F. G. Tricomi. Iztegral Equatiozs. Dover Publications, New York, 1985.  Home/Search   Document Not in Database   Summary   Related Articles

....is the identity operator, we can again write f as a single operator times g: f (I K) g. 30) The two varieties of Fredholm integral equations actually have quite different properties. For instance, if k and g are step functions, then the solution f to equation (29) must also be a step function [36], yet we can say nothing about the form of f in equation (27) This fact would seem to indicate that 2nd kind equations are in some sense nicer to work with than 1st kind equations. This is indeed the case. Fortunately, the integral equations we encounter in radiative transfer are of the 2nd ....

....f, it can be evaluated separately to k2(z , z) f k(z , k(y, z) ds, 42) which is evidently the kernel of K 2. In general, we can obtain the kernel corresponding to the operator K by a recurrence relation: h( z) k( h t(y, z) ds. 43) The kernel k is called the iterated ker tel of order t [36]. From the Neumann series we see that the kernel of the operator M can be obtained by summing the iterated kernels of 11 orders up to m( z) k( z) 44) where k0 is taken to be the identity. Finally, we can express the tth approximation f, in terms of the kernel m: f, f m ( g(y) ds. ....

F. G. Tricomi. Iztegral Equatiozs. Dover Publications, New York, 1985.

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