THE CAUCHY PROBLEM FOR A HIGH-ORDER ORDINARY DIFFERENTIAL EQUATION INVOLVING THE BESSEL OPERATOR AND LOWER-ORDER TERMS
This paper investigates the Cauchy problem for a high-order ordinary differential equation involving
the Bessel operator with a spectral parameter. This type of problem presents significant challenges
and has received limited attention in the literature due to the lack of appropriate analytical tools. The
main objective of the study is to solve the Cauchy problem by employing a transmutation operator.
As the transmutation operator, the generalized Erd‘elyi-Kober fractional operator is utilized. When
this operator is applied, the considered problem is transformed into an equation without degeneration
and without a lower-order term. A key advantage of the proposed approach is that it leads to an
explicit solution of the formulated problem. Despite the significant progress in modern computational
techniques, obtaining exact solutions for boundary value problems of ordinary differential equations
remains an important and relevant challenge. Such solutions provide a deeper understanding of the
qualitative behavior of the described processes and phenomena, reveal the intrinsic properties of
the underlying mathematical models, and can also serve as benchmark examples for asymptotic and
numerical methods.
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