Fractional Differential Equations Involving the Spherical Bessel Function j\(_0\): Analytical Solutions Via Laplace Transform
Jorge Olivares Funes *
Department of Mathematics, Universidad de Antofagasta, Antofagasta, Chile.
Pablo Martin
Department of Physics, Universidad de Antofagasta, Antofagasta, Chile.
Elvis Valero Kari
Universidad Mayor de San Andres, Bolivia.
Maria Teresa Veliz Aviles
Department of Physics, Universidad de Antofagasta, Antofagasta, Chile.
*Author to whom correspondence should be addressed.
Abstract
This work addresses the resolution of fractional differential equations whose nonhomogeneous part is given by the spherical Bessel function \(J_0 (x)\). By using the fractional derivative in the sense of Caputo and the Laplace transform, a general analytical solution is obtained in terms of the generalised hypergeometric functions \( _2 F_3\), revealing a recurrent structure in the solutions. Furthermore, particular cases for integer and fractional orders are presented, highlighting the appearance of special functions such as the sine integral and Fresnel functions. The results confirm the close relationship between fractional calculus and Bessel functions, proposing new perspectives for applications in mathematical physics.
Keywords: Fractional calculus, differential equations, Caputo, Laplace transform