Solution of an Initial Boundary Value Problem for a Multidimensional Fourth-Order Equation Containing the Bessel Operator

Shakhobiddin Karimov () and Yorkinoy Tulasheva
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Shakhobiddin Karimov: Department of Applied Mathematics and Informatics, Fergana State University (FSU), Murabbiylar Street, 3A, Fergana 150100, Uzbekistan
Yorkinoy Tulasheva: Department of Mathematical Analysis, Namangan State University (NamSU) 159, Uychi St., Namangan 160100, Uzbekistan

Mathematics, 2024, vol. 12, issue 16, 1-11

Abstract: In the present work, the transmutation operator approach is employed to construct an exact solution to the initial boundary-value problem for multidimensional free transverse equation vibration of a thin elastic plate with a singular Bessel operator acting on geometric variables. We emphasize that multidimensional Erdélyi–Kober operators of a fractional order have the property of a transmutation operator, allowing one to transform more complex multidimensional partial differential equations with singular coefficients acting over all variables into simpler ones. If th formulas for solutions are known for a simple equation, then we also obtain representations for solutions to the first complex partial differential equation with singular coefficients. In particular, it is successfully applied to the singular differential equations, particularly when they involve operators of the Bessel type. Applying this operator simplifies the problem at hand to a comparable problem, even in the absence of the Bessel operator. An exact solution to the original problem is constructed and analyzed based on the solution to the supplementary problem.

Keywords: fourth-order equation; plate vibration equation; Bessel operator; transmutation operator; Erdélyi–Kober operator (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2024
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