The Solution of a Class of Integral Equations

The solution of some integral equations of Wiener-Hopf type

“…Now, if k is any function whose Fourier transform has this property, it is not hard to prove that k itself satisfies an ordinary differential equation with linear coefficients. This last is exactly the hypothesis made by Latta [6], who also reduced the solution of (1.3) (when E is a single interval) to an ordinary differential equation, but in an entirely different way.…”

“…In this case, it will have to be Theorem 2 that is used rather than Theorem 1, for it is known ( [7]; see also [6], [11]) that if g is continuous, / tends to look like (1 -x2)~1/2 near the ends of the interval (-1, 1) and so will not be in L2 in general. What is important about this is not the warning that Theorem 2 must be used instead of Theorem 1, but that the following calculations are the same no matter which of the two theorems is used.…”

“…The method to be used is what I have earlier called the general method of Wiener and Hopf [12], [13], In the present context, the method has points of contact both with the method of separation of variables for the solution of partial differential equations and with Latta's method for solving certain integral equations ( [6], see also [10]). …”

The solution of an integral equation of the first kind on a finite interval

“…If we put T = e 'x4>(x, y) into (1.2) we obtain d2<l>/dx2 + d2<t>/dy2 -\2<t> = 0 (1. 3) which itself admits a simple physical interpretation. Thus if we replace the convection term 2\ dT/dx in (1.2) by an endothermic release of heat at all points in the fluid proportional to the local temperature then the temperature satisfies Eq.…”

“…It was found at an advanced stage of this paper that both Rvachev [1] and Gill [2] had also solved the associated partial differential equation and it appears that Rvachev had the solution of (1.1) within his grasp. Neither author pursued the analysis given here however, and since the equation has received attention from Latta [3] and Pearson [4] it would appear that an explicit solution is of some interest. Further, integral equations of the first kind frequently arise in applied mathematics, and (unless they are of a simple Cauchy type) are often difficult to solve exactly.…”