Abstract

In this paper, we generate asymmetric Fourier kernels as solutions of ODE′s. These kernels give many previously known kernels as special cases. Several applications are considered.

Highlights

  • In a previous paper [1], we indicated how Fourier kernels could be generated as solutions of ordinary differential equations and we generated a large number of hitherto unknown Fourier kernels

  • F A2A(A) k(A,x) dA gt(x) in x > i. This is a new set of dual integral equations which have not been considered previously

  • We propose to consider such dual integral equations subsequently

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Summary

INTRODUCTION

In a previous paper [1], we indicated how Fourier kernels could be generated as solutions of ordinary differential equations and we generated a large number of hitherto unknown Fourier kernels. In each one of these cases the corresponding solution of equation (1) is a Fourier kernel. We notice from equation (3) that (disregarding the contribution from x (R)) the operator is symmetric if u (and v) satisfy any one of the following five conditions:. We shall show that in each one of the above cases, the corresponding solutions of equation (1), which are bounded at infinity, generate Fourier-like kernels. Taking the Laplace Transform of (9c), changing the order of integration, and substituting, we get the integral of a rational function of A, from zero to infinity. An appropriate representation of u in this case would be u(x,t) f kt(A,x)[A(A)cosA2t + A sinAt] dA and we would require f f(x) These equations are inverted with the help of equtions (9) and substitution gives u. After re-scaling, we would get the equation

Ou y
The square root singflarity at
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