Hammerstein integral equivalent of Riccati's equation

✍ Scribed by J. D. Pulfer; M. A. Whitehead

• Publisher: John Wiley and Sons
• Year: 1974
• Tongue: English
• Weight: 566 KB
• Volume: 8
• Category: Article
• ISSN: 0020-7608
• DOI: 10.1002/qua.560080504

✦ Synopsis

Abstract

A transformation exists which allows the general Riccati equation

to be written in a simpler form:

The transformed equation has the equivalent nonlinear Hammerstein integral equation

if the kernel N(r, r′) satisfies three conditions:

and

and

A solution of the nonlinear integral equation is devised by repeatedly integrating the Hammerstein equation. During this procedure the kernel generates an equation that contains only coefficients of β(r)^0^ and β(r)^1^. As a result, after truncating at the end of the __n__th cycle, it is a simple matter to write down a Padé‐type approximation: all coefficients in this approximation are capable of being evaluated in terms of simple algebraic formulations of P(r), R(r), and integrals over P(r).

The zeroes of the denominator of the Padé‐type approximation define the points where singularities occur in β(r).