Rent full text

Let (a,b) be a finite interval and 1/p, q, r∈L1[a,b]. We show that a general solution (in the weak sense) of the equation (pu′)′+qu = λru on (a,b) can be constructed in terms of power series of the spectral parameter λ. The series converge uniformly on [a,b] and the corresponding coefficients are constructed by means of a simple recursive procedure. We use this representation to solve different types of eigenvalue problems. Several numerical tests are discussed. Copyright © 2014 John Wiley & Sons, Ltd.

show abstract

“…Indeed, Equation can be written as

{(p u^{'})}^{'} + (q - λ^{*} r) u = (λ - λ^{*}) ru,

Section: Spps Representation For Solutions Of the Sturm–liouville Equmentioning

confidence: 99%

Spectral parameter power series method for discontinuous coefficients

Blancarte

Campos

Khmelnytskaya

2014

Math. Meth. Appl. Sci.

Self Cite

View full text Add to dashboard Buy / Rent full text

show abstract

“…13) we obtainR(x) = x Re 2R 1 (c 1 ) + c1 0 q(s)R(s)ds + ix Im 2R 1 (c 2 ) +c2 0 q(s)R(s)ds = x Re (o(x n ) + o(x n+1 )) + i Im (o(x n ) + o(x n+1 )) = o(x n+1 ).…”

mentioning

confidence: 98%

Analytic approximation of transmutation operators and related systems of functions

Kravchenko

Torba

2016

Bol. Soc. Mat. Mex.

Self Cite

View full text Add to dashboard Buy / Rent full text

In [27] a method for approximate solution of Sturm-Liouville equations and related spectral problems was presented based on the construction of the Delsarte transmutation operators. The problem of numerical approximation of solutions and eigendata was reduced to approximation of a primitive of the potential by a finite linear combination of certain specially constructed functions obtained from the generalized wave polynomials introduced in [15], [26]. The method allows one to compute both lower and higher eigendata with an extreme accuracy.Since the solution of the approximation problem is the main step in the application of the method, the properties of the system of functions involved are of primary interest. In [27] two basic properties were established: the completeness in appropriate functional spaces and the linear independence. In this paper we present a considerably more complete study of the systems of functions. We establish their relation with another linear differential second-order equation, find out certain operations (in a sense, generalized derivatives and antiderivatives) which allow us to generate the next such function from a previous one. We obtain the uniqueness of the coefficients of expansions in terms of such functions and a corresponding generalized Taylor theorem, as well as formulas for exact expansion coefficients involving the operations mentioned above. We also construct the invertible integral operators transforming powers of the independent variable into the functions under consideration and establish their commutation relations with differential operators. We present some error bounds for the solution of the approximation problem depending on the smoothness of the potential and show that these error bounds are close to optimal in order. Also, we provide a rigorous justification of the alternative formulation of the proposed method allowing one to make use of the known initial values of the solutions at the left endpoint of the spectral problem.

show abstract

“…In [33] it was mentioned that for equation (1.2) it is also possible to construct the SPPS representation of a general solution starting from a non-vanishing particular solution for some λ = λ 0 . Such procedure is called spectral shift and has already proven its usefulness for numerical applications [9,21,33].…”

Section: Spectral Shift Techniquementioning

confidence: 99%

“…As was shown in a number of recent publications the SPPS representation provides an efficient and accurate method for solving initial value, boundary value and spectral problems (see [7], [8], [16], [21], [23], [22], [24], [25], [31], [32], [33], [35], [47]). In this paper we demonstrate this fact in application to equation (1.1).…”

Section: Introductionmentioning

confidence: 99%

Spectral parameter power series for Sturm-Liouville equations with a potential polynomially dependent on the spectral parameter and Zakharov-Shabat systems

Kravchenko

Torba

Velasco-García

2015

Journal of Mathematical Physics

Self Cite

View full text Add to dashboard Buy / Rent full text

Abstract. A spectral parameter power series (SPPS) representation for solutions of Sturm-Liouville equations of the formis obtained. It allows one to write a general solution of the equation as a power series in terms of the spectral parameter λ. The coefficients of the series are given in terms of recursive integrals involving a particular solution of the equation (pu 0 ) +qu0 = 0. The convenient form of the solution of ( * ) provides an efficient numerical method for solving corresponding initial value, boundary value and spectral problems. A special case of the Sturm-Liouville equation ( * ) arises in relation with the Zakharov-Shabat system. We derive an SPPS representation for its general solution and consider other applications as the one-dimensional Dirac system and the equation describing a damped string. Several numerical examples illustrate the efficiency and the accuracy of the numerical method based on the SPPS representations which besides its natural advantages like the simplicity in implementation and accuracy is applicable to the problems admitting complex coefficients, spectral parameter dependent boundary conditions and complex spectrum.

show abstract

Spectral parameter power series for fourth-order Sturm–Liouville problems

Cited by 23 publications

References 22 publications

Spectral parameter power series method for discontinuous coefficients

Spectral parameter power series method for discontinuous coefficients

Analytic approximation of transmutation operators and related systems of functions

Spectral parameter power series for Sturm-Liouville equations with a potential polynomially dependent on the spectral parameter and Zakharov-Shabat systems

Contact Info

Product

Resources

About