The Solution to Some Hypersingular Integral Equations
- 1 Department of Mathematics, Kansas State University, Manhattan, KS 66506, United States
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.
References
DOI: https://doi.org/10.3844/jmssp.2024.45.48
Copyright: © 2024 Alexander G. Ramm. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
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Keywords
- Integral Equations with Hypersingular Kernels
- Laplace Transform