The Mittag-Leffler–Bessel Integral Transform: Theory, Operational Calculus, and Applications to Fractional Differential Equations

A new integral transform, called the Mittag-Leffler–Bessel transform (MLB transform), is introduced and systematically studied. The transform is defined by an integral on the half-line whose kernel is the product of a fractional power weight, the two-parameter Mittag-Leffler function Eα,μ+1 evaluated at a negative argument, and the Bessel function of the first kind Jμ of order μ, with real parameters α > 0 and μ > –3/4. The kernel combines a power weight, a function central to fractional calculus, and a function central to cylindrical problems into a single operator. The main results are: (i) a complete convergence theory determining the parameter region (α, μ) in which the transform is well-defined; (ii) the fundamental identification of the MLB-transform as a weighted composition of the Hankel transform with a Mittag-Leffler multiplier, allowing inversion to be inherited from the Hankel theory; (iii) a full operational calculus including linearity, scaling, action on t-derivatives, and interaction with Caputo fractional derivatives; (iv) an inversion theorem; (v) a Parseval-type identity; (vi) a convolution theorem adapted to the Bessel structure; (vii) applications to linear fractional differential equations of relaxation and oscillation type; and (viii) numerical examples illustrating each result. All theorems are accompanied by full proofs, and all results are illustrated with numerical graphs