Eigenfunctions and Fundamental Solutions of the Fractional Two-Parameter Laplacian

Abstract EN

We deal with the following fractional generalization of the Laplace equation for rectangular domains (x,y)∈(x0,X0)×(y0,Y0)⊂ℝ+×ℝ+, which is associated with the Riemann-Liouville fractional derivatives Δα,βu(x,y)=λu(x,y), Δα,β:=Dx0+1+α+Dy0+1+β, where λ∈ℂ, (α,β)∈[0,1]×[0,1].

Reducing the left-hand side of this equation to the sum of fractional integrals by x and y, we then use the operational technique for the conventional right-sided Laplace transformation and its extension to generalized functions to describe a complete family of eigenfunctions and fundamental solutions of the operator Δα,β in classes of functions represented by the left-sided fractional integral of a summable function or just admitting a summable fractional derivative.

A symbolic operational form of the solutions in terms of the Mittag-Leffler functions is exhibited.

The case of the separation of variables is also considered.

An analog of the fractional logarithmic solution is presented.

Classical particular cases of solutions are demonstrated.