###### Abstract  
The time-dependent diffusion equation is studied, where the diffusion coefficient itself depends simultaneously on space and time. First, a family of novel, nontrivial analytical solutions is constructed in one space dimension with the classical self-similar Ansatz. Then, the analytical solution for two different sets of parameters is reproduced by 18 explicit numerical methods. Fourteen of these time integrators are recent unconditionally stable algorithms, which are often much more efficient than the mainstream explicit methods. Finally, the adaptive time-step version of some of these algorithms are created and tested versus widespread algorithms, such as the Runge-Kutta-Fehlberg solver. diffusion; heat conduction; analytical solution; explicit time integration; unconditionally stable numerical methods; adaptive step size controllers +
Footnote †: journal:  
0000-0001-9999-4000-0000-0000-0000-0000-0000-0000-0000-000  
spatial inhomogeneities, such as narrow ribbon channels [6]. Such systems are created when molecules diffuse through carbon nanotubes [7], systems of channels like in zeolites [8], or even in cell membranes [9]. Diffusion equations with space-dependent and time-dependent coefficients were also used to model the movement of molecular species in water by [PERSON] at al. [10] and [PERSON] [11], respectively. Previously, we investigated the regular and irregular diffusion equations [12; 13]. For the regular diffusion equation (such as Equation (1)) with the traveling wave, self-similar, traveling profile, or with some generalized self-similar trial functions, numerous new types of analytic solutions were found. These solutions have a much more complicated structure than the well-known Gaussian (plus an error function in the most general case). Our new formulas contain Kummer's or Whittaker functions with quadratic arguments with a new free parameter which results in a larger variety of solutions. These solutions can have a drastically different rate of decay than the fundamental Gaussians. We found solutions which have some oscillatory behavior with a rapid power-law decay both in space and time. In this work, we modify the PDE (1) to have a diffusion coefficient which is non-constant in two senses. We introduce a new variable, which is a combination of the space and time variable: \(\eta=\frac{x}{\mu^{2}}\ \in\mathbb{R}\). The diffusion coefficient has the simplest power-law dependence on this variable: \(\overline{D}(\eta)=\ D\eta^{m}\), where \(D\) is a constant, whose physical dimension depends on the concrete value of \(\beta\) and \(m\). Inserting it into the diffusion equation, we obtain  
\[\frac{\partial u(x,t)}{\partial t}=\ D\frac{\partial}{\partial x}\bigg{(}\eta ^{m}\frac{\partial u(x,t)}{\partial x}\bigg{)}=\ D\bigg{(}m\eta^{m-1}\frac{ \partial\eta}{\partial x}\frac{\partial u(x,t)}{\partial x}+\eta^{m}\frac{ \partial^{2}u(x,t)}{\partial x^{2}}\bigg{)} \tag{2}\]  
Equations (1) and (2), and similar kinds of equations, are most frequently solved numerically [14], and several methods are proposed for this purpose. Most of them can be classified as either explicit or implicit schemes. Both kinds have a major advantage and a disadvantage compared to one another. The widely used explicit methods, such as the FTCS (forward time central space), require a fairly short time to execute a time step and they are easily parallelizable. However, they are unstable, and the solution is expected to blow up if the time-step size exceeds the so-called CFL (Courant-Friedrichs-Lewy) limit. In other words, their stability region is limited [15]. In our case, due to the space dependence of the diffusion coefficient, this limit is rather small, and the stiffness ratio is high. On the other hand, due to the time dependence of the coefficient, it is changing in time, so using the traditional explicit methods is very risky.