Numerical inversion of the multiplicative Laplace transform via multiplicative Laguerre polynomials

In this manuscript, we introduce the multiplicative Laguerre polynomials (MLPs) that arise as one of the solutions of the multiplicative Sturm-Liouville equation \frac{d^*}{dx}\left(e^{x\omega(x)}\odot\frac{d^*y}{dx}\right)\oplus\left(e^{n\omega(x)}\odot y\right)=1,\quad x>0, , where \omega(x)=x^{\alpha}e^{-x} with \alpha>-1 . Here, \frac{d^*}{dx}f(x) denotes the multiplicative derivative of the function f at x, defined by \lim_{h\to 0}\left(\frac{f(x*h)}{f(x)}\right)^{1/h}, whenever this limit exists. We compute the multiplicative Laplace transform of the multiplicative Laguerre polynomials and establish the multiplicative version of Tricomi’s formula. Furthermore, we introduce two numerical methods for approximating the inverse multiplicative Laplace transform, based on properties of the multiplicative Laguerre polynomials. We illustrate the obtained results with some examples related to the solution of nonlinear classical second-order differential equations.