Eigenfunctions and Fundamental Solutions of the Fractional Two‐Parameter Laplacian
SEMYON B. YAKUBOVICH · International Journal of Mathematics and Mathematical Sciences · 2010
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), , 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.