Physical Science: New Insights and Developments Vol. 3

The study addresses the solution of a nonhomogeneous linear differential equation of fractional order α, equal to the modified Bessel function of order zero (I₀(x)), under the initial condition (f(0) = 0) and with (0 < α < 1). The Caputo definition of the fractional derivative is adopted, which is widely used in the analysis of physical and chemical phenomena such as viscoelasticity, anomalous diffusion, and electrical circuits. By means of the Laplace transform and its inverse, analytical solutions are obtained for the specific cases (α = 1/4, 1/2, 3/4), expressed in terms of hypergeometric functions ₂F₁. These solutions combine a fractional power of the independent variable (x) with a special function, reflecting the non-integer nature of the differential operator. Furthermore, regular patterns are observed in the parameters of the hypergeometric functions as α varies, suggesting a generalizable structure for other fractional values. The work demonstrates that fractional differential equations can be systematically solved using classical tools of integral and transform calculus, connecting fractional derivatives with families of special functions.