Eigenfunctions and fundamental solutions of the fractional two-parameter Laplacian (Q963560)

Summary: We deal with the following fractional generalization of the Laplace equation for rectangular domains \((x,y)\in (x_0,X_0)\times (y_0,Y_0)\subset\mathbb R_+\times\mathbb R_+\), which is associated with the Riemann-Liouville fractional derivatives \(\Delta^{\alpha,\beta}u(x,y)=\lambda u(x,y)\), \(\Delta^{\alpha,\beta}:=D_{x_0+}^{1+\alpha}+ D_{y_0+}^{1+\beta}\), where \(\lambda\in\mathbb C\), \((\alpha,\beta)\in [0,1]\times[0,1]\). 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 \(\Delta^{\alpha,\beta}\) 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.