D. J. Smith

Bio: D. J. Smith is an academic researcher from University of Birmingham. The author has contributed to research in topics: Separable partial differential equation & Exponential integrator. The author has an hindex of 1, co-authored 1 publications receiving 10 citations.

The Fourth-Order Type Linear Ordinary Differential Equations

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.

Finding all bessel type solutions for linear differential equations with rational function coefficients

Abstract: A linear differential equation with rational function coefficients has a Bessel type solution when it is solvable in terms of Bv(f), Bv+1(f). For second order equations, with rational function coefficients, f must be a rational function or the square root of a rational function. An algorithm was given by Debeerst, van Hoeij, and Koepf, that can compute Bessel type solutions if and only if f is a rational function. In this paper we extend this work to the square root case, resulting in a complete algorithm to find all Bessel type solutions.

Solving differential equations in terms of bessel functions

Abstract: For differential operators of order 2, this paper presents a new method that combines generalized exponents to find those solutions that can be represented in terms of Bessel functions.

Quasi-separation of the biharmonic partial differential equation

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.

Solution bases for the fourth-order Bessel-type and Laguerre-type differential equations

Abstract: The fourth-order Bessel-type and Laguerre-type linear ordinary differential equations are prototypes of structured linear differential equations of higher even-order, which naturally extend the second-order Bessel and Laguerre equations defined on the positive half-line of the real field R. Whilst the Laguerre-type equation arose from a search for all orthogonal polynomial generated by a linear differential equation, the present authors derived the Bessel-type equations and functions in 1994 by applying a generalized limit process to the Laguerre-type case. Due to their close relationship, the Laguerre- and Bessel-type functions of the same order share many important properties as, for example, orthogonality or a generalized hypergeometric representation. In this article, we first survey the most recent achievements in our study of the fourth-order Bessel equation which led to explicit representations of four linear independent solutions. Our main purpose then is to show how these techniques carry over to...