Group Classification and Symmetry Reduction of a (1+1)-Dimensional Porous Medium Equation

Abstract

In this paper, we present Lie symmetry analysis of a generalized (1+1)-dimensional porous medium equation characterized by parameters m and d. Through group classification, we examine how these parameters influence the Lie symmetry structure of the equation. Our analysis establishes conditions under which the equation admits either a three-dimensional or a five-dimensional Lie algebra. Using the obtained symmetry algebras, we construct optimal systems of one-dimensional subalgebras. Subsequently, we derive invariant solutions corresponding to each subalgebra, providing explicit formulas in relevant parameter regimes. These solutions deepen our understanding of the nonlinear diffusion processes modeled by porous medium equations and offer valuable benchmarks for analytical and numerical studies.

1. Introduction

The study of Lie symmetries in partial differential equations (PDEs) has long been recognized as a powerful analytical framework for understanding complex physical phenomena [1,2,3,4,5,6,7,8]. The Lie symmetry method, pioneered by Sophus Lie in the late 19th century, provides a systematic approach to identifying transformations that leave differential equations invariant, facilitating their reduction to simpler forms and often leading to exact solutions. This approach offers deep insights into the structural properties of PDEs and their solution spaces. Over the years, symmetry methods have been successfully applied to a wide range of nonlinear equations arising in fluid dynamics, heat transfer, and diffusion processes [9,10]. For instance, Hussain and Usman [11] applied symmetry classification to the Barenblatt–Gilman equation, a nonlinear PDE modeling nonequilibrium countercurrent capillary impregnation, deriving a system of one-dimensional subalgebras and group-invariant solutions. Their work highlights the power of symmetry reductions in addressing complex nonlinear problems, providing analytical solutions that serve as benchmarks for numerical studies.

A notable class of diffusion equations that has attracted extensive attention is the family of porous medium equations (PMEs), which arise in various physical and biological contexts, such as groundwater flow, gas dynamics, and heat conduction in materials with memory effects. Sophocleous and Tracinà [12] investigated a variable coefficient Burgers system, a related nonlinear PDE, using Lie symmetry analysis to reduce the system to ordinary differential equations and derive exact solutions, including those arising from hidden symmetries. Similarly, Charalambous and Sophocleous [13] performed a Lie group classification for a class of compound KdV–Burgers equations with time-dependent coefficients. These studies emphasize the adaptability of symmetry methods to PDEs with variable coefficients, a feature relevant to the parameter-dependent PME considered here. Additionally, Sinkala [14] revisited the group classification of the general nonlinear heat equation $u_t={\left(K\left(u\right)u_x\right)}_x$, systematically deriving admissible forms of $K\left(u\right)$ and their corresponding Lie symmetry algebras.

As presented in Aronson and Graveleau [15], the flow of an ideal gas through a homogeneous porous medium, under isentropic conditions, is described by a nonlinear diffusion process. Let $u=u(t,z)$ denote the gas density, where $z=(z_1,\dots ,z_d)\in {\mathbb{R}}^d$ and $t\in \mathbb{R}$. The governing equation for this process is the porous medium equation

$${\displaystyle \frac{\partial u}{\partial t}}=\Delta \left(u^m\right),$$

(1)

where $\Delta ={\sum }_{j=1}^d{\partial }^2/\partial z_j^2$ is the Laplacian operator and m is a constant parameter characterizing the nonlinearity of the medium. This equation arises in various physical contexts, including heat conduction in plasma, groundwater infiltration, and gas flow in porous materials.

In porous medium flows, the pressure v is typically related to the density u through the equation of state for an ideal gas. Specifically, due to the isentropic assumption, the pressure is proportional to a power of the density:

$$v={\displaystyle \frac{m}{m-1}}{u}^{m-1}.$$

(2)

By substituting this relation into the porous medium Equation (1), one obtains the so-called pressure formulation of the flow, which governs the evolution of v:

$${\displaystyle \frac{\partial v}{\partial t}}=(m-1)v\Delta v+{|\nabla v|}^2.$$

(3)

This pressure equation provides an alternative yet equivalent representation of the original porous medium equation.

To further simplify the analysis and reduce the problem to a one-dimensional setting, we consider radially symmetric solutions of the form $v=v(t,x)$, where $x=\left|z\right|$. In this case, the Laplacian and gradient operators simplify accordingly, and Equation (3) reduces to

$${\displaystyle \frac{\partial v}{\partial t}}=(m-1)v\left({\displaystyle \frac{{\partial }^{2}v}{\partial x^2}}+{\displaystyle \frac{d-1}{x}}{\displaystyle \frac{\partial v}{\partial x}}\right)+{\left({\displaystyle \frac{\partial v}{\partial x}}\right)}^2.$$

(4)

This radially symmetric pressure equation (4) is the focus of our study. It captures the essential features of nonlinear diffusion in a spherically symmetric setting and serves as a rich model for investigating the role of symmetry in the analysis and classification of exact solutions.

Bluman and Kumei [2] presented a comprehensive framework for applying Lie symmetry methods to nonlinear diffusion equations, demonstrating how symmetry reductions lead to physically meaningful and analytically tractable solutions. Studies of PMEs through Lie symmetry methods, such as those by Barenblatt [16] and Vázquez [17], have provided rich structures of self-similar solutions and offered significant insights into the evolution of diffusion processes. Recent research has expanded the application of symmetry methods to generalized forms of the PME containing arbitrary parameters. For instance, Avdonina and Trachevsky [18] investigated conservation laws and similarity solutions for diffusion equations with nonlinear source terms, while Antontsev and Diaz [19] examined the effects of variable exponents on the symmetry structure of PMEs. Similarly, Mbusi et al. [20] applied Lie point symmetry analysis combined with an ansatz method to a generalized (1+1)-dimensional system, deriving novel exact solutions, including solitary, cnoidal, and snoidal waves, and constructing conservation laws.

Classification of Lie algebras associated with a given differential equation is a key aspect of Lie symmetry analysis. Ovsiannikov [4] developed a systematic procedure for group classification that enables the identification of parameter-dependent symmetry structures. Popovych and Ivanova [21] applied this technique to nonlinear diffusion equations, showing how specific parameter values lead to the emergence of additional symmetries, a methodology we adopt in our classification of the PME. Nadjafikhah [22] applied similar techniques to the inviscid Burgers’ equation.

The construction of optimal systems of subalgebras has played a crucial role in classifying group-invariant solutions. Olver [5] and Amata et al. [23] propose algorithms for systematically generating optimal systems of Lie subalgebras, enabling the classification of inequivalent subgroups and the derivation of representative invariant solutions. Khalique et al. [24] applied these methods to the modified equal-width equation, constructing an optimal system of one-dimensional subalgebras and deriving exact solutions, as well as conservation laws using the multiplier and Noether approaches. Similarly, Agnus et al. [25] utilized Lie point symmetries to perform similarity reductions of the (1+2)-dimensional Calogero–Degasperis equation, obtaining exact solutions and demonstrating the reduction to solvable ordinary differential equations.

In this paper, we apply Lie symmetry analysis to the radially symmetric pressure formulation of the porous medium equation, given in (4), which depends on arbitrary parameters. We perform a complete group classification of this equation with respect to the parameters m and d, identifying the distinct symmetry structures that arise for different values of these parameters. By conducting a comprehensive group classification, we investigate how the symmetry structure of the equation depends on the parameter’s value. Our analysis reveals that the equation admits either a three-dimensional or a four-dimensional Lie algebra, depending on the specific parameter setting. We then construct optimal systems of one-dimensional subalgebras and use them to obtain invariant solutions. Through this systematic approach, we provide a detailed classification of the equation’s symmetries and corresponding exact solutions, contributing to a deeper understanding of nonlinear diffusion phenomena modeled by the porous medium equation.

The remainder of this paper is organized as follows. In Section 2, we carry out a complete group classification of the porous medium equation with respect to the parameters m and d, identifying the distinct symmetry structures that arise. Section 3 is devoted to constructing optimal systems of one-dimensional subalgebras for each symmetry case, which serve as the foundation for reducing the PDE to ODEs. In Section 4, we apply the method of symmetry reduction to derive group-invariant solutions corresponding to each element of the optimal systems. Finally, in Section 5, we summarize our findings and highlight their relevance to the analysis of nonlinear diffusion phenomena modeled by porous medium equations.

2. Group Classification of Equation (4)

Consider the infinitesimal generator of the Lie group of point transformations for Equation (4), defined as

$$X=\tau (t,x,v){\displaystyle \frac{\partial }{\partial t}}+\xi (t,x,v){\displaystyle \frac{\partial }{\partial x}}+\eta (t,x,v){\displaystyle \frac{\partial }{\partial v}},$$

(5)

where $\tau$, $\xi$, and $\eta$ are the infinitesimals corresponding to transformations in t, x, and v, respectively. For the equation to remain invariant under these transformations, the following condition must hold:

$$X^{\left(2\right)}{\left.\left[v_t-(m-1)v\left(v_{xx}+{\displaystyle \frac{d-1}{x}}{v}_x\right)-v_x^2\right]\right|}_{\left(4\right)}=0,$$

(6)

where $X^{\left(2\right)}$ is the second prolongation of X. Its explicit form

$$X^{\left(2\right)}=X+{\eta }^t{\displaystyle \frac{\partial }{\partial v_t}}+{\eta }^x{\displaystyle \frac{\partial }{\partial v_x}}+{\eta }^{xx}{\displaystyle \frac{\partial }{\partial v_{xx}}}+{\eta }^{xt}{\displaystyle \frac{\partial }{\partial v_{xt}}}+{\eta }^{tt}{\displaystyle \frac{\partial }{\partial v_{tt}}}$$

can be found in [2].

If the derivative eliminated from the Equation (4) is $v_t$, then the invariance condition (6) results into a polynomial in $v_x$, $v_{tt}$, $v_{tx}$, and $v_{xx}$.

By equating the coefficients of this polynomial to zero, we obtain the following system of determining equations for $\tau$, $\xi$, and $\eta$, which we solve to classify the symmetries of the equation.

$$\begin{array}{cc}\hfill {\tau }_v+\lambda v{\tau }_{vv}& =0,\hfill \end{array}$$

(7)

$$\begin{array}{cc}\hfill 2x{\tau }_x+x{\xi }_v+(d-1)\lambda v{\tau }_v+\lambda vx{\xi }_{vv}+(d-1){\lambda }^2{v}^2{\tau }_{vv}+2\lambda vx{\tau }_{xv}& =0,\hfill \end{array}$$

(8)

$$\begin{array}{cc}\hfill {\tau }_v+\lambda v{\tau }_{vv}& =0,\hfill \\ \hfill 3(1-d)\lambda v{\tau }_x+x{\tau }_t-2x{\xi }_x+x{\eta }_v+(2-2d){\lambda }^2{v}^2{\tau }_{xv}-2\lambda vx{\xi }_{xv}& \hfill \end{array}$$

(9)

$$\begin{array}{cc}\hfill +\lambda vx{\eta }_{vv}-\lambda vx{\tau }_{xx}& =0,\hfill \end{array}$$

(10)

$$\begin{array}{cc}\hfill {\tau }_v& =0,\hfill \end{array}$$

(11)

$$\begin{array}{cc}\hfill {\xi }_v+{\tau }_x+\lambda v{\tau }_{xv}& =0,\hfill \\ \hfill (d-1)\lambda v\xi +(d-1)\lambda x\eta +x^2{\xi }_t+\lambda vx{\xi }_x+(d-1)\lambda vx{\tau }_t& \hfill \end{array}$$

(12)

$$\begin{array}{cc}\hfill +(1-d^2){\lambda }^2{v}^2{\tau }_x+2x^2{\eta }_x& =0,\hfill \end{array}$$

(13)

$$\begin{array}{cc}\hfill {\tau }_x& =0,\hfill \end{array}$$

(14)

$$\begin{array}{cc}\hfill \eta x-2vx{\xi }_x+(d-1)\lambda v^2{\tau }_x+vx{\tau }_t-\lambda v^{2}x{\tau }_{xx}& =0,\hfill \end{array}$$

(15)

$$\begin{array}{cc}\hfill (d-1)\lambda v{\eta }_x+x\left(\lambda v{\eta }_{xx}-{\eta }_t\right)& =0,\hfill \end{array}$$

(16)

where $\lambda =m-1$.

From Equations (11) and (14), we deduce that

$$\tau (t,x,v)=\alpha \left(t\right).$$

Substituting this result into Equations (8) and (12), we solve for $\xi$ and obtain

$$\xi (t,x,v)=\beta (t,x).$$

At this point, only Equations (10), (13), (15), and (16) remain. They simplify to

$$\begin{array}{cc}\hfill x{\alpha }_t-2x{\beta }_x+x\left(2{\beta }_x-{\alpha }_t\right)& =0,\hfill \\ \hfill d\lambda vx{\alpha }_t-(d-1)\lambda v\beta -\lambda vx{\alpha }_t-(d-1)\lambda vx\left({\alpha }_t-2{\beta }_x\right)& \hfill \end{array}$$

(17)

$$\begin{array}{cc}\hfill +\lambda vx{\beta }_x-d\lambda vx{\beta }_x+4v{x}^2{\beta }_{xx}+3\lambda v{x}^2{\beta }_{xx}+x^2{\beta }_t& =0,\hfill \end{array}$$

(18)

$$\begin{array}{cc}\hfill \lambda \left(v\left(x{\alpha }_t-2x{\beta }_x\right)-vx\left({\alpha }_t-2{\beta }_x\right)\right)& =0,\hfill \end{array}$$

(19)

$$\begin{array}{cc}\hfill 2(d-1)\lambda v^2{\beta }_{xx}+x\left(v\left({\alpha }_{tt}-2{\beta }_{tx}\right)+2\lambda v^2{\beta }_{xxx}\right)& =0,\hfill \end{array}$$

(20)

respectively.

Solving Equations (17) and (19), we obtain

$$\eta (t,x,v)=v\left(2{\beta }_x-{\alpha }_t\right).$$

With this result, Equations (18) and (20) simplify further to

$$\begin{array}{cc}\hfill x^2{\beta }_t+v\left[3\lambda x^2{\beta }_{xx}+4x^2{\beta }_{xx}+d\lambda x{\beta }_x-d\lambda \beta -\lambda x{\beta }_x+\lambda \beta \right]& =0,\hfill \end{array}$$

(21)

$$\begin{array}{cc}\hfill v\left[x{\alpha }_{tt}-2x{\beta }_{tx}\right]+v^2\left[2d\lambda {\beta }_{xx}-2\lambda {\beta }_{xx}+2\lambda x{\beta }_{xxx}\right]& =0.\hfill \end{array}$$

(22)

Equations (21) and (22) are solved by treating them as polynomials in v. The resulting four determining equations are easily solved, yielding

$$\alpha \left(t\right)=k_1+k_{2}t,$$

$$\beta (t,x)={\displaystyle \frac{k_3{x}^{3-d}}{d^2-5d+6}}+k_4+k_{5}x,d\ne 2,d\ne 3,$$

(23)

provided that

$${\displaystyle \frac{2k_3(dm+d-4m-2)x^{3-d}}{d-3}}+k_4(m-1)(1-d)=0.$$

(24)

How many symmetries Equation (4) admits depends on parameter specifications in Equation (24) that constitute a solution of the equation.

Case 1: m and d arbitrary

For arbitrary values of m and d, Equation (24) is solved by setting $k_3=k_4=0$, which yields

$$\tau =k_1+k_{2}t,\xi =k_{5}x,\eta =v(2k_5-k_2).$$

(25)

In this case, the porous media equation admits a three-parameter Lie group of point transformations, with infinitesimal generators given by

$$X_1={\displaystyle \frac{\partial }{\partial t}},X_2=t{\displaystyle \frac{\partial }{\partial t}}-v{\displaystyle \frac{\partial }{\partial v}},X_3=x{\displaystyle \frac{\partial }{\partial x}}+2v{\displaystyle \frac{\partial }{\partial v}}.$$

(26)

The Lie algebra spanned by the generators in (26) constitutes the principal Lie algebra of the porous media Equation (4).

Remark 1.

In particular, for the cases $d=2$ and $d=3$, the porous media Equation (4) also admits the symmetries in (26).

Case 2: $d=1$ and $m=-{\displaystyle \frac{1}{3}}$

For $k_3$ and $k_4$ arbitrary, Equation (24) is solved when we set $d=1$ and $m=-{\displaystyle \frac{1}{3}}$. This results in a five-parameter Lie group of point transformations with its infinitesimal generators given by

$$\begin{array}{cc}\hfill X_1& ={\displaystyle \frac{\partial }{\partial t}},X_2=t{\displaystyle \frac{\partial }{\partial t}}-v{\displaystyle \frac{\partial }{\partial v}},X_3={\displaystyle \frac{\partial }{\partial x}},\hfill \\ \hfill X_4& =x{\displaystyle \frac{\partial }{\partial x}}+2v{\displaystyle \frac{\partial }{\partial v}},X_5=x^2{\displaystyle \frac{\partial }{\partial x}}+4xv{\displaystyle \frac{\partial }{\partial v}},\hfill \end{array}$$

(27)

that is those in (26) and two additional ones.

3. Optimal System of One-Dimensional Subalgebras of Equation (4)

A well-known standard procedure allows us to classify all the one-dimensional subalgebras into subsets of conjugate subalgebras. This involves using the adjoint representation map, which introduces a conjugate relation in the set of all one-dimensional subalgebras.

For each generator $X_i$, the adjoint action $Ad\left(exp\left(\epsilon X_i\right)\right)$ is defined by the formula

$$Ad\left(exp\left(\epsilon X_i\right)\right)X_j=X_j-\epsilon \left[X_i,X_j\right]+{\displaystyle \frac{1}{2}}{\epsilon }^2\left[X_i,\left[X_i,X_j\right]\right]-\cdots$$

(28)

which introduces an equivalence relationship between elements of the Lie algebra generated by the infinitesimal generators.

3.1. Optimal System of Equation (4) for m and d Arbitrary

The commutation relations of the infinitesimal generators (26) are given in

Table 1. Commutator table for the infinitesimal generators in (26).

$[·,·]$	${\mathit{X}}_1$	${\mathit{X}}_2$	${\mathit{X}}_3$
$X_1$	0	$X_1$	$2X_1$
$X_2$	$-X_1$	0	0
$X_3$	$-2X_1$	0	0

, where the entry in row i and column j represents the commutation relations between the infinitesimal generator $X_i$ and $X_j$ defined by

$$\left[X_i,X_j\right]=X_i{X}_j-X_j{X}_i.$$

(29)

Using the adjoint action defined by (28) and the commutator table (Table 1), we construct the adjoint table (

Table 2. Adjoint table for the infinitesimal generators (26). The entry in row i and column j is $Ad(exp\left(\epsilon X_i\right))X_j$.

Ad $(exp\left(\mathit{\epsilon }{\mathit{X}}_{\mathit{i}}\right)){\mathit{X}}_{\mathit{j}}$	${\mathit{X}}_1$	${\mathit{X}}_2$	${\mathit{X}}_3$
$X_1$	$X_1$	$X_2-\epsilon X_1$	$X_3-2\epsilon X_1$
$X_2$	$e^{\epsilon }{X}_1$	$X_2$	$X_3$
$X_3$	$e^{2\epsilon }{X}_1$	$X_2$	$X_3$

) for the infinitesimal generators (26).

We now consider a general element of the Lie algebra generated by the infinitesimal generators in (26)

$$X=a_1{X}_1+a_2{X}_2+a_3{X}_3,$$

(30)

where $a_i$ are arbitrary constants. Our goal is to simplify this general element as much as possible using adjoint actions under different cases.

Case 3.1 (a): $a_3\ne 0$. Without loss of generality, we scale X such that $a_3=1$. Then,

$$X=a_1{X}_1+a_2{X}_2+X_3.$$

Applying the adjoint actions $Ad(exp\left(\epsilon X_1\right))$, with $\epsilon ={\displaystyle \frac{a_1}{2+a_2}}$, $a_2\ne -2$, we obtain

$$\tilde{X}=Ad\left(exp\left(\epsilon X_1\right)\right)X=a_2{X}_2+X_3$$

No further simplification is possible.

Case 3.1 (b): $a_3=0, a_2\ne 0$. Here, the general element reduces to

$$X=a_1{X}_1+X_2$$

Applying the adjoint actions $Ad(exp\left(a_1{X}_1\right))$, we obtain

$$\tilde{X}=Ad\left(exp\left(a_1{X}_1\right)\right)X=X_2.$$

Case 3.1 (c): $a_3=a_2=0, a_1\ne 0$.

In this case, the general element trivially reduces to $X_1$.

The simplification of a general element of the form (30) under adjoint actions is summarized in the tree diagram below:

Thus, the optimal system of one-dimensional subalgebras is given by

$$\left\{X_1,X_2,\alpha X_2+X_3\right\},$$

(31)

where $\alpha$ is a constant.

3.2. Optimal System of Equation (4) for $d=1$ and $m=-{\displaystyle \frac{1}{3}}$

The commutator table for the infinitesimal generators $〈X_1,X_2,X_3,X_4,X_5〉$ in (27) is given in

Table 3. Commutator table for the infinitesimal generators in (27).

$[·,·]$	${\mathit{X}}_1$	${\mathit{X}}_2$	${\mathit{X}}_3$	${\mathit{X}}_4$	${\mathit{X}}_5$
$X_1$	0	$X_1$	0	0	0
$X_2$	$-X_1$	0	0	0	0
$X_3$	0	0	0	$X_3$	$2X_4$
$X_4$	0	0	$-X_3$	0	0
$X_5$	0	0	$-2X_4$	0	0

.

Using the adjoint action defined in (28) and the commutator relations summarized in Table 3, we derive the adjoint representation table (

Table 4. Adjoint table for the infinitesimal generators (27). The entry in row i and column j is $Ad(exp\left(\epsilon X_i\right))X_j$.

Ad $(exp\left(\mathit{\epsilon }{\mathit{X}}_{\mathit{i}}\right)){\mathit{X}}_{\mathit{j}}$	${\mathit{X}}_1$	${\mathit{X}}_2$	${\mathit{X}}_3$	${\mathit{X}}_4$	${\mathit{X}}_5$
$X_1$	$X_1$	$X_2-\epsilon X_1$	$X_3$	$X_4$	$X_5$
$X_2$	$e^{\epsilon }{X}_1$	$X_2$	$X_3$	$X_4$	$X_5$
$X_3$	$X_1$	$X_2$	$X_3$	$X_4-\epsilon X_3$	${\epsilon }^2{X}_3-2\epsilon X_4+X_5$
$X_4$	$X_1$	$X_2$	$e^{\epsilon }{X}_3$	$X_4$	$e^{-\epsilon }{X}_5$
$X_5$	$X_1$	$X_2$	$X_3+2\epsilon X_4+{\epsilon }^2{X}_5$	$X_4+\epsilon X_5$	$X_5$

) corresponding to the infinitesimal generators listed in (27).

As we did in the previous case, we consider a general element of the Lie algebra generated by the infinitesimal generators in (27).

$$X=a_1{X}_1+a_2{X}_2+a_3{X}_3+a_4{X}_4+a_5{X}_5,$$

where $a_i$ are arbitrary constants, and then simplify this general element as much as possible using adjoint actions under different cases.

Case 3.2 (a): $a_5\ne 0$. Without loss of generality, we scale X such that $a_5=1$. Then,

$$X=a_1{X}_1+a_2{X}_2+a_3{X}_3+a_4{X}_4+X_5.$$

(32)

Applying the adjoint actions $Ad(exp\left({\epsilon }_1{X}_3\right))$ and $Ad(exp\left({\epsilon }_2{X}_4\right))$ in turn, with the parameters ${\epsilon }_1={\displaystyle \frac{a_4}{2}}$, ${\epsilon }_2=ln\left({\displaystyle \frac{1}{\sqrt{\beta }}}\right)$, where $\beta =a_3-{\displaystyle \frac{a_4^2}{4}}$, $a_3>{\displaystyle \frac{a_4^2}{4}}$, we obtain

$$\begin{array}{cc}& \tilde{X}=Ad\left(exp\left({\epsilon }_1{X}_3\right)\right)X=a_2{X}_2+\left(a_3-{\displaystyle \frac{a_4^2}{4}}\right)X_3+X_5\hfill \\ & \tilde{\tilde{X}}=Ad\left(exp\left({\epsilon }_2{X}_4\right)\right)\tilde{X}=a_2{X}_2+\beta \left(X_3+X_5\right),\hfill \end{array}$$

No further simplification is possible.

Case 3.2 (b): $a_5=0,a_4\ne 0$. Here, the general element reduces to

$$X=a_1{X}_1+a_2{X}_2+a_3{X}_3+X_4.$$

Applying the adjoint action $Ad(exp\left(\epsilon X_1\right))$, with $\epsilon ={\displaystyle \frac{a_1}{a_2}}$, $a_2\ne 0$, we obtain

$$\tilde{X}=Ad\left(exp\left(\epsilon X_1\right)\right)X=a_2{X}_2+X_4.$$

No further simplification is possible.

Case 3.2 (c): $a_5=a_4=0, a_3\ne 0$. In this case, the general element is

$$X=a_1{X}_1+a_2{X}_2+X_3.$$

In this case, simplification is achieved using the same adjoint action as in the previous case. We obtain

$$\tilde{X}=Ad\left(exp\left(\epsilon X_1\right)\right)X=a_2{X}_2+X_3,$$

where $\epsilon ={\displaystyle \frac{a_1}{a_2}}$, $a_2\ne 0$, with no further simplification possible.

Case 3.2 (d): $a_5=a_4=a_3=0,a_2\ne 0$. In this case, the general element

$$X=a_1{X}_1+X_2$$

reduces as follows:

$$\tilde{X}=Ad\left(exp\left(a_1{X}_1\right)\right)X=X_2,$$

Case 3.2 (e): $a_5=a_4=a_3=a_2=0,a_1\ne 0$.

In this case, the general element trivially reduces to $X_1$.

The simplification of a general element of the form (32) under adjoint actions is summarized in the tree diagram below:

Thus, the optimal system of one-dimensional subalgebras is given by

$$\left\{X_1,X_2,\alpha X_2+X_3,\alpha X_2+X_4,X_2+\beta \left(X_3+X_5\right)\right\},$$

(33)

where $\alpha$ and $\beta$ are constants.

4. Group-Invariant Solutions for Equation (4)

This section is devoted to constructing exact solutions for two specific cases of Equation (4). We derive the group-invariant solutions through symmetry reductions, using elements of the optimal system of Lie subalgebras. This process involves solving the characteristic equations associated with each symmetry operator in the optimal system to identify similarity variables, thereby reducing the PDE to an ODE. The resulting ODEs are often nonlinear and challenging to solve analytically; however, we obtain general or particular solutions in some cases. The following subsections explore these solutions for arbitrary parameters m and d, as well as the specific case of $d=1$ and $m=-{\displaystyle \frac{1}{3}}$.

4.1. Solutions of (4) for Arbitrary m and d

We use the symmetries $X_1$, $X_2$, and $X_3+\alpha X_2$ from the optimal system (31).

Case 4.1 (a): Symmetry $X_1={\displaystyle \frac{\partial }{\partial t}}$.

The characteristic equations

$${\displaystyle \frac{dt}{1}}={\displaystyle \frac{dx}{0}}={\displaystyle \frac{dv}{0}}$$

(34)

yield invariants $J_1=x$ and $J_2=v$, leading to the invariant solution $v(t,x)=w\left(x\right)$. Substituting into (4), we obtain the ODE

$$w^{\prime }{\left(x\right)}^2+(m-1)w\left(x\right)\left[w^{″}\left(x\right)+{\displaystyle \frac{(d-1)w^{\prime }\left(x\right)}{x}}\right]=0.$$

(35)

Solving (35), we find

$$w\left(x\right)=c_1{x}^{{\displaystyle \frac{(1-d)(m-1)}{m}}}{\left(c_2{x}^{d-1}+mx\right)}^{{\displaystyle \frac{m-1}{m}}},$$

(36)

where $c_1$ and $c_2$ are arbitrary constants, yielding the group-invariant solution

$$v(t,x)=c_1{x}^{{\displaystyle \frac{(1-d)(m-1)}{m}}}{\left(c_2{x}^{d-1}+mx\right)}^{{\displaystyle \frac{m-1}{m}}}.$$

(37)

Case 4.1 (b): Symmetry $X_2=t{\displaystyle \frac{\partial }{\partial t}}-v{\displaystyle \frac{\partial }{\partial v}}$.

The characteristic equations

$${\displaystyle \frac{dt}{t}}={\displaystyle \frac{dx}{0}}={\displaystyle \frac{dv}{-v}}$$

(38)

give invariants $J_1=x$ and $J_2=tv$, resulting in $v(t,x)={\displaystyle \frac{w\left(x\right)}{t}}$. Substituting into (4), we obtain

$$x{w}^{\prime }{\left(x\right)}^2+w\left(x\right)\left[(m-1)x{w}^{″}\left(x\right)+(m-1)(d-1)w^{\prime }\left(x\right)+x\right]=0.$$

(39)

A particular solution of (39) is

$$w\left(x\right)={\displaystyle \frac{x^2}{2\left[d(1-m)-2\right]}},d(1-m)\ne 2$$

(40)

leading to

$$v(t,x)={\displaystyle \frac{x^2}{2\left[d(1-m)-2\right]t}}.$$

(41)

Case 4.1 (c): Symmetry $X=\alpha X_2+X_3=(\alpha +2)t{\displaystyle \frac{\partial }{\partial t}}+x{\displaystyle \frac{\partial }{\partial x}}-\alpha v{\displaystyle \frac{\partial }{\partial v}}$.

The characteristic equations

$${\displaystyle \frac{dt}{(\alpha +2)t}}={\displaystyle \frac{dx}{x}}={\displaystyle \frac{dv}{-\alpha v}}$$

(42)

yield invariants $J_1=x{\left[(\alpha +2)t\right]}^{-{\displaystyle \frac{1}{\alpha +2}}}$ and $J_2=v{x}^{\alpha }$, giving

$$v(t,x)=x^{-\alpha }w\left(y\right),y=x{\left[(\alpha +2)t\right]}^{-{\displaystyle \frac{1}{\alpha +2}}}.$$

(43)

Substituting (43) into (4), we obtain

$$K_1{\left[w\left(y\right)\right]}^2+\left[y^{3+\alpha }+K_{2}yw\left(y\right)\right]w^{\prime }\left(y\right)+y^2\left[w^{\prime }{\left(y\right)}^2+(m-1)w\left(y\right)w^{″}\left(y\right)\right]=0,$$

(44)

where

$$\begin{array}{cc}\hfill K_1& ={\alpha }^{2}m+\alpha \left(d-2+(2-d)m\right),\hfill \end{array}$$

(45)

$$\begin{array}{cc}\hfill K_2& =1+d(m-1)-(1+2\alpha )m.\hfill \end{array}$$

(46)

A particular solution of (44) is

$$w\left(y\right)={\displaystyle \frac{(2+\alpha )y^{2+\alpha }}{2\left(d(1-m)-2\right)}},d(1-m)\ne 2$$

(47)

yielding the invariant solution

$$v(t,x)={\displaystyle \frac{x^2}{2\left(d(1-m)-2\right)t}}.$$

(48)

4.2. Solutions of (4) for $d=1$, $m=-{\displaystyle \frac{1}{3}}$

For $d=1$ and $m=-{\displaystyle \frac{1}{3}}$, Equation (4) simplifies to

$$v_t=v_{xx}-{\displaystyle \frac{4}{3}}v{v}_{xx}.$$

(49)

We use the symmetries $X_1$, $X_2$, and $\alpha X_2+X_4$ from the optimal system (31) to derive the invariant solutions of (49).

Case 4.2 (a): Symmetry $X_1={\displaystyle \frac{\partial }{\partial t}}$.

The characteristic equations

$${\displaystyle \frac{dt}{1}}={\displaystyle \frac{dx}{0}}={\displaystyle \frac{dv}{0}}$$

(50)

yield invariants $J_1=x$ and $J_2=v$, giving $v(t,x)=w\left(x\right)$. Substituting into (49), we obtain

$$w^{\prime }{\left(x\right)}^2-{\displaystyle \frac{4}{3}}w\left(x\right)w^{″}\left(x\right)=0.$$

(51)

The solution of (51) is

$$w\left(x\right)=c_1{(x-4c_2)}^4,$$

(52)

yielding

$$v(t,x)=c_1{(x-4c_2)}^4.$$

(53)

Case 4.2 (b): Symmetry $X_2=t{\displaystyle \frac{\partial }{\partial t}}-v{\displaystyle \frac{\partial }{\partial v}}$.

The invariants $J_1=x$ and $J_2=tv$ give $v(t,x)=w\left(x\right)/t$. Substituting into (49), we obtain

$$w\left(x\right)\left[3-4w^{″}\left(x\right)\right]+3w^{\prime }{\left(x\right)}^2=0.$$

(54)

The solution of (54) is

$$w\left(x\right)={\displaystyle \frac{{\left(c_1^2{(c_2+x)}^2+48\right)}^2}{256c_1^2}},c_1\ne 0,$$

(55)

where $c_1$ and $c_2$ are arbitrary constants, yielding the group-invariant solution

$$v(t,x)={\displaystyle \frac{{\left(c_1^2{(c_2+x)}^2+48\right)}^2}{256c_1^{2}t}}.$$

(56)

Case 4.2 (c): Symmetry $\alpha X_2+X_3=\alpha t{\displaystyle \frac{\partial }{\partial t}}+{\displaystyle \frac{\partial }{\partial x}}-\alpha v{\displaystyle \frac{\partial }{\partial v}},\alpha \ne 0$.

The invariants $J_1=x-{\displaystyle \frac{1}{\alpha }}lnt$ and $J_2=tv$ give

$$v(t,x)={\displaystyle \frac{w\left(y\right)}{t}},y=x-{\displaystyle \frac{1}{\alpha }}lnt.$$

(57)

Substituting (57) into (49), we obtain

$$\alpha w\left(y\right)\left[3-4w^{″}\left(y\right)\right]+3w^{\prime }\left(y\right)\left[\alpha w^{\prime }\left(y\right)+1\right]=0.$$

(58)

Solving (58) completes the solution.

Case 4.2 (d): Symmetry $\alpha X_2+X_4=\alpha t{\displaystyle \frac{\partial }{\partial t}}+x{\displaystyle \frac{\partial }{\partial x}}+(2-\alpha )v{\displaystyle \frac{\partial }{\partial v}},\alpha \ne 0$.

The invariants $J_1=x{t}^{-1/\alpha }$ and $J_2=v{t}^{1-2/\alpha }$ give

$$v(t,x)=t^{2/\alpha -1}w\left(x{t}^{-1/\alpha }\right).$$

(59)

Substituting (59) into (49), we obtain

$$w\left(y\right)\left[3\alpha -4\alpha w^{″}\left(y\right)-6\right]+3w^{\prime }\left(y\right)\left[\alpha w^{\prime }\left(y\right)+y\right]=0.$$

(60)

Solving (60) completes the solution.

Case 4.2 (e): Symmetry $X_2+\beta (X_3+X_5)=t{\displaystyle \frac{\partial }{\partial t}}+\beta (x^2+1){\displaystyle \frac{\partial }{\partial x}}-v(1-4\beta x){\displaystyle \frac{\partial }{\partial v}}$.

The invariants

$$\begin{array}{cc}\hfill J_1& =texp\left(-{\displaystyle \frac{{tan}^{-1}\left(x\right)}{\beta }}\right),\hfill \end{array}$$

(61)

$$\begin{array}{cc}\hfill J_2& ={\displaystyle \frac{v}{{(x^2+1)}^2}}exp\left({\displaystyle \frac{{tan}^{-1}\left(x\right)}{\beta }}\right),\hfill \end{array}$$

(62)

where $\beta \ne 0$, give

$$v(t,x)={(x^2+1)}^{2}exp\left(-{\displaystyle \frac{{tan}^{-1}\left(x\right)}{\beta }}\right)w\left(y\right),y=texp\left(-{\displaystyle \frac{{tan}^{-1}\left(x\right)}{\beta }}\right).$$

(63)

Substituting (63) into (49), we obtain

$$(1+16{\beta }^2)w{\left(y\right)}^2+\left[3{\beta }^2+6yw\left(y\right)\right]w^{\prime }\left(y\right)-3y^2{w}^{\prime }{\left(y\right)}^2+4y^{2}w\left(y\right)w^{″}\left(y\right)=0.$$

(64)

A particular solution of (64) is

$$w\left(y\right)={\displaystyle \frac{3}{16y}},$$

(65)

yielding the invariant solution

$$v(t,x)={\displaystyle \frac{3{(x^2+1)}^2}{16t}}.$$

(66)

5. Conclusions

In this work, we have conducted comprehensive Lie symmetry analysis of a (1+1)-dimensional porous medium equation characterized by parameters m and d. Through systematic group classification, we identified distinct Lie algebra structures that emerge depending on these parameters. We have established that for arbitrary m and d, the equation admits a three-dimensional Lie algebra, whereas for the specific case $d=1$ and $m=-{\displaystyle \frac{1}{3}}$, it admits a five-dimensional Lie algebra. We constructed optimal systems of one-dimensional subalgebras for both cases and utilized these to perform symmetry reductions and derive invariant solutions. These explicit solutions obtained enhance our theoretical understanding of the porous media equation under varying parameter regimes and provide valuable analytical benchmarks for validating numerical methods. The insights gained through this study have practical implications, particularly in modeling and analyzing gas flows and diffusion processes in porous media, and open avenues for future theoretical exploration and applied research in nonlinear PDE analysis.

Future research on the porous medium equation could pursue several directions. First, numerical simulations can be developed to validate the stability and physical relevance of the derived group-invariant solutions, particularly in the context of nonlinear diffusion phenomena such as gas flow in porous media. Second, investigating nonclassical or conditional symmetries may reveal additional invariant solutions, enriching the mathematical structure of the equation and potentially uncovering novel physical behaviors. These directions offer opportunities to deepen theoretical insights and enhance practical applications in fields like fluid dynamics and environmental modeling.