On inversion of Laplace transform of function, involving hyperbolic tangent

F.G. Khushtova

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Abstract: The paper examines the inverse of the Laplace transform with a hyperbolic tangen function. This function arises when solving a boundary value problem in a bounded domain governed by the heat
equation, subject to boundary conditions of the second and third kind.
Aim. To determine the inverse Laplace transform of a function that emerges from solving a boundary value problem, specifically a second or third type condition, associated with the heat equation.
Results. Using the residue theorem and the theory of a complex variable functions, we derive the inverse transform, suitable for large and small time values. In the first case, the inverse transform is expressed as a series of exponential functions with constant coefficients; in the second case, as a series of Laplace convolutions of special functions.
Conclusion and deduction. The derived results constitute a basis for constructing a solution to the boundary value problem for the heat equation in a bounded domain with a second-order condition on one of the boundaries and a third-order condition on the other, in a form suitable for small time values. In the context of mathematical physics, a solution to a similar problem is derived via separation of variables suitable for characterizing heat transfer processes for large time values. However, this proves inconvenient given sufficiently small temporal values, due to poor convergence properties pertaining to the Fourier series expansion involving eigenfunctions of the problem.

Keywords: Laplace transform, residue theorem, Jordan’s lemma, hyperbolic tangent, probability integral, Laguerre polynomials

For citation. Khushtova F.G. On inversion of Laplace transform of function, involving hyperbolic tangent. News of the Kabardino-Balkarian Scientific Center of RAS. 2025. Vol. 27. No. 6. Pp. 30–38. DOI: 10.35330/1991-6639-2025-27-6-30-38

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Information about the authors

Fatima G. Khushtova, Candidate of Physics and Mathematics, Researcher, Department of Fractional calculus, Institute of Applied Mathematics and Automation – branch of the Kabardino-Balkarian Scientific Center of the Russian Academy of Sciences;
89 A, Shortanov street, Nalchik, 360000, Russia;
khushtova@yandex.ru, ORCID: https://orcid.org/0000-0003-4088-3621, SPIN-code: 6803-4959