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ON THE NUMERICAL SOLUTION OF A CERTAIN CLASS OF SECOND ORDER DIFFERENTIAL EQUATIONS

IP.com Disclosure Number: IPCOM000128146D
Original Publication Date: 1977-Dec-31
Included in the Prior Art Database: 2005-Sep-15
Document File: 20 page(s) / 44K

Publishing Venue

Software Patent Institute

Related People

Stylianos D. Danieiopoulos: AUTHOR [+3]

Abstract

Among the second order differential equations With singular points, there ABSTRACT is a class of equations that occur frequently in mathematical physics, which It is shown that the usual numerical methods for solving differential have parameters that can assume arbitrarily large values. If such a parameter equations (boundary value problems) may fail to give satisfactory approximations is contained in a term which also contains the singularity, the discretization of the solution in cases in which the differential equation has a singularity error may become too large to allow meaningful computation of the numerical (e.g. the Schrodinger equation with centrifugal potential). The difficulties solution throughout the entire interval of interest. In order to be more can be bypassed by solving the equivalent integral equation. It is shown that specific, we consider an equation of the general form for this class of equations, appropriate Green's functions can always be con- . (1.1) [Equation ommitted] structed and that the numerical solution of the integral equation can be obtained (using Simpson's rule)"with a global error which is 0(h4). The corresponding and note that through the transformation x error when the differential equation is solved directly is [Equation ommitted] we can rewrite it in the [Equation ommitted] A further transformation [Equation ommitted] allows the original equation (1.1) to be written in the form [Equation ommitted]

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THIS DOCUMENT IS AN APPROXIMATE REPRESENTATION OF THE ORIGINAL.

ON THE NUMERICAL SOLUTION OF A CERTAIN CLASS OF SECOND ORDER DIFFERENTIAL EQUATIONS

Stylianos D. Danieiopoulos

TR 77-04 November 1977

1. Introduction

Among the second order differential equations With singular points, there ABSTRACT is a class of equations that occur frequently in mathematical physics, which It is shown that the usual numerical methods for solving differential have parameters that can assume arbitrarily large values. If such a parameter equations (boundary value problems) may fail to give satisfactory approximations is contained in a term which also contains the singularity, the discretization of the solution in cases in which the differential equation has a singularity error may become too large to allow meaningful computation of the numerical (e.g. the Schrodinger equation with centrifugal potential). The difficulties solution throughout the entire interval of interest. In order to be more can be bypassed by solving the equivalent integral equation. It is shown that specific, we consider an equation of the general form for this class of equations, appropriate Green's functions can always be con- . (1.1)

(Equation Omitted)

structed and that the numerical solution of the integral equation can be obtained (using Simpson's rule)"with a global error which is 0(h4). The corresponding and note that through the transformation x error when the differential equation is solved directly is

(Equation Omitted)

we can rewrite it in the

(Equation Omitted)

A further transformation

(Equation Omitted)

allows the original equation (1.1) to be written in the form

(Equation Omitted)

The zeros of the function a(x) are the singularities of the differential equation (1.2). We shall focus our attention on equations of the form (1.2) with functions

>-2-

North Carolina State University Page 1 Dec 31, 1977

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ON THE NUMERICAL SOLUTION OF A CERTAIN CLASS OF SECOND ORDER DIFFERENTIAL EQUATIONS

F(x) which, after all simplifications are done, are reduced to the form and due to (1.3) we deduce

(Equation Omitted)

where p is a constant and q(x) is a function that is bounded over the interval, from which we can further obtain .

(Equation Omitted)

Because of the singularity of F(x), the right-hand side of equation (1.2) p does not satisfy a Lipschitz condition in the closed interval [a,b]. Initial where we have used-x0 = ph. When the computation of uk is started, according or boundary value problems involving the differential equation (1.2) with to formula (1.5), the value of p may be very small. Its first value, and prob- functions F(x) as given in (1.3), are solved, in practice, in an interval [x0,b] ably a few more, are less than 1, and this causes the error term to be very that excludes the point a (see for example
[8]). large. For large values of p, which is a possibility in many applications, the The error term for most numerical (one-step or multi-step) methods is give...