The Solution to Some Hypersingular Integral Equations

Abstract

The solution to integral equations $b(t) = f(t) + {\int_0^t {({t - s})} ^{\lambda - 1}}b(s)ds$ is given explicitly for λ <0 for the first time. For λ <0 the kernel of the integral equation is hypersingular and the integral diverges classically. Therefore, the above equation was considered as an equation that did not make sense. The author gives a definition of the divergent integral in the above equation. The Laplace transform is used in this definition and in a study of this equation. Sufficient conditions are given for a function F(p) to be a Laplace transform of a function f(t) or of a tempered distribution f. These results are new and their proofs are also novel.

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