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W. N. Everitt

Bio: W. N. Everitt is an academic researcher. The author has contributed to research in topics: Differential equation & Bessel function. The author has an hindex of 2, co-authored 2 publications receiving 12 citations.

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TL;DR: In this paper, the authors report on the properties of the fourth-order Bessel-type linear ordinary differential equation, on the generated self-adjoint differential operators in two associated Hilbert function spaces, and on the generalisation of the classical Hankel integral transform.
Abstract: This survey paper reports on the properties of the fourth-order Bessel-type linear ordinary differential equation, on the generated self-adjoint differential operators in two associated Hilbert function spaces, and on the generalisation of the classical Hankel integral transform. These results are based upon the properties of the classical Bessel and Laguerre secondorder differential equations, and on the fourth-order Laguerre-type differential equation. From these differential equations and their solutions, limit processes yield the fourth-order Bessel-type functions and the associated differential equation.

10 citations

Proceedings ArticleDOI
01 May 2007
TL;DR: In this paper, the authors report on the properties of the fourth-order Bessel-type linear ordinary differential equation, on the generated self-adjoint differential operators in two associated Hilbert function spaces, and on the generalisation of the classical Hankel integral transform.
Abstract: This survey paper reports on the properties of the fourth-order Bessel-type linear ordinary differential equation, on the generated self-adjoint differential operators in two associated Hilbert function spaces, and on the generalisation of the classical Hankel integral transform. These results are based upon the properties of the classical Bessel and Laguerre secondorder differential equations, and on the fourth-order Laguerre-type differential equation. From these differential equations and their solutions, limit processes yield the fourth-order Bessel-type functions and the associated differential equation.

3 citations


Cited by
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TL;DR: In this paper, a new proof of the existence of this solution base is given, on using the advanced theory of special functions in the complex plane, which leads to the development of analytical properties of these solutions, in particular the series expansions of all solutions at the regular singularity at the origin.

12 citations

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TL;DR: In this article, the authors report on the recent advancements in the search for explicit representation, in classical special functions, of the solutions of the fourth-order ordinary differential equations named Bessel-type, Jacobi-type and Legendre-type.
Abstract: This note reports on the recent advancements in the search for explicit representation, in classical special functions, of the solutions of the fourth-order ordinary differential equations named Bessel-type, Jacobi-type, Laguerre-type, Legendre-type.

10 citations

Journal ArticleDOI
TL;DR: In this article, the authors proved that S M has a complete eigenpacket, which is reflected in a distributional orthogonality on which the expansion theorems are based.
Abstract: In connection with the fourth-order Bessel-type differential equation two expansion theorems are established, the convergence being pointwise or in an L 2-setting. If the positive parameter M tends to zero, these two expansion theorems reduce to the classical Hankel transform of order zero. In a previous article, the authors have proved that in one of the introduced Lebesgue–Stieltjes Hilbert function spaces, the differential expression x −1 L M gives rise to exactly one self-adjoint operator S M . In this article, it is proved, together with the corresponding expansion theorems, that S M has a complete eigenpacket. The orthogonality property of this eigenpacket is reflected in a distributional orthogonality on which the expansion theorems are based. †This paper is dedicated to the achievements and memory of Professor Gunter Hellwig.

8 citations

Journal ArticleDOI
TL;DR: In this article, the authors considered analytical and numerical solutions to the Dirichlet boundary-value problem for the biharmonic partial differential equation on a disc of finite radius in the plane.
Abstract: In this paper, we consider analytical and numerical solutions to the Dirichlet boundary-value problem for the biharmonic partial differential equation on a disc of finite radius in the plane. The physical interpretation of these solutions is that of the harmonic oscillations of a thin, clamped plate. For the linear, fourth-order, biharmonic partial differential equation in the plane, it is well known that the solution method of separation in polar coordinates is not possible, in general. However, in this paper, for circular domains in the plane, it is shown that a method, here called quasi-separation of variables, does lead to solutions of the partial differential equation. These solutions are products of solutions of two ordinary linear differential equations: a fourth-order radial equation and a second-order angular differential equation. To be expected, without complete separation of the polar variables, there is some restriction on the range of these solutions in comparison with the corresponding separated solutions of the second-order harmonic differential equation in the plane. Notwithstanding these restrictions, the quasi-separation method leads to solutions of the Dirichlet boundary-value problem on a disc with centre at the origin, with boundary conditions determined by the solution and its inward drawn normal taking the value 0 on the edge of the disc. One significant feature for these biharmonic boundary-value problems, in general, follows from the form of the biharmonic differential expression when represented in polar coordinates. In this form, the differential expression has a singularity at the origin, in the radial variable. This singularity translates to a singularity at the origin of the fourth-order radial separated equation; this singularity necessitates the application of a third boundary condition in order to determine a self-adjoint solution to the Dirichlet boundary-value problem. The penultimate section of the paper reports on numerical solutions to the Dirichlet boundary-value problem; these results are also presented graphically. Two specific cases are studied in detail and numerical values of the eigenvalues are compared with the results obtained in earlier studies.

5 citations

Journal ArticleDOI
09 Jun 2013
TL;DR: Asymptotic integration has turned out to be a powerful method to determine the deficiency indices and spectra of higher order differential operators as discussed by the authors, and the general method is by now well established and illustrate typical results and properties via examples.
Abstract: Asymptotic integration has turned out to be a powerful method to determine the deficiency indices and spectra of higher order differential operators. Since the general method is by now well established we shall only outline this method and illustrate typical results and properties via examples. In addition to the calculation of deficiency indices, the location and multiplicity of the absolutely continuous spectrum will be found as well as showing the absence of singular continuous spectrum. Finite singular points will also be considered. For unbounded coefficients new results arise from competing terms of the operators. Mathematics Subject Classification (2010). Primary 34A30, 34B20, 34L05; Secondary 34B05.

5 citations