Eigenfunctions and Fundamental Solutions of the Fractional Two-Parameter Laplacian

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 derivativesΔu x, y λu x, y ,Δ : D1 α x0 D 1 β y0 , 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 Δ 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.