Research Article Open Access

The Solution to Some Hypersingular Integral Equations

Alexander G. Ramm1
  • 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

Gel’fand, I. M., & Shilov, G. E. (1964). Generalized Functions. Volume I: Properties and Operations (1st ed., Vol. 1). Academic Press.
Hörmander, L. (1998). The Analysis of Linear Partial Differential Operators I (1st ed.). Springer Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-96750-4
Ramm, A. G. (2022). The Navier-Stokes Problem (2nd ed.). Springer Nature.
Ramm, A. G. (2023). Analysis of the Navier-Stokes problem (2nd ed.). Springer Nature.
Schiff, J. L. (1999). The Laplace Transform (1st ed.). Springer. https://doi.org/10.1007/978-0-387-22757-3

Journal of Mathematics and Statistics
Volume 20 No. 1, 2024, 45-18

DOI: https://doi.org/10.3844/jmssp.2024.45.18

Submitted On: 2 January 2024 Published On: 25 May 2024

How to Cite: Ramm, A. G. (2024). The Solution to Some Hypersingular Integral Equations. Journal of Mathematics and Statistics, 20(1), 45-18. https://doi.org/10.3844/jmssp.2024.45.18

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Keywords

  • Integral Equations with Hypersingular Kernels
  • Laplace Transform