Lawrence E. Levine

Polynomial solutions of certain classes of ordinary differential equations

Several classes of ordinary differential equations which have polynomial solutions are studied. In particular, generalizations of the Hermite, Laguerre, Legendre, and Chebyshev equations are given for which such solutions exist. These solutions turn out to be generalizations of well‐known polynomials and enjoy similar properties.

Polynomial solutions of the classical equations of Hermite, Legendre, and Chebyshev

The classical differential equations of Hermite, Legendre, and Chebyshev are well known for their polynomial solutions. These polynomials occur in the solutions to numerous problems in applied mathematics, physics, and engineering. However, since these equations are of second order, they also have second linearly independent solutions that are not polynomials. These solutions usually cannot be expressed in terms of elementary functions alone. In this paper, the classical differential equations of Hermite, Legendre, and Chebyshev are studied when they have a forcing term x M on the right-hand side. It will be shown that for each equation, choosing a certain initial condition is a necessary and sufficient condition for ensuring a polynomial solution. Once this initial condition is determined, the exact form of the polynomial solution is presented.

Families of separable partial differential equations

In this paper the authors investigate classes of partial differential equations (mostly in two independent variables, of order at most 4) which can be solved by the well‐known technique of separation of variables. In particular, several classical ordinary differential equations, some of which have been generalized (by means of certain parameters), are used to assist in the construction of such partial differential equations. The process gives rise to (potentially) many families of equations which appear to be difficult to solve. The article not only illustrates the technique used in the determination of solutions to such equations, but also contrasts the effect of the nature of solutions as certain parameters effect them. In addition, the paper indicates how other classes of solvable equations might be obtained; equations with more than two variables and of any order. Only analytical methods are discussed; however it would not be difficult to use computing machinery as the number of independent variables ...

Classroom note: Polynomial solutions of the Laguerre equation and other differential equations near a singular point

It can be shown that if a differential equation is analytic near a point, then it is always possible to select a forcing term along with initial conditions that will ensure the solution to the new non-homogeneous equation is a polynomial that is the finite, truncated portion of the (infinite) series solution of the original equation. It turns out that this result can be extended to expansions about a singular point. The conditions under which such a polynomial truncation can be accomplished about a singular point are presented in the Appendix. A brief algorithm is described that enables one to choose the appropriate forcing term and initial conditions. Following this, an example involving Laguerres equation is presented.

Generalized multitime expansions for equations with slowly varying coefficients

The successive terms in a uniformly valid multitime expansion of the solutions of constant coefficient differential equations containing a small parameter ϵ may be obtained without resorting to secularity conditions if the time scales ti=ϵit(i=0,1,…) are used. Similar results have been achieved in some cases for equations with variable coefficients by using nonlinear time scales generated from the equations themselves. This paper extends the latter approach to the general second order ordinary differential equation with slowly varying coefficients and examines the restrictions imposed by the method.

Correcting for Initial Conditions in Multitime Problems

A method is developed which yields a uniformly valid expansion of the solution of certain perturbed differential equations. This involves the use of a two-term expansion. In contrast to other approaches, this one yields an approximation that can be made to have any desired degree of accuracy.

A Special Topics Course in Perturbation Methods

A special topics course dealing with perturbation methods in applied mathematics which was recently taught at Stevens Institute of Technology is described. Since the course enrolment was comprised of undergraduate and graduate mathematics students as well as graduate students in engineering, an unusual course philosophy had to be developed and implemented. The approach taken to achieve this in both teaching techniques and selection of subject matter is discussed in detail. Finally, conclusions are drawn from the students’ and the instructors experiences with the course which hopefully will be of value to those considering offering special topics courses in the future.