Sturm–Liouville problems with boundary conditions depending quadratically on the eigenparameter

Math Methods in App Sciences

Porter

2009

111

260

We consider a recently discovered representation for the general solution of the Sturm-Liouville equation as a spectral parameter power series (SPPS). The coefficients of the power series are given in terms of a particular solution of the Sturm-Liouville equation with the zero spectral parameter. We show that, among other possible applications, this provides a new and efficient numerical method for solving initial value and boundary value problems. Moreover, due to its convenient form the representation lends itself to numerical solution of spectral Sturm-Liouville problems, effectively by calculation of the roots of a polynomial. We discuss examples of the numerical implementation of the SPPS method and show it to be equally applicable to a wide class of singular Sturm-Liouville problems as well as to problems with spectral parameter dependent boundary conditions. IntroductionIn the recent work [14] a representation was obtained for solutions of the equation (puin terms of a known non-trivial solution of the equationwhere p, q, u, u 0 are complex-valued functions of the real variable x satisfying certain smoothness conditions, and λ is an arbitrary complex constant. The general solution of (1) has the form of a power series with respect to the spectral parameter λ. The fact that under certain conditions u is a complex analytic 1

Section: Singular Problemsunclassified

Spectral parameter power series for Sturm–Liouville problems

Math Methods in App Sciences

Porter

2009

111

260

“…In this case together with equation and boundary condition , the eigenfunction must satisfy a second boundary condition of the form

β_{1} u (b) MathClass-bin- β_{2} u^{MathClass-rel'} (b) MathClass-rel= ϕ (λ) ((), β_{1}^{MathClass-rel'} u (b) MathClass-bin- β_{2}^{MathClass-rel'} u^{MathClass-rel'} (b)) MathClass-punc,

where ϕ is a complex‐valued function of the variable λ and β 1 , β 2 ,

β_{1}^{MathClass-rel'}

β_{2}^{MathClass-rel'}

are complex numbers. For some special forms of the function ϕ such as ϕ ( λ ) = λ or ϕ ( λ ) = λ 2 + c 1 λ + c 2 , results were obtained , concerning the regularity of the problem , , and ; we will not dwell upon the details. In general, the presence of the spectral parameter in boundary conditions introduces additional considerable difficulties both in theoretical and numerical analysis of the problems.…”

Section: Solution Of Sturm–liouville Problemsmentioning

confidence: 99%

“…/ D or '. / D 2 C c 1 C c 2 , results were obtained [35], [38] concerning the regularity of the problem (21), (22), and (26); we will not dwell upon the details. In general, the presence of the spectral parameter in boundary conditions introduces additional considerable difficulties both in theoretical and numerical analysis of the problems.…”

Section: Solution Of Sturm-liouville Problemsmentioning

confidence: 99%

Eigenvalue problems, spectral parameter power series, and modern applications

Khmelnytskaya

Math Methods in App Sciences

Rosu

2014

In the present review, we deal with the recently introduced method of spectral parameter power series (SPPS) and show how its application leads to an explicit form of the characteristic equation for different eigenvalue problems involving Sturm–Liouville equations with variable coefficients. We consider Sturm–Liouville problems on finite intervals; problems with periodic potentials involving the construction of Hill's discriminant and Floquet–Bloch solutions; quantum‐mechanical spectral and transmission problems as well as the eigenvalue problems for the Zakharov–Shabat system. In all these cases, we obtain a characteristic equation of the problem, which in fact reduces to finding zeros of an analytic function given by its Taylor series. We illustrate the application of the method with several numerical examples, which show that at present, the SPPS method is the easiest in the implementation, the most accurate, and efficient. We emphasize that the SPPS method is not a purely numerical technique. It gives an analytical representation both for the solution and for the characteristic equation of the problem. This representation can be approximated by different numerical techniques and leads to a powerful numerical method, but most important, it offers a different insight into the spectral and transmission problems. Copyright © 2014 John Wiley & Sons, Ltd.

“…where α is an arbitrary complex number, φ is a complex-valued function of the variable λ and β 1 , β 2 , β ′ 1 , β ′ 2 are complex numbers. For some special forms of the function φ such as φ(λ) = λ or φ(λ) = λ 2 + c 1 λ + c 2 , results were obtained [19], [64] concerning the regularity of the problem; we will not dwell upon the details. Notice that the SPPS approach is applicable as well to a more general Sturm-Liouville equation (pu ′ ) ′ + qu = λru.…”

Section: Dispersion Relations For Spectral Problems and Approximate Smentioning

confidence: 99%

Transmutations and Spectral Parameter Power Series in Eigenvalue Problems

Operator Theory, Pseudo-Differential Equations, and Mathematical Physics

Torba

2012

We give an overview of recent developments in Sturm-Liouville theory concerning operators of transmutation (transformation) and spectral parameter power series (SPPS). The possibility to write down the dispersion (characteristic) equations corresponding to a variety of spectral problems related to Sturm-Liouville equations in an analytic form is an attractive feature of the SPPS method. It is based on a computation of certain systems of recursive integrals. Considered as families of functions these systems are complete in the L2-space and result to be the images of the nonnegative integer powers of the independent variable under the action of a corresponding transmutation operator. This recently revealed property of the Delsarte transmutations opens the way to apply the transmutation operator even when its integral kernel is unknown and gives the possibility to obtain further interesting properties concerning the Darboux transformed Schrödinger operators.We introduce the systems of recursive integrals and the SPPS approach, explain some of its applications to spectral problems with numerical illustrations, give the definition and basic properties of transmutation operators, introduce a parametrized family of transmutation operators, study their mapping properties and construct the transmutation operators for Darboux transformed Schrödinger operators. (2010). Primary 34B24, 34L16, 65L15, 81Q05, 81Q60; Secondary 34L25, 34L40. Mathematics Subject Classification∞ k=0 there exists a transmutation operator T such that T[x k ] = ϕ k , i.e., the functions ϕ k are the images of the usual powers of the independent variable. Moreover, it was shown how this operator can be constructed and how it is related to the "canonical" transformation operator considered, e.g., in [51, Chapter 1]. This result together with the practical formulas for calculating the functions ϕ k makes it possible to apply the transmutation technique even when the integral kernel of the operator is unknown. Indeed, now it is easy to apply the transmutation operator to any function approximated by a polynomial.Deeper understanding of the mapping properties of the transmutation operators led us in [46] to the explicit construction of the transmutation operator for a Darboux transformed Schrödinger operator by a known transmutation operator for the original Schrödinger operator as well as to several