## 1 Introduction  
Fractional calculus has many advantages in various areas of investigation [1; 2; 3; 4; 5; 6]. Abstract integro-differential equations have been discussed in fundamental monographs [7; 8]. The classical theory of heat conduction is based on the assumption that the heat flux vector \(\mathbf{q}(\mathbf{x},t)\) at a point \(\mathbf{x}\) at time \(t\) is proportional to the temperature gradient \(\mathrm{grad}\,T(\mathbf{x},t)\) at the same point \(\mathbf{x}\) and at the same time \(t\). This is the content of the standard (local) Fourier law  
\[\mathbf{q}(\mathbf{x},t)=-k\,\mathrm{grad}\,T(\mathbf{x},t), \tag{1}\]  
where \(k\) is the thermal conductivity coefficient. The time-nonlocal generalization of the Fourier law describing \"long-tail\" memory is formulated as [9; 10]  
\[\mathbf{q}(\mathbf{x},t)=-\frac{k}{\Gamma(\alpha)}\,\frac{\partial}{\partial t }\,\int_{0}^{t}(t-\tau)^{\alpha-1}\,\mathrm{grad}\,T(\mathbf{x},\tau)\, \mathrm{d}\tau,\quad 0<\alpha\leq 1, \tag{2}\]  
\[\mathbf{q}(\mathbf{x},t)=-\frac{k}{\Gamma(\alpha-1)}\,\int_{0}^{t}(t-\tau)^{ \alpha-2}\,\mathrm{grad}\,T(\mathbf{x},\tau)\,\mathrm{d}\tau,\quad 1<\alpha\leq 2, \tag{3}\]  
where \(\Gamma(\alpha)\) is the gamma function and can be written in terms of the Riemann-Liouville time-fractional integrals and derivatives:  
\[\mathbf{q}(\mathbf{x},t)=-k\,D_{RL}^{1-\alpha}\,\mathrm{grad}\,T(\mathbf{x},t ),\quad 0<\alpha\leq 1, \tag{4}\]  
\[\mathbf{q}(\mathbf{x},t)=-k\,I^{\alpha-1}\,\mathrm{grad}\,T(\mathbf{x},t), \quad 1<\alpha\leq 2. \tag{5}\]  
Here,\[I^{\alpha}\,f(t)=\frac{1}{\Gamma(\alpha)}\,\int_{0}^{t}(t-\tau)^{\alpha-1}\,f( \tau)\,\mathrm{d}\tau,\quad\alpha>0, \tag{6}\]  
is the Riemann-Liouville fractional integral, and  
\[D_{RL}^{\alpha}\,f(t)=\frac{1}{\Gamma(m-\alpha)}\,\frac{\mathrm{d}^{m}}{ \mathrm{d}t^{m}}\biggl{[}\int_{0}^{t}(t-\tau)^{m-\alpha-1}\,f(\tau)\,\mathrm{d }\tau\biggr{]},\quad m-1<\alpha<m, \tag{7}\]  
is the Riemann-Liouville fractional derivative of order \(\alpha\)[11; 12]. The Caputo fractional derivative is defined as  
\[D_{\mathbb{C}}^{\alpha}\,f(t)=\frac{1}{\Gamma(m-\alpha)}\,\int_{0}^{t}(t-\tau) ^{m-\alpha-1}\,\frac{\mathrm{d}^{m}f(\tau)}{\mathrm{d}\tau^{m}}\,\mathrm{d} \tau,\quad m-1<\alpha<m. \tag{8}\]  
Recall the equations of the Laplace transform for fractional integrals and derivatives [11; 12]:  
\[\mathcal{L}\{I^{\alpha}\,f(t)\}=\frac{1}{s^{\alpha}}\,f^{*}(s), \tag{9}\]  
\[\mathcal{L}\{D_{RL}^{\alpha}\,f(t)\}=s^{\alpha}\,f^{*}(s)-\sum_{k=0}^{m-1}D^{ k}I^{m-\alpha}f\bigl{(}0^{+}\bigr{)}s^{m-1-k},\quad m-1<\alpha<m, \tag{10}\]  
\[\mathbf{q}(\mathbf{x},t)=-\frac{k}{\Gamma(\alpha)}\,\frac{\partial}{\partial t }\,\int_{0}^{t}(t-\tau)^{\alpha-1}\,\mathrm{grad}\,T(\mathbf{x},\tau)\, \mathrm{d}\tau,\quad 0<\alpha\leq 1, \tag{2}\]0  
where the asterisk marks the transform and \(s\) denotes the Laplace transform variable. The constitutive equations for the heat flux (4) and (5) in combination with the law of conservation of energy lead to the time-fractional heat conduction equation with the Caputo derivative  
\[\mathbf{q}(\mathbf{x},t)=-\frac{k}{\Gamma(\alpha)}\,\frac{\partial}{\partial t }\,\int_{0}^{t}(t-\tau)^{\alpha-1}\,\mathrm{grad}\,T(\mathbf{x},\tau)\, \mathrm{d}\tau,\quad 0<\alpha\leq 1, \tag{2}\]1  
where \(a\) can be taken as a counterpart of the thermal diffusivity coefficient, and the Caputo time-derivative \(D_{\mathbb{C}}^{\alpha}f(t)\) is denoted as  
\[\mathbf{q}(\mathbf{x},t)=-\frac{k}{\Gamma(\alpha)}\,\frac{\partial}{\partial t }\,\int_{0}^{t}(t-\tau)^{\alpha-1}\,\mathrm{grad}\,T(\mathbf{x},\tau)\, \mathrm{d}\tau,\quad 0<\alpha\leq 1, \tag{2}\]2  
The generalization of Equation (12)  
\[\mathbf{q}(\mathbf{x},t)=-\frac{k}{\Gamma(\alpha)}\,\frac{\partial}{\partial t }\,\int_{0}^{t}(t-\tau)^{\alpha-1}\,\mathrm{grad}\,T(\mathbf{x},\tau)\, \mathrm{d}\tau,\quad 0<\alpha\leq 1, \tag{2}\]3  
has been considered in several publications [13; 14; 15; 16; 17; 18]. Two special versions of Equation (13) with integer values of the time derivative are well known.