The authors find scale-invariant solutions of the partial differential equation (PDE) \[ {\frac {\partial^{\alpha} u(x,t)}{\partial t^{\alpha}}} = a {\frac {\partial^{\beta} u(x,t)} {\partial x^{\beta}}} , \quad x > 0, \;t > 0, \quad a=\text{const}> 0, \tag{1} \] involving simultaneously time- and space-fractional derivatives of orders \(\alpha > 0\), \(\beta > 0\) resp., both assumed in the known Riemann-Liouville sense. Their study has been stimulated by very recent papers devoted to the time-fractional diffusion equation (obtained by replacing the first order time-derivative in the diffusion equation by a derivative of order \(\alpha > 0\)) and to the space-fractional diffusion equation (where the second order space-derivative is replaced by an inverse Riesz potential of order \(\beta > 0\)).
To this aim, the authors use the method of Lie group analysis, that allows to determine special types of solutions, invariant under some subgroup of the full symmetry group of the given ordinary or partial differential equation. In the case of equation (1), the scale-invariant solutions can be found by solving an equivalent ordinary differential equation (ODE) of fractional order, \[ \left(P_{\beta/\alpha}^{1+\gamma-\alpha,\alpha} v \right) (z) = a z^{-\beta} \left(D_1^{-\beta,\beta} v \right) (z), \quad z > 0, \tag{2} \] obtained by the transformation \(u(x,t)=t^{\gamma} v(z)\), \(z = x t^{-\alpha/\beta}\) and involving the Erdélyi-Kober (E-K) fractional derivatives, depending on the orders \(\alpha,\beta\) and the parameter \(\gamma\) of the scaling group. These fractional order derivatives are defined as follows [cf. V. Kiryakova, Generalized fractional calculus and applications, Longman, Harlow (1994; Zbl 0882.26003)]: \[ \begin{aligned} (P_{\delta}^{\tau,\alpha} g) (z) &:= \left(\prod_{j=0}^{\eta-1} (\tau+j-{1 \over \delta} z {d \over dz})\right) (K_{\delta}^{\tau+\alpha,\eta-\alpha} g) (z), \quad \delta>0,\;\alpha>0,\\ (D_{\delta}^{\tau,\beta} g) (z) &:= \left(\prod_{j=1}^{\kappa} (\tau+j+ {1 \over \delta} z {d \over dz})\right) (I_{\delta}^{\tau+\beta,\kappa-\beta} g) (z), \quad \delta>0,\;\beta>0, \end{aligned} \] where \(K_{\delta}^{\tau,\alpha}\) and \(I_{\delta}^{\tau,\beta}\) are the E-K left- and right-sided fractional integrals and \(\eta,\kappa\) are the smallest integers greater than \(\alpha,\beta\) respectively.
The exact form of the scale-invariant solutions of the PDE (1) and the corresponding fractional ODE (2) is presented in terms of the so-called Wright functions \[ W_{(\mu,a),(\nu,b)} (z) := \sum_{k=0}^{\infty} {z^k \over \Gamma(a+\mu k) \Gamma(b+\nu k)},\qquad \mu,\nu \in R,\quad a,b \in C, \tag{3} \] for which the authors establish some basic properties (entireness, integral representations, asymptotics), necessary for proving the main result.
We can emphasize that functions (3) can be viewed as two-indices generalization of the Mittag-Leffler functions, as well as a modification of the two-indices Mittag-Leffler type functions of M. Dahrbashyan [Akad. Nauk Arm. SSR, Izv., Ser. Fiz.-Mat Nauk 13, No 3, 21-63 (1960; Zbl 0187.36902)], or useful example of the “multiple” Mittag-Leffler type functions, related to the “multiple Dzhrbashyan-Gelfond-Leontiev operators” in the book of V. Kiryakova (loc. cit.).