A note on the numerical solution of certain non-linear integral equations

1956 ◽  
Vol 236 (1207) ◽  
pp. 529-534 ◽  
Keyword(s):  
Numerical Solution ◽  
Integral Equations ◽  
Asymptotic Form ◽  
Essential Singularity ◽  
Non Linear ◽  
Linear Integral ◽  
Hydrodynamical Problem ◽  
Linear Integral Equations

Methods are described for the numerical solution of two non-linear integral equations occurring in a hydrodynamical problem. In each case the existence of an essential singularity of the solution requires the application of special techniques. The asymptotic form of the solutions for large x is determined.

scholarly journals Non-linear integral equations for Heun functions

1969 ◽  
Vol 16 (4) ◽  
pp. 281-289 ◽  
Author(s):  
B. D. Sleeman
Keyword(s):  
Integral Equations ◽  
Heun Equation ◽  
Mathieu Functions ◽  
Special Cases ◽  
Mathieu’S Equation ◽  
Heun Functions ◽  
Non Linear ◽  
Linear Integral ◽  
Image Position ◽  
Linear Integral Equations

Some years ago Lambe and Ward (1) and Erdélyi (2) obtained integral equations for Heun polynomials and Heun functions. The integral equations discussed by these authors were of the formFurther, as is well known, the Heun equation includes, among its special cases, Lamé's equation and Mathieu's equation and so (1.1) may be considered a generalisation of the integral equations satisfied by Lamé polynomials and Mathieu functions. However, integral equations of the type (1.1) are not the only ones satisfied by Lamé polynomials; Arscott (3) discussed a class of non- linear integral equations associated with these functions. This paper then is concerned with discussing the existence of non-linear integral equations satisfied by solutions of Heun's equation.

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