2014
DOI: 10.1002/mma.3282
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Abstract: 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.

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Cited by 7 publications
(6 citation statements)
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“…Note that q does not need to be continuous on [0, b] for the equality (3.13) to hold, the condition (1.2) is sufficient. Indeed, the functions ϕ k are defined by the same formulas (3.6) (their validity under the condition (1.2) can be verified similarly to [4]), the SPPS representation (3.11) and the integral representation (4.1) hold, and the proof from Section 3 can be easily repeated. Now let additionally q satisfy (4.2).…”
Section: A Fourier-legendre Representation Of the Kernel R(x T)mentioning
confidence: 89%
“…Note that q does not need to be continuous on [0, b] for the equality (3.13) to hold, the condition (1.2) is sufficient. Indeed, the functions ϕ k are defined by the same formulas (3.6) (their validity under the condition (1.2) can be verified similarly to [4]), the SPPS representation (3.11) and the integral representation (4.1) hold, and the proof from Section 3 can be easily repeated. Now let additionally q satisfy (4.2).…”
Section: A Fourier-legendre Representation Of the Kernel R(x T)mentioning
confidence: 89%
“…Clearlyc 0 = 1. Then by (6) we have recursivelyc (4,5) X (5,0) X (5,2) X (5,4) Figure 1: Construction of X (  ) .…”
Section: Formal Powersmentioning
confidence: 99%
“…Since its appearance in 2008, consequences of this SPPS (spectral parameter power series) representation have been investigated in many directions. These include completeness properties of the "formal powers" used to define the coefficients of the power series [21,22]; relationship to transmutation operators, Darboux and other transformations, and Goursat problems [17,25,27,28]; extension to other number systems (quaternions, etc) [8,9,27] and equations of higher order [15]; relaxation of regularity conditions on the coefficients of the differential equation [5,11]. Further, there have appeared numerous applications to problems in physics and engineering [10,16,18,19,29] as well as in complex analysis [6,24].…”
Section: Introductionmentioning
confidence: 99%
“…The proof is by induction, similarly to the proof of Proposition 5 from [7]. We left the details to the reader.…”
Section: Discontinuous Coefficientsmentioning
confidence: 95%