Expansion in Series of Bessel Functions and Transmutations for Perturbed Bessel Operators

Chebli, Houcine; Fitouchi, A.; Hamza, Mongi

doi:10.1006/jmaa.1994.1058

Cited by 25 publications

(21 citation statements)

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“…In an interesting research reported in [11] and [16] there appears a representation of solutions of Sturm-Liouville equations in the form of Neumann series of Bessel functions different to the representation obtained in the present work. The representation from [11] and [16] does not possess the uniformity with respect to ω to the difference from our representation, and the convergence of the series which is guaranteed on a certain interval of x for holomorphic q only is achieved due to the exponential decay of j n (z) when n → ∞. Apart from that previous work, to our best knowledge, the Neumann series of Bessel functions have not been used to represent solutions of a general linear differential equation.…”

Section: Introductioncontrasting

confidence: 75%

Representation of solutions to the one-dimensional Schrödinger equation in terms of Neumann series of Bessel functions

Kravchenko

Navarro

Torba

2017

Applied Mathematics and Computation

151

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A new representation of solutions to the equation −y ′′ + q(x)y = ω 2 y is obtained. For every x the solution is represented as a Neumann series of Bessel functions depending on the spectral parameter ω. Due to the fact that the representation is obtained using the corresponding transmutation operator, a partial sum of the series approximates the solution uniformly with respect to ω which makes it especially convenient for the approximate solution of spectral problems. The numerical method based on the proposed approach allows one to compute large sets of eigendata with a nondeteriorating accuracy.

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Section: Introductioncontrasting

confidence: 75%

Representation of solutions to the one-dimensional Schrödinger equation in terms of Neumann series of Bessel functions

Kravchenko

Navarro

Torba

2017

Applied Mathematics and Computation

151

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“…This follows the framework of J. L. Lions [16] and generalizes some results known in the scalar case (see [6] and [22]). We use two techniques to study these operators; the first one is the Fourier transform associated with the spectral decomposition of (∆ α +q).…”

Section: Introductionsupporting

confidence: 54%

Partial differential equations with matricial coefficients and generalized translation operators

Mahmoud

2000

Trans. Amer. Math. Soc.

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Abstract. Let ∆α be the Bessel operator with matricial coefficients defined on (0, ∞) bywhere α is a diagonal matrix and let q be an n × n matrix-valued function.In this work, we prove that there exists an isomorphism X on the space of even C ∞ , C n -valued functions which transmutes ∆α and (∆α + q). This allows us to define generalized translation operators and to develop harmonic analysis associated with (∆α + q). By use of the Riemann method, we provide an integral representation and we deduce more precise information on these operators.

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“…The SPPS representation allows us to obtain a result on mapping properties of the transmutation operator corresponding to the operator L, which was studied, e.g., in [11], [39] and [40]. We show in Section 6 how the transmutation operator acts on certain powers of the independent variable.…”

Section: Introductionmentioning

confidence: 99%

Spectral parameter power series for perturbed Bessel equations

Castillo-Pérez

Kravchenko

Torba

2013

Applied Mathematics and Computation

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A spectral parameter power series (SPPS) representation for regular solutions of singular Bessel type Sturm-Liouville equations with complex coefficients is obtained as well as an SPPS representation for the (entire) characteristic function of the corresponding spectral problem on a finite interval. It is proved that the set of zeros of the characteristic function coincides with the set of all eigenvalues of the Sturm-Liouville problem. Based on the SPPS representation a new mapping property of the transmutation operator for the considered perturbed Bessel operator is obtained, and a new numerical method for solving corresponding spectral problems is developed. The range of applicability of the method includes complex coefficients, complex spectrum and equations in which the spectral parameter stands at a first order linear differential operator. On a set of known test problems we show that the developed numerical method based on the SPPS representation is highly competitive in comparison to the best available solvers such as SLEIGN2, MATSLISE and some other codes and give an example of an exactly solvable test problem admitting complex eigenvalues to which the mentioned solvers are not applicable meanwhile the SPPS method delivers excellent numerical results.Comment: 25 pages, 5 figures, 7 table

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Expansion in Series of Bessel Functions and Transmutations for Perturbed Bessel Operators

Cited by 25 publications

References 0 publications

Representation of solutions to the one-dimensional Schrödinger equation in terms of Neumann series of Bessel functions

Representation of solutions to the one-dimensional Schrödinger equation in terms of Neumann series of Bessel functions

Partial differential equations with matricial coefficients and generalized translation operators

Spectral parameter power series for perturbed Bessel equations

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