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1.   Introduction

It is of common knowledge, that the most effective method for constructing solutions of nonlinear partial differential equations (PDEs) of mathematical physics is the symmetry reduction method. The method can be both classical ^[1] and non-classical ^{[2,3,4,5,6,7]}. In these cases the construction of a proper ansatz (by which we mean a general form of an invariant solution) boils down to solving a quasilinear first-order differential equation, therefore an ansatz includes one arbitrary function and the initial equation reduces to a single differential equation with fewer independent variables, especially an ordinary differential equation (ODE). Reductions of differential equations to algebraic equations were considered in detail in ^[8] and ^[9].

Papers ^[10] and ^[11] presented a concept of conditional symmetry of evolution equations, which is a natural generalization of nonclassical symmetry. By using this method one can reduce nonlinear evolution equations with two independent variables to a system of ODEs. In ^[11] Zhdanov proved the theorem on the connection between the generalized conditional symmetry and reduction of evolutionary equations to a system of ODEs. It is worth pointing out that the number of differential equations in this system is equal to the number of unknown functions. The approach is used to construct exact solutions of nonlinear diffusion equations in ^[12]. The relationship between the generalized conditional symmetry of evolution equations and compatibility for overdetermined system of differential equations is studied in ^[13].

An approach on symmetry reduction of evolutionary equations is well developed. However, the problem of reducing of non-evolutionary equations is significantly less studied. The relation between the compatibility and reduction of partial differential equations in two independent and one dependent variables has been studied in ^[14]. Svirshchevskii ^[15] put forward the inverse symmetry reduction method for evolution equations of the form

where $u = u(t, x)$ and $K[u]: = K\left (t, x, u, \tfrac{\partial u}{\partial x}, \ldots, \tfrac{\partial^s u}{\partial x^s} \right)$. One can use this approach if $K[u]\partial_u$ is the generalized infinitesimal symmetry of a linear homogeneous ODE. In ^[16] the generalization of Svirshhchevskii's method was proposed. It provides an explicit characterization of all nonlinear differential operators that leave a given subspace of functions invariant. It turns out that the inverse symmetry reduction method is also applicable to non-evolutionary differential equation ^[17,18,19]. We are looking for the ansatz reducing PDEs by solving ODEs that do not necessarily have to be linear.

More specifically, let $u_{(k)}$ denote the set of all $k$th order partial derivatives of $u = u(t, x)$ with respect to $(t, x)$. Suppose that a generalized vector field

is a generalized symmetry of an ordinary differential equation

where $H$ is a smooth function of its arguments, and $t$ plays the role of a parameter ^[17]. Then an ansatz reducing the equation

to a system of ODEs can be constructed via finding the general solution of the ODE (1.2) ^[17]. Integrating the reduced system, one obtain exact solutions of PDEs (1.3). Exact solutions of differential equations are useful for understanding physical phenomena described by these equations and testing approximate and numerical methods for solving them. The method can be applied to non-evolutionary equations and even to ODEs. Moreover, in the framework of our approach one can construct an ansatz that reduces non-evolutionary PDEs to a system of ODEs and also the number of equations is smaller than the number of unknown functions. This enabled us to find solutions depending on arbitrary functions, which will be shown in Section 3.1. It is worth pointing out that the suggested method is applicable in multi-dimensional case as well ^[17,20,21]. We see that the ODE (1.2) includes the parametric variable $t$, apart from dependent and independent variables $u$ and $x$. This allowed us to construct integrable PDEs. For example the equation

admits the symmetry operator $X = \eta\partial_u = \big(u_t-u_{xxx}+3\tfrac{u_{xx}u}{u_x}\big)\partial_u$ provided that $a(t, x)$ satisfies the KdV equation

as was shown in ^[17]. Then it is clear that the method is related to the inverse scattering transformation method. We emphasize that $\eta$ depends on $u_t$ and the condition of invariance of Eq (1.4) with respect to the group with generator $X$ in classical sense leads to a determining equation which is the well-known integrable KdV equation. This approach belongs to a class of nonclassical symmetry methods in a sense that it produces results unobtainable within the classical methods. The idea is to use generalized symmetries of ODEs for constructing solutions of evolutionary equations. In this paper we consider a nonlinear evolutionary equation that describes transport phenomena in heterogeneous medium and apply a reduction method based on the symmetries of second-order nonlinear ODEs.

Extended symmetry analysis of porous medium equations with absorption and convection terms, including nonclassical symmetries, was carried out in papers ^[22,23,24]. Those equations included a factor dependent on a spatial variable $x$. We also studied diffusion-type equations for which the right-hand side involves a spatial variable and found solutions which are not invariant in classical sense by using the suggested method. We present the results obtained for the model medium with exponential and polynomial heterogeneity. Within the method applied, nonlinear transport equation is reduced to a system of two ODEs. After integrating (solving) the system of ODEs, we obtain exact solution of the initial equation. Right now main two differences from classical symmetry reduction should become apparent. With classical symmetry reduction there is only one reduced equation, not a system, and the solution from classical reduction will always be classically invariant. Since the method applied differs from the classical Lie method, it is not suitable for constructing algorithms for the generation of new solutions, or production of conservation laws. Its only advantage is the preservation of the reduction property. In addition, it doesn't ensure that none of the obtained solutions could also be found within the classical method. Therefore there is a very important question of distinguishing truly new solutions obtained within the method proposed. Many new local and nonlocal symmetries have been found for nonlocally related PDE systems ^[25]. Nonlocal symmetries alongside Bäcklund transformations helped finding solutions for the (2+1)-dimensional KdV–mKdV equation ^[26].

Based on the fact that a set of point and generalized symmetry operators (of the ODE) form a Lie algebra, we distinguish a class of diffusion equations whose solutions, obtained with the help of the aforementioned approach, cannot be obtained through the classical Lie method. Furthermore, it can be used to construct a large class of nonlinear evolution equation all of which are reduced to systems of ODEs by the same ansatz and possess solutions which are not invariant in the classical Lie sense. We emphasize that evolutionary equations are widely used and referenced in mathematical biology (see for instance ^[27] and references given there).

The organization of the article is as follows: We give the definition of generalized symmetry of an ODE and outline and explanation of the method in Section 2. In Section 3.1 we discuss applying of the method to PDEs. We find the classes of nonlinear evolutionary equations for which the method can be applied in the Section 3.2. In Section 3.3 we show the application of the method for finding solutions and obtain the solutions which cannot be constructed by the classical Lie method for the modified diffusion equations. The Theorem on a sufficient condition for the solution to be invariant one in classical sense is given too. In Section 4 we discuss the obtained results and provide some conclusions.

2.   Materials and methods

Consider the differential equation (1.2). Let us denote by $L$ the set of all differential consequences of (1.2) with respect to $t$ and $x$. Let

We treat (1.2) as it was a PDE which does not include the partial derivatives with respect to $t$.

Definition 1. We say that a generalized vector field

is a generalized infinitesimal symmetry of equation (1.2) if the condition

holds, where

is the standard $p$th prolongation ($p$th extension) of $X$ and $D_x = \tfrac{\partial}{\partial x}+u_1\tfrac{\partial}{\partial u} + u_{1t}\tfrac{\partial}{\partial u_t} + u_{2}\tfrac{\partial}{\partial u_1}+\ldots$ is the operator of total derivative with respect to $x$, $u_0 = u$, $u_i = \tfrac{\partial^i u }{\partial x^i}$, $D_x^0 = 1$, $D_x^{j+1} = D_x(D_x^j)$, $i, \, j\in {\mathbb N}$.

For a more elaborate take on the basics of symmetry methods please refer to ^[28]. The invariance property (2.1) ensures a reduction of the equation

to a system of ordinary differential equations. Let

where $F$ is a smooth function of $(x, t, \varphi_1, \ldots, \varphi_p)$, be the general solution of Eq (1.2). Then the ansatz (2.3), where $\varphi_1(t), \ldots, \varphi_p(t)$ are unknown functions of the variable $t$, reduces partial differential equation (2.2) to the system of $k_1$ ordinary differential equations, where $k_1\le p$ ^[17]. For evolutionary equations $k_1 = p$, and for non-evolutionary equations, this number may be less than $p$. This property is illustrated by Eqs (3.9) and (3.13). The necessary computation was made in Maple^TM with the addition of a software Jets for differential calculus on jet spaces and diffieties, which was created by H. Baran and M. Marvan and is distributed under the GNU General Public License. The tool and the user guide can be found at https://jets.math.slu.cz/.

3.   Results

3.1.   On application of the generalized symmetry reduction method to differential equations with two independent variables

In this subsection we show how to apply the method to PDE's. At first we consider differential equations obtained with the help of symmetry operators of Eq (1.4). One can find the solution $a = \frac{2}{x^2}$ of Eq (1.5) when $\frac{\partial a}{\partial t} = 0$ (the stationary Korteweg–de Vries equation). Then the equation

admits the three-dimensional Lie algebra with basic operators $Q_1 = u_t\partial_u$, $Q_2 = \big(u_{xxx}-3\tfrac{u_{xx}u}{u_x}\big)\partial_u$, $Q_3 = u\partial_u$ in this case. Equation (3.1) is integrable by quadratures and thus we obtain the ansatz

where $\varphi_1(t)$, $\varphi_2(t)$ are unknown functions, which reduces nonlinear evolution equation

with $\lambda = {\rm const}$, to the system of two ODEs

The solution of this system has the form

Substituting (3.4) into (3.2) one can obtain the solution of (3.3). Note that for $a = w_x$, where $w$ is a solution of the stationary Calogero–Bogoyavlenskii–Shiff equation

Eq (1.4) is also integrable by quadratures ^[29]. The reduction method can also be applied for reducing the Cauchy problem for Eq (3.3) with an initial condition $u(0, x) = f(x)$ to the Cauchy problem for a system of ODEs.

Our next goal is to show that the method is applicable to non-evolutionary type PDEs. For this purpose we first consider the following differential equation:

It is invariant with respect to the two-parameter Lie group of point transformations with generators

From this it immediately follows that the Eq (3.5) is integrable by quadratures. To obtain non-evolutionary equations we look for symmetry operators of the form $Y = \eta(t, x, u, u_x, u_t, u_{xt})\partial_u$. We prove that the Eq (3.5) admits the generalized infinitesimal symmetries given by

as well. We emphasize that the coefficient $\eta_2$ depends explicitly on the (parametric) variable $t$ and the derivative $u_{tx}$. This enabled us to find non-evolutionary nonlinear differential equations that can be reduced to a system of ODEs by appropriate ansatz. One can use linear combinations or commutators of the operators $Y_1$, $Y_2$ and $Y_3$. We consider the symmetry operator $Y_2-Y_3$. Then the corresponding non-evolutionary differential equation takes the form

By using symmetry properties we integrate Eq (3.5) and thus obtain the ansatz

where $\omega = x^{3/2}$, $k(t) = \frac{\sqrt{2}}{3\sqrt{3(t+1)}}$, and $\varphi_1(t)$ and $\varphi_2(t)$ are unknown functions, which reduces (3.6) to the system of ODEs

The application of the same method to non-evolutionary nonlinear hyperbolic equation

with $F$ being an arbitrary function of $u_{x_1}-u$, yields the solution

parameterized by an arbitrary function $\varphi_1(x_2)$. This solution is obtained by using the generalized infinitesimal symmetries $Q_1 = u_{x_1x_2}\partial_u$ and $Q_2 = u_{x_1}F(u_{x_1}-u)\partial_u$ of the ODE $u_{x_1x_1}-u_{x_1} = 0$.

It is clear that an ODE generating an ansatz may be nonlinear. For example, the equation

admits the generalized infinitesimal symmetries $Q_1 = \tfrac{u_{x_1x_2}}{u_{x_1}^2}\partial_u$ and $Q_2 = F(u+\ln u_{x_1})\partial_u$, where $F$ is an arbitrary function of $u+\ln u_{x_1}$. Thus, it admits the generalized infinitesimal symmetry $Q = Q_1-Q_2$ as well. The general solution of (3.11) gives the ansatz

where $\Phi_1(x_2)$ and $\Phi_2(x_2)$ are arbitrary smooth functions, which reduces the nonlinear PDE

to the single ODE

From this we obtain the family of solutions of (3.13)

which is parameterized by the arbitrary smooth function $\Phi_2(x_2)$. Therefore, the method allows us to construct solutions, which contain arbitrary functions. We recall that second order in time nonlinear partial differential equations, which have only one higher symmetry and have a solution depending on one arbitrary function were called partially integrable equations in ^[30].

Note that setting $F(u+\ln u_{x_1}) = {\rm e}^{-2(u+\ln u_{x_1})}$ in (3.13), we obtain the equation $u_{x_1x_2} = {\rm e}^{-2u}$ which is transformed to the classical Liouville equation $v_{xt} = {\rm e}^v$ by means of the change of variables $v = -2u$, $x = -2x_1$, $t = x_2$.

It is obvious that Eq (3.11) is invariant with respect to the Lie group of translations in $t$, which is generated by the vector field $Q_3 = u_t\partial_u$. Therefore, the generalized vector field $Q_3+Q_1-Q_2$ is a generalized infinitesimal symmetry of Eq (3.11). Then the ansatz

reduces the partial differential equation with three independent variables of evolution type

to the system of two partial differential equations with two independent variables

At the same time we do not obtain such reduction for the non-evolutionary equation

Note that the generalized vector field $Q_3 = u_{tt}\partial_u$ is not an infinitesimal symmetry of Eq (3.11). In ^[16], it was shown that the reduction method can be applied to partial differential equations of the form $u_{tt} = Q[u]$ or the more general form $T[u] = Q[u]$, where $T$ is a linear ordinary differential operator in $t$. Each of ansatzes used in ^[16] in this case to reduce partial differential equations is a solution of a linear ordinary differential equation whose coefficients do not depend on $t$ and, therefore, the generalized vector fields $u_{tt}\partial_u$ and $T[u]\partial_u$ are the generalized symmetries of this equation. These facts substantiate the reduction of considered partial differential equations $u_{tt} = Q[u]$ and $T[u] = Q[u]$.

The above examples demonstrate rather strikingly that the method can be applied to non-evolutionary type PDE. The main idea of applying the method to evolutionary equations can be illustrated by the example of the Korteweg–de Vries equation. One can verify that the equation

is invariant with respect to the generalized vector field $X = (u_{xxx}+uu_x)\partial_u$. Note that we can obtain any desired coefficients for the terms of Eq (1.5) by rescaling dependent variable and independent variables $t$ and $x$. We have proved that the Eq (3.19) is invariant with respect to a two-parameter group of contact transformations. A basis of the corresponding Lie algebra consists of the generalized vector fields

where $h_1$ and $h_2$ are arbitrary smooth functions. The advantage of using the approach lies in the fact that a linear combination $\alpha_1X+\alpha_2Q_1+\alpha_3Q_2$ with arbitrary real constants $\alpha_1$, $\alpha_2$ and $\alpha_3$ is also a symmetry operator of (3.19) and, therefore, the method can be applied to the nonlinear differential equation

Then the ansatz

generated by (3.19) reduces the PDE (3.22) to the system of two ODEs

Equation (3.22) in the general form is invariant with respect to the two-dimensional Lie algebra $\langle\partial_t, \partial_x\rangle$. Since the derivatives $\frac{\partial u}{\partial \varphi_1}$ and $\frac{\partial u}{\partial \varphi_2}$ are linearly independent, one can easily see that among solutions of (3.22) constructed with the ansatz (3.23), there are those that are invariant with respect to no one-parameter Lie group with generator in $\langle\partial_t, \partial_x\rangle$. Hence we conclude that the method enables us to construct solutions to equations from class (3.22) that are not invariant in the classical Lie sense.

3.2.   The classes of evolutionary equations for which the method can be applied

Next we consider a nonlinear evolutionary equation which can be used for describing diffusion processes in heterogeneous medium and for which the method can be applied. We are looking for the second-order ODEs of the form

(which belong to the class (1.2)) admitting a generalized symmetry with a generator of the form $X = \big(\tfrac{H(x)}{u}\big)_{xx}\partial_u$ corresponding to the right hand side of our diffusion equation. Note that the Eq (3.25) is invariant with respect to the translation group $t' = t+a$, where $a$ is a group parameter and therefore it admits the symmetry operator $X_1 = u_t\partial_u$. From this it follows that the method is applicable to any equation from the class $k_1u_t+k_2\big(\tfrac{H(x)}{u}\big)_{xx} = 0$, where $k_1, k_2$ are arbitrary constants. We assume $H$ and $U$ are some sufficiently smooth functions and $H$ is nonzero. Function $U$ should satisfy the determining equation

as stated in Definition 1, for a given function $H$, with $L$ being the set of all differential consequences of Eq (3.25) with respect to the variable $x$. For the sake of being able to split the equation above into an overdetermined system of differential equations, we restrict our search to a function $U = \sum_{i, j\in\mathbb{Z}}A_{ij}(x)u^iu_x^j$, that is a power series in both $u$ and $u_x$. We focus not a complete classification but rather particular $H$ and $U$ for which this method may produce nonclassical solutions. Due to a great number of determining equations and cases to consider we will move past the explanation of calculations and present some of the results in the following proposition.

Proposition 1. An equation $u_{xx} = U(x, u, u_x)$ admits the LBS operator$X = \big(\frac{H(x)}{u}\big)_{xx}\partial_u$if H(x) and U have the form

a) $H(x) = \mathrm{e}^{\beta x}$, $U = 3\tfrac{u_{x}^2}{u}-3\beta u_{x}+\beta^2u$,

b) $H(x) = (x+\gamma)^{\alpha}$, $U = 3\tfrac{u_{x}^2}{u}+\tfrac{3-3\alpha}{x+\gamma}u_{x}+\tfrac{(\alpha-2)(\alpha-1)}{(x+\gamma)^2}u$,

c) $H(x) = (x+\gamma)^{\alpha}$, $U = 3\tfrac{u_{x}^2}{u}+\tfrac{1-3\alpha}{x+\gamma}u_{x}+\tfrac{(\alpha-2)\alpha}{(x+\gamma)^2}u$,

d) $H(x) = (x+\gamma)^{-2}$, $U = 3\frac{u_{x}^2}{u}+\tfrac{7}{x+\gamma}u_{x}+\tfrac{8}{(x+\gamma)^2}u$,

e) $H(x) = (x+\gamma)^{-2}$, $U = 3\tfrac{u_{x}^2}{u}+\tfrac{10}{x+\gamma}u_{x}+\tfrac{15}{(x+\gamma)^2}u$.

Terms $\alpha$, $\beta$, $\gamma$ are all constants.

It must be noted that because of the said restrictions this is not a complete classification. Solutions of ODEs can be used as ansatzes which produce a reduction of the equation

to a system of two ODEs. Equation (3.26) can be written in the form

where $D(x, u) = -\tfrac{H(x)}{u^2}$, $P(x, u) = -\tfrac{H'(x)}{u^2}$, $Q(x, u) = \tfrac{H''(x)}{u}$. Assumption of constancy $D$ is not always justified – often the diffusion coefficient depends on the concentration of the diffusant, concentration gradient, spatial coordinate and time of the diffusion experiment (and sometimes - from all these parameters together). In ^[31] the reaction-diffusion equation of the typical form

is considered. $u = u(x, t)$ is a state variable and describes density/concentration of a substance, a population at point $x$: $x\in \Omega\subset\mathbb{R}^n$ ($\Omega$ is an open set). $\Delta$ denotes the Laplace operator. The second term, $f(u)$ is a smooth function, and describes processes with really "change" the present $u$, i.e., something happens to it (birth, death, chemical reaction), not just diffuse in the space. It is also possible, that the reaction term depends not only on $u$, but also on the first derivative of $u$, i.e., $\nabla u$ and/or explicitly on $x$. Instead of a scalar equation, one can also introduce systems of reaction diffusion equations, which are of the form

where $u(x, t)\in \mathbb{R}^m$ ^[31]. The system of equations

is used in ^[32] to describe the densities of two biological species, considering nonlinear movements of the individuals of populations. Here $K(x)$ is environmental potential describing the heterogeneity of environment. Equations (3.26) and (3.27) belong to the class of Eq (3.29) and the method is applicable to any equation from this class or (3.30). Note that Eq (3.26) can be used for describing fast diffusion in heterogeneous medium if $H(x)\not = const$ and homogeneous medium if $H(x) = const$. We use the fact that the equation

also allows reduction with the same ansatz, for $\eta$ such that $\eta\partial_u$ is the symmetry of the ODE. Note that if we take $H = -1$ and $\eta = 0$ then we get the well-known equation describing nonlinear diffusion processes, which possesses an infinite generalized symmetry. We consider separate cases from Proposition 1 depending on the type of heterogeneity of the medium $H(x)$ and the ODE (which is not unique for the choice of $H$), calculate the full contact symmetry of the ODE and obtain the reduced equations. From now on $A_i$ are arbitrary smooth functions on their two arguments. We put $\gamma = 0$ in these cases.

(i) When $H(x) = {\rm e}^{\beta x}$ and

the most general formula for $\eta(x, u, u_x)$ is $\eta = \sum_{i = 1}^2 {{\rm e}^{-i\beta x}}u^3 A_i \left(2\frac{{\rm e}^{\beta x}}{u^3}\Big(u-\frac{u_x}{\beta}\Big), \frac{{\rm e}^{2\beta x}}{u^3}\Big(2\frac{u_x}{\beta}-u\Big) \right)$. From what already has been shown it follows that the ansatz

which is general solution of (3.31), reduces the equation

to the system of ODEs

(ii) When $H(x) = x^{\alpha}, \; \alpha\neq 0$, and $u_{xx} = 3\tfrac{u_x^2}{u} +\tfrac{3-3\alpha}{x} u_x+ \tfrac{(\alpha-2)(\alpha-1)}{x^2}u$ we obtain, in a similar way, the equation

and the ansatz

Substituting (3.36) into (3.37) yields the reduced system

Analysis similar to that in the first two cases gives the following results:

(iii) When $H(x) = x^{\alpha}, \; \alpha\neq -2$ and $u_{xx} = 3\tfrac{u_x^2}{u} +\tfrac{1-3\alpha}{x}u_x +\tfrac{(\alpha-2)\alpha}{x^2}u$, the equation

is reduced by the ansatz

to the system

(iv) When $H(x) = x^{-2}$ and $u_{xx} = 3\tfrac{u_x^2}{u} +\tfrac{7}{x} u_x+ \tfrac{8}{x^2}u$, the equation

is reduced by the ansatz

to the system

(v) When $H(x) = x^{-2}$ and $u_{xx} = 3\tfrac{u_x^2}{u} +\tfrac{10}{x} u_x+ \tfrac{15}{x^2}u$, the equation

is reduced by the ansatz

to the system

We calculated all $\eta(x, u, u_x)$ for which the method can be applied. In general, $\eta(x, u, u_x)$ is a nonlinear function on its arguments. Note that the nonlinear terms are essential in realistic models ^[33]. In special cases we can obtain from $\eta(x, u, u_x)$ the terms describing external forces, absorbent rate, sources or sinks and convection (advection) processes. We show that the well known nonlinearity $\lambda\, u\ln u$ ^[34] can be obtained in the framework of the approach by using the stationary solution of equation

Indeed, it is obvious that the vector field $Q = \big(\tfrac{u_x}{u}\big)_{x}\partial_u$ is the generalized symmetry of ordinary differential equation

The solution of (3.39) satisfies condition $u_t = 0$ and is the stationary solution of (3.38) The contact symmetry of (3.39) is given by vector field $Q_1 = \eta_1\partial_u = (x\, u\, F_1(\omega_1, \omega_2)+u\, F_2(\omega_1, \omega_2))\partial_u$, where $F_1$, $F_2$ are arbitrary smooth functions on two variables $\omega_1 = \tfrac{u_x}{u}$, $\omega_2 = \ln u- \tfrac{xu_x}{u}$. Imposing conditions $\tfrac{\partial\eta_1}{\partial x} = 0$ and $\tfrac{\partial\eta_1}{\partial u_x} = 0$ yields $F_1 = \lambda \omega_1$, $F_2 = \lambda\omega_2$, where $\lambda$ is an arbitrary real constant. Then we have the diffusion equation

This equation possessing point nonclassical symmetry has been obtained in ^[34]. If $F_1 = \lambda \omega_1$, $F_2 = \lambda\omega_2+h(\omega_1)$, where $h$ is a smooth function then we obtain the generalization of (3.40)

The ansatz

generated by the stationary solution of (3.38) reduces Eq (3.41) to system of two ODEs

Applying the method to the two-component diffusive system we have

Next consider the equation

when $H(x) = 1$. It is obvious that the vector field $Q = \big(\tfrac{1}{u}\big)_{xx}\partial_u$ is the generalized symmetry of ordinary differential equation

This equation is also invariant with respect to the Lie group of point symmetry given by vector field $Q_2 = (ku^2+k_1u+u_x(k_2+\tfrac{k_3}{u}))\partial_u$, where $k, k_1, k_2, k_3$ are real constants. From the stationary solution of (3.43) we obtain the ansatz

which reduces nonlinear diffusion equation

to the system of ordinary differential equations

In the framework of this approach one can obtain the two-component diffusive system

where $k_4, k_5, k_6, k_7, k_8, k_9$ are real constants. Note that the method based on stationary solutions, in general, is not applicable to the non-evolutionary equations. Although the approach described here is purely mathematical, however, it is sufficiently general to be applied to differential equations used in various fields of science including biological. For example, the diffusion coefficients which are used in ^[35] and are proportional to $C^{\beta}z^m$, where the values of $\beta$, $m$ may be positive, negative or zero, $z$ is the scaled height and $C$ is the scaled concentration of the diffusing material, closely correlate with results described in (ii), (iii), (iv) and (v). There is a plausible physical justification for an inverse relation between the diffusion coefficient and the concentration $C$ ^[35]. The relevant nonlinear differential equation is demonstrated as a model of turbulent dispersion in the atmosphere in ^[35], thermal wave propagation in plasma physics and fluid flow in porous media ^[36]. The approach can be also applied to the systems of Lotka–Volterra type which are used in mathematical biology ^[27] and equations describing fast diffusion processes. Here we use the method which rather belong to nonclassical symmetry approach since it provides solutions that cannot be obtained by using the classical Lie group method. In general $A_i$ may depend on parametric variable $t$ and the corresponding diffusion equation may be used for simulation of nonstationary media. It is obvious that the method is applicable to the ODE $(\tfrac{H}{u})_{xx}+\eta(x, u, u_x) = 0$. The corresponding ansatz reduces an ODE to a system of algebraic (not differential) equations in this case. For example ansatz (3.32) reduces the ODE

to the system of two algebraic equations

One can easily choose such $A_1$, $A_2$ that the system (3.47) will not have solutions. In general, the method ensures the reduction of ODEs to a system of algebraic equations but does not guarantee the existence of even one solution of reduced system and consequently solution of overdetermined system given by the ODE under study and the ODE possessing the corresponding generalized symmetry. It means that the ordinary differential equation is reduced to the system of algebraic equations but the appropriate overdetermined system of ordinary differential equations is not compatible.

3.3.   How to construct solutions using a $2^{{nd}}$ order ODE

In this subsection we show the application of the method for finding solutions and obtain the solutions which cannot be constructed by the classical Lie method for the modified diffusion equations. We modify our original diffusion equation by some selected characteristics of point symmetries representing some physical properties of nonlinearity on heterogeneity. Let's start with the equation

The ansatz is

We will consider equation

which is obtained from (3.33) by letting

where $I_1 = 2\frac{{\rm e}^{\beta x}}{u^3}\Big(u-\frac{u_x}{\beta}\Big)$, $I_2 = \frac{{\rm e}^{2\beta x}}{u^3}\Big(2\frac{u_x}{\beta}-u\Big)$. By choosing only those five of all eight characteristics of the symmetry operators it is easy to solve the reduced equations. Note, that the Eq (3.48) can be linearized using a point change of variables, but we don't do such thing, because it would change also the term $\Big(\frac{\mathrm{e}^{\beta x}}{u}\Big)_{xx}$. The property of reduction for Eq (3.50) is not limited to linear ODEs, it is also valid for nonlinear ODEs. For example, Korteweg–de Vries equation admits reduction with an ansatz obtained from a solution to an ODE which is an arbitrary linear combination of higher-order symmetries. One can also use any linear combination of higher-order symmetries of KdV to produce the ansatz. After substituting the ansatz (3.49) into (3.50) we obtain reduced equations:

Now, the evolutionary equation (3.50) admits the following symmetry operators:

Depending on the choice of $a_i$ the reduced system is integrable and we can construct the solutions to the modified evolutionary equation.

For $a_2\neq0$, $\gamma = \pm\sqrt{(a_2\beta+2a_1)^2- 4\beta^2a_3}$, $\gamma\neq 0$, $\delta = \tfrac{ a_1 a_5 - a_4 \beta + a_2 a_5 \beta}{a_1^2 + a_1 a_2 \beta - a_3 \beta^2}$, $a_1^2 + a_1 a_2 \beta - a_3 \beta^2\neq 0$ the reduced system (3.51) and (3.52) has the following solution:

Substituting $\varphi_1$, $\varphi_2$ into (3.49) gives the solution of (3.50) with the restrictions above.

The construction of solutions in the rest of the cases runs as before.

The imposition of condition $a_2 = 0$ on (3.50) slightly simplifies the reduced system as well as its solution, which then is:

$\gamma$ now representing $\pm2\sqrt{a_1^2-\beta^2a_3}\neq 0.$ Equating in (3.50) a nonzero term $a_3$ to $(\tfrac{a_2}{2}+\tfrac{a_1}{\beta})^2$ leads to quite a different solution:

Similarly, putting in (3.50) $a_2 = 0$ and $a_3 = \tfrac{a_1^2}{\beta^2}$ results in solutions:

Lastly, for $a_1 = 0$ and $a_2+\beta a_3 = 0$ in (3.50):

When $\exists_{i\in\{4, 5\}}:a_i\neq 0$ and $\exists_{j\in\{1, 2, 3, 4, 5\}, j\neq i}:a_j\neq 0$ the maximal Lie invariance algebra of the Eq (3.50) is one-dimensional and is spanned by $\partial_t$. The presented solutions are clearly not invariant under translations of the variable $t$. When exactly one of the constants $a_4$, $a_5$ is nonzero and $a_1 = a_2 = a_3 = 0$ or $a_3$ is nonzero and both $a_4$ and $a_5$ are zeros, the maximal Lie invariance algebra of the Eq (3.50) is two-dimensional, and invariance under one-parameter symmetry group with the generator $\alpha_1Y_1+\alpha_2Y_2$ must be verified from the definition, that is solution $u = u(x, t)$ is invariant when there exist real numbers $\alpha_1, \; \alpha_2$, at least one nonzero, that $(\alpha_1Y_1+\alpha_2Y_2)(u-u(x, t))\big|_{u = u(x, t)} = 0$. Otherwise, the solution is not invariant. Instead of checking the invariance by the definition, we will compare them with the invariant solutions in the class (3.49). Functions $\varphi_i$ for invariant solutions are as follows.

For $a_3\neq 0$, $a_4 = a_5 = 0$

for $a_4\neq 0$, $a_1 = a_2 = a_3 = a_5 = 0$

and for $a_5\neq 0$, $a_1 = a_2 = a_3 = a_4 = 0$

where $c_1$, $c_2$, $\alpha_1$, $\alpha_2$ are all constants. By plain comparison, in those 3 cases, none of the five solutions in general form obtained from the reduced equations is invariant under $\alpha_1Y_1+\alpha_2Y_2$ ($Y_2$ depending on the choice of nonzero $a_i$).

One does not always have to solve the reduced equations to determine if the solution is or isn't invariant. Let's for example take equation

which is obtained from (3.33) by letting

It admits the same ansatz as in the previous example,

and is reduced to a system

The Eq (3.53) possesses three symmetry operators when $a_6 = a_7 = 0$ and only one symmetry operator ($\partial_t$) when both $a_6\neq 0$ and $a_7\neq 0$. Let's consider the case $a_6 = 0$, $a_7\neq 0$ with two-dimensional Lie algebra. Here $Q_1 = \partial_t, \; Q_2 = t\partial_t+\tfrac{1}{2}u\partial_u$ is the basis of the algebra for Eq (3.53) (with $a_6 = 0$). Invariance criterion in terms of functions $\varphi_i$ after splitting with respect to the powers of $\mathrm{e}^{\beta x}$ is

Reduced equations do have an explicit solution but it is invariant, because $\varphi_1 = 0$. They also have an implicit solution. At this point all we need is to solve the reduced equations for the derivatives of $\varphi_i$ and substitute those into the system above. The result is another system,

The two equations are very similar. After dividing the $i$th equation by $\beta\varphi_1\varphi_i$ and subtracting one from another we have

meaning

so the Eq (3.54), where both $\varphi_i$ are nonzero solutions of the reduced equations, wouldn't be an invariant solution.

Now we will find the solution to the system of the reduced equation. But firstly, we will show how to find a symmetry of the reduced equations having the symmetry of the PDE. We consider a case, when the symmetry operator $Q$ of the PDE is admitted also by the ODE (3.48) (we treat the ODE as a PDE that does not include the derivative $u_t$) and condition

holds, where $Q^{[1]}$ is the first prolongation of operator $Q$, $f_1(I_1, I_2)$, $f_2(I_1, I_2)$ are arbitrary smooth functions,

are the first integrals of Eq (3.48). For an independent variable we impose the condition

where $m(t)$ is arbitrary smooth function. One can construct the symmetry operator for reduced system in the form $\tilde{Q} = m(t)\partial_t+f_1(\varphi_1, \varphi_2)\partial_{\varphi_1}+f_2(\varphi_1, \varphi_2)\partial_{\varphi_2}$ if the conditions (3.55) and (3.56) are fulfilled. Obviously, we obtain a nontrivial symmetry for a system of reduced equations, if $f_1$, $f_2$ and $m$ are not identically zeros. It's pretty obvious that $Q_1$ i $Q_2$ satisfy all the conditions, namely for $Q_1$ we have $f_1 = 0$, $f_2 = 0$ and $m = 1$, and for $Q_2$ we have $f_1 = -I_1$, $f_2 = -I_2$ and $m = t$. Thus we conclude that the ODE system

is invariant with respect to 2-parameter Lie group of point transformations whose Lie algebra is given by basic elements

which are obtained from $Q_1$, $Q_2$. It means that the system is solvable in quadratures. A point transformations

maps the symmetry operators $X_1, X_2$ into

The transformed system can be simplified into

We can easily solve for $Z$,

which in initial coordinates is an algebraic equation

Solution

is trickier to utilize once going back to the original coordinates. If we were to use $W_T = \tfrac{2}{{\rm e}^Z\beta^2T^2}$, this equation would imply a differential equation

that can be alternatively written as

The only invariant solution of the form $u(x, t) = \pm\tfrac{{\rm e}^{\beta x}}{\sqrt{\varphi_1 {\rm e}^{\beta x}+\varphi_2}}$ is the one where $\varphi_i = \tfrac{c_i}{\alpha_1+\alpha_2t}$, $c_i, \alpha_j = {\rm const}$, $i, j = 1, 2$. In that case $\tfrac{\varphi_1}{\varphi_2} = \tfrac{c_1}{c_2} = {\rm const}$, and it is visible that such $\varphi_i$ do not satisfy Eqs (3.57) to (3.60).

If we take $a_8 = -\beta$, then

but most importantly the integral in (3.58) can be easily calculated,

From this we obtain

Taking the equations above into account, we see that

Because $W = t$, $\varphi_2 = {\rm e}^Z$, $\varphi_1 = T\varphi_2$, the solution of the reduced system in original coordinates is

The solutions we constructed have the property that corresponding solutions $\varphi_1$, $\varphi_2$ are not (identically) zero. At least we can say that there exist values of $a_8$ such that the proposed method allows us to construct non-invariant solutions, i.e., solutions that cannot be obtained by the classical Lie method. In conclusion, we have found $\varphi_i$'s for which the solution $u(x, t) = u(x, \varphi_1(t), \varphi_2(t))$ is not invariant under the point symmetries of the evolution equation. Consider the equation

A basis of its Lie algebra $A_3$ of point symmetries consists of the vector fields

The solution to the reduced equations

is

We call ansatz (3.54) together with solutions (3.64) a particular solution of (3.61).

Theorem 1. Any particular solution of (3.61) given by (3.54) and (3.64) is invariant with respect to a one-parameter Lie invariance group of (3.61).

Proof of Theorem 1. Note that in the general case, the derivatives $\tfrac{\partial u}{\partial \varphi_1}$ and $\tfrac{\partial u}{\partial \varphi_2}$ are linearly independent. Otherwise, for some $\beta_1$, $\beta_2$, equation $\beta_1\tfrac{\partial u}{\partial \varphi_1}+\beta_2\tfrac{\partial u}{\partial \varphi_2} = 0$ would be true, and therefore $u$ would be dependent on only one constant,

which is impossible, because it is the general solution of second-order ODE (3.48) Moreover the derivatives $\tfrac{\partial u}{\partial c_1}$ and $\tfrac{\partial u}{\partial c_2}$ are linearly independent since the pair ($\varphi_1$, $\varphi_2$) is the general solution of (3.63) and has the form (3.64).

The action of the operator $X = \xi_j(x, u)\tfrac{\partial}{\partial x_j}+\eta(x, u)\partial_u$, is the following

where $x = (x_1, \ldots, x_n)$ for some integer $n$, $h(x)$ is a differentiable function. Then we show that

where $W_2 = \mathrm{span}\{\tfrac{\partial u}{\partial c_1}, \tfrac{\partial u}{\partial c_2}\}$ and $Q\in A_3$. To prove this, it is enough to show this property for each of the basis elements $X_1$, $X_2$, $X_3$.

By direct computation we show that for $X_1 = \partial_t$

for $X_2 = 2t\partial_t+u\partial_u$

and for $X_3 = 2\partial_x+\beta u\partial_u$

where $f(x, t)$ is the solution of (3.61) given by (3.54) and (3.64).

From the fact that any three vectors in two-dimensional vector space are linearly dependent, it follows that for any special solution (3.54) and (3.64) there can be selected $\alpha_1$, $\alpha_2$, $\alpha_3$ such that $X \big(f(x, t)-u)\big)\big|_{u = f(x, t)} = 0$, where $X = \alpha_1X_1+\alpha_2X_2+\alpha_3X_3$ and not all $\alpha_i$ are equal to zero.

We conclude that every solution given by (3.54) and (3.64) can be found using a classical method of invariant solutions with respect to a one-parameter Lie group $\alpha_1X_1+\alpha_2X_2+\alpha_3X_3$.

This theorem is only sufficient but not necessary condition for a solution to be invariant in a classical sense. In fact, let us consider the two-dimensional abelian subalgebra of Lie algebra $A_3$, with basis elements $Q_1$, $Q_2$, where

(these operators clearly commute, $\left[Q_1, Q_2\right] = 0$). Because

and

the solution $u = f(x, t)$ is invariant with respect to a linear combination $Q_1-\beta Q_2$. On the other hand, it is obvious that this solution cannot be obtained with a classical method using just any two-dimensional subalgebra, because not every two-dimensional algebra has the aforementioned properties, for example $\{X_1, X_3\}$. No nonzero linear combination of $X_1$ and $X_3$ leaves the solution invariant.

4.   Discussion

We have constructed solutions of nonlinear evolution equations which can be used for describing the diffusion processes in heterogeneous medium by using the method based on the generalized symmetry of ODEs ^[17]. We show that the method gives us the possibility to obtain solutions which are not invariant ones in the classical Lie sense. We use the generalized symmetry of the second-order ODEs. The corresponding ansatzes reduce nonlinear diffusion equations to systems of two ODEs. One can obtain the solutions which can not be constructed by the classical Lie method in the cases when the dimension of the invariance Lie algebra is equal to 1 or 2. When the Lie algebra of the Lie invariance group of studied diffusion equation is three-dimensional then the solutions obtained by using our method could also be obtained via the classical Lie symmetry method as follows from Theorem 1. We have found wide classes of diffusion-type equations (and exact solutions) for which this method can be used. It is also shown that the method is applicable to non-evolutionary PDEs and enables us to construct solution depending on an arbitrary function for nonlinear hyperbolic (wave-type) equation. In fact, we show that the approach extends the applicability of the symmetry method for constructing exact solutions to PDEs. Exact solutions of differential equations are useful for understanding physical phenomena described by these equations and testing approximate and numerical methods for solving them.

Acknowledgments

This study received no funding.

Conflict of interest

The authors declare there is no conflict of interest.