Keywords: Lie symmetries, heat equation, constitutive functions, radial geometry.
MSC (2020): 35K57, 35B06, 35K65, 22E70.

1 Introduction↩︎

Lie symmetry analysis has become one of the most powerful analytical tools for the investigation of nonlinear partial differential equations arising in mathematical physics, continuum mechanics, fluid dynamics, heat transfer, and reaction–diffusion phenomena. Since the pioneering work of Sophus Lie, continuous transformation groups have provided a systematic framework for studying differential equations through their intrinsic symmetry properties [1]–[12]. In particular, Lie symmetries enable the reduction of partial differential equations to ordinary differential equations, the construction of invariant solutions, the derivation of conservation laws.

For nonlinear evolution equations, an important aspect of Lie theory is the group classification problem. Instead of analyzing a single equation with fixed coefficients, it is more important to examine a class of equations containing arbitrary constitutive functions or parameters and seeks to determine all special forms for which the equations admit symmetry extensions beyond the principal Lie algebra. Such classifications play a crucial role in the systematic organization of nonlinear models because the admitted symmetry structure is closely related to the existence of exact solutions, similarity reductions and conservation properties.

In diffusion and heat-transfer models, constitutive functions usually describe real physical properties, such as diffusivity, thermal capacity, conductivity, or nonlinear reaction sources. For this reason, classifying these constitutive relations is not just a mathematical exercise; it can also reveal which physical mechanisms are consistent with richer symmetry structures. At the same time, arbitrary constitutive functions make the determining equations much harder to analyze. The infinitesimals get linked with unknown nonlinear coefficient functions, leading to complicated functional-differential relations.

The problem becomes even more complex in radial geometries. In these cases, the geometry itself adds extra terms that can create singular or exceptional cases for certain values of the radial parameter. These effects can change the admitted Lie algebra, give rise to resonant symmetry structures, and produce similarity reductions with very different behavior. The situation is further complicated when nonlinear reaction sources are included. These sources create additional links between the symmetry generators and the constitutive functions, making the classification problem much harder than in the source-free case.

With these ideas in mind, this work focuses on the Lie symmetry classification of generalized radial heat equations with nonlinear reaction sources. The determining equations are obtained using the classical Lie invariance criterion and then studied systematically through suitable constitutive transformations and compatibility conditions. The analysis identifies admissible subclasses that allow nontrivial extensions of symmetry. For these subclasses, the corresponding Lie algebras are constructed, and optimal systems of one-dimensional subalgebras are obtained. Finally, the admitted symmetries are used to derive similarity reductions and group-invariant solutions of the governing equation.

Motivated by the generalized nonlinear heat equation studied by Nauryz’s [13] for media with varying cross-section geometry, we consider an extended radial heat-transfer model with nonlinear reaction effects. In particular, Nauryz investigated the source-free equation \[C(u)u_t= \frac{1}{z^{\nu}} \left(z^{\nu}K(u)u_z\right)_z, \label{eq:nauryzmodel}\tag{1}\] where \(C(u)\) and \(K(u)\) denote temperature-dependent thermal coefficients, and \(\nu>0\) is a geometric parameter describing the effect of the varying cross-section or radial geometry. Building on this framework, we study the following reaction-diffusion extension: \[\begin{align} \label{Rad:01} C(u)u_t = \frac{1}{z^{\nu}} \left(z^{\nu}K(u)u_z\right)_{z} +R(u). \end{align}\tag{2}\] This equation provides a flexible model for heat transfer and diffusion processes in radial geometries. Here, \(C(u)\) represents the thermal capacity, \(K(u)\) denotes the thermal conductivity, and \(R(u)\) describes a nonlinear reaction source. The function \(R(u)\) allows the model to include additional physical effects such as internal heat generation, absorption, chemical reactions, or other source-driven processes.

Expanding the diffusion term in 2 , we obtain \[\begin{align} C(u)u_t = K(u)u_{zz} + K'(u)u_z^2 + \frac{\nu}{z}K(u)u_z + R(u). \label{eq:expandedpresent95model} \end{align}\tag{3}\] In this expanded form, the first two terms on the right-hand side describe nonlinear diffusion and the contribution due to the temperature dependence of the thermal conductivity. The term involving \(\dfrac{\nu}{z}\) captures the effect of radial geometry, where different values of \(\nu\) correspond to different spatial configurations.

The presence of arbitrary nonlinear functions \(C(u)\), \(K(u)\), and \(R(u)\) makes the equation rich from both physical and mathematical points of view. These functions allow the model to describe a wide range of nonlinear diffusion and heat-transfer phenomena. At the same time, they make the Lie symmetry classification problem more challenging, since the admitted symmetries depend strongly on the specific forms of these constitutive functions.

The main difference from the work of Nauryz is the inclusion of the additional constitutive function \(R(u)\). In the source-free case, the Lie symmetry classification mainly depends on the relationship between \(C(u)\) and \(K(u)\). In contrast, the present model requires the symmetry generators to be compatible not only with \(C(u)\) and \(K(u)\), but also with the nonlinear reaction source \(R(u)\). This additional coupling makes the determining equations more restrictive and substantially enriches the classification problem. Consequently, the present study may lead to new admissible subclasses, new symmetry extensions, and new similarity reductions that do not appear in the source-free model.

For a broader discussion of the physical background, related nonlinear heat-transfer models, and earlier applications of Lie symmetry methods to generalized heat equations, the references cited in the work of Nauryz’s can be seen.

2 Lie Symmetry Formulation↩︎

We consider the governing equation \[C(u)u_t = K(u)u_{zz} + K'(u)u_z^2 + \frac{\nu}{z}K(u)u_z + R(u). \label{eq:governing95expanded}\tag{4}\] where \(C(u)\), \(K(u)\), and \(R(u)\) are smooth functions of \(u\), and \(\nu\) is a constant parameter. Equivalently, we define \[G(z,t,u,u_z,u_t,u_{zz}) = C(u)u_t - K(u)u_{zz} - K'(u)u_z^2 - \frac{\nu}{z}K(u)u_z - R(u), \label{eq:G}\tag{5}\] so that the governing equation is simply \[G(z,t,u,u_z,u_t,u_{zz})=0. \label{eq:G95zero}\tag{6}\] Let a one-parameter local Lie group of point transformations act on the variables \((z,t,u)\) as \[\bar z=Z(z,t,u;\varepsilon),\qquad \bar t=T(z,t,u;\varepsilon),\qquad \bar u=U(z,t,u;\varepsilon), \label{eq:group95transformations}\tag{7}\] where \(\varepsilon\) is the group parameter. The identity transformation is recovered at \(\varepsilon=0\): \[Z(z,t,u;0)=z,\qquad T(z,t,u;0)=t,\qquad U(z,t,u;0)=u.\] For small \(\varepsilon\), the transformations have the infinitesimal form \[\begin{align} \bar z &= z+\varepsilon \xi(z,t,u)+O(\varepsilon^2), \\ \bar t &= t+\varepsilon \tau(z,t,u)+O(\varepsilon^2), \\ \bar u &= u+\varepsilon \eta(z,t,u)+O(\varepsilon^2). \end{align}\] The corresponding infinitesimal generator is therefore \[\mathcal{X} = \xi(z,t,u)\frac{\partial}{\partial z} + \tau(z,t,u)\frac{\partial}{\partial t} + \eta(z,t,u)\frac{\partial}{\partial u}. \label{eq:infinitesimal95generator}\tag{8}\] Since the equation 5 contains derivatives up to second order, the second prolongation of \(\mathcal{X}\) is required. It is given by \[\mathcal{X}^{(2)} = \mathcal{X} + \eta^z\frac{\partial}{\partial u_z} + \eta^t\frac{\partial}{\partial u_t} + \eta^{zz}\frac{\partial}{\partial u_{zz}}, \label{eq:second95prolongation}\tag{9}\] where the relevant prolonged coefficients are \[\begin{align} \eta^z &= D_z(\eta)-u_zD_z(\xi)-u_tD_z(\tau), \tag{10} \\ \eta^t &= D_t(\eta)-u_zD_t(\xi)-u_tD_t(\tau), \tag{11} \\ \eta^{zz} &= D_z(\eta^z)-u_{zz}D_z(\xi)-u_{zt}D_z(\tau). \tag{12} \end{align}\] Here \(D_z\) and \(D_t\) denote the total derivative operators with respect to \(z\) and \(t\), respectively. The invariance condition for the differential equation is \[\mathcal{X}^{(2)}G\big|_{G=0}=0. \label{eq:invariance95condition}\tag{13}\] Substituting these expressions into 13 , we obtain \[\begin{align} 0 &= \xi\left( \frac{\nu}{z^2}K(u)u_z \right) \nonumber\\ &\quad + \eta\left( C'(u)u_t - K'(u)u_{zz} - K''(u)u_z^2 - \frac{\nu}{z}K'(u)u_z - R'(u) \right) \nonumber\\ &\quad + \eta^z \left( -2K'(u)u_z - \frac{\nu}{z}K(u) \right) + C(u)\eta^t - K(u)\eta^{zz}, \qquad G=0. \label{eq:invariance95expanded} \end{align}\tag{14}\] Finally, on the solution manifold \(G=0\), we may use \[u_t = \frac{1}{C(u)} \left[ K(u)u_{zz} + K'(u)u_z^2 + \frac{\nu}{z}K(u)u_z + R(u) \right]. \label{eq:ut95replacement}\tag{15}\] Substitution of 15 into 14 , followed by splitting with respect to the independent derivatives, yields the determining equations for the infinitesimals \(\xi(z,t,u)\), \(\tau(z,t,u)\), and \(\eta(z,t,u)\).

\[\begin{align} &\tag{16}\xi_{u}=\,0, \tau_{z}=\,0, \tau_{u}=\,0\\[1mm] &\tag{17}\frac{\eta\, C'\,K}{C}-\eta \,K'-\tau_{t}\, K+2\,\xi_z\, K=0,\\[1mm] &\tag{18}K'\eta_u+\tau_{t} K'-2K'\xi_z+\eta K''+\eta_{uu}K -\frac{\eta \,C'\,K'}{C}=0,\\[1mm] &\tag{19}C\eta_t - K\eta_{zz} - \frac{\nu K}{z}\eta_z + R\eta_u - \tau_t R + \eta \left( \frac{C'}{C}R - R' \right)=0,\\[1mm] &C\,\xi_t-K\,\xi_{zz}+2K\,\eta_{zu} +2\eta_z \,K' -\frac{\nu\,\eta\, C'\,K}{z\,C} -\frac{\nu\,\xi\, K}{z^{2}}\nonumber\\[1mm] \tag{20}&-\frac{\nu\,\xi_z\, K}{z}+\frac{\nu\,\eta K'}{z} +\frac{\nu\,\tau_{t}\, K}{z}=0. \end{align}\]

The determining equation 17 can be simplified as: \[\begin{align} \tag{21}&\tau_{t}-2\,\xi_{z}=\,\eta\,M(u),\quad F(u)=\,\left(\frac{C'}{C}-\frac{K'}{K}\right),\\ \tag{22} &\eta=\,\frac{1}{F(u)}\,(\tau_{t}-2\,\xi_{z})=\,G(u)\, S(z,t), \end{align}\] and 22 is the first simplification in the classification process. For convenience, we introduce the notation \[\begin{align} \tag{23}&A(u)=\frac{C(u)}{K(u)}, \quad B(u)=\frac{R(u)}{K(u)},\quad F(u)=\frac{C'(u)}{C(u)}-\frac{K'(u)}{K(u)},\\[1mm] \tag{24}&G(u)=\frac{1}{F(u)},\quad S(z,t)=\tau_t-2\xi_z. \end{align}\]

Substituting \(\eta\) from 22 into 18 gives \[\begin{align} \label{rad:4}S\left[ G'' +\frac{K'}{K}G' +\left( \frac{K''}{K} -\frac{C'}{C}\frac{K'}{K} \right)G \right]=0 \end{align}\tag{25}\] and 19 becomes \[\begin{align} CS_tG -KS_{zz}G -\frac{\nu K}{z}S_zG +RSG' -\tau_tR +SG\left( \frac{C'}{C}R-R' \right)=0 \end{align}\] divide by \(K\) further simplifies the above equation into \[\begin{align} \label{rad:5}AS_tG -S_{zz}G -\frac{\nu}{z}S_zG +BSG' -\tau_tB +SB -SGB'=0. \end{align}\tag{26}\] The equations 25 and 26 will serve as pivots for classification. Finally, from 22 , we have \[\begin{align} \eta_z=S_zG=-2\xi_{zz}G, \eta_{zu}=S_zG'=-2\xi_{zz}G' \end{align}\] Substituting these expressions into 20 and simplifying gives \[\begin{align} \label{rad:6} A(u)\xi_t -\Lambda(u)\xi_{zz} -\frac{\nu}{z^2}\xi +\frac{\nu}{z}\xi_z=0, \end{align}\tag{27}\] where \(\Lambda(u)=1+4G'(u)+4\frac{K'}{K}G(u)\). The equation 27 is the main classification equation between \(\xi\) and the constitutive functions \(C,K\). To obtain symmetries beyond the kernel, we need to impose the following; \[\begin{align} \label{rad:9} \Lambda(u)=\,a\, A(u)+b, \end{align}\tag{28}\] where \(a\) and \(b\) are constants. This will split 27 into a pair of partial differential equations in \(\xi\) as follows: \[\begin{align} \tag{29}\xi_t -a\,\xi_{zz}=0,\\ \tag{30}b\,\xi_{zz} +\frac{\nu}{z^2}\xi -\frac{\nu}{z}\xi_z=0. \end{align}\]

The equation 30 is the standard Euler’s equation and can be solved followed by the restrictions from 29 will give the following splitting cases:

Compared to the source-free equation examined in [13], the addition of the reaction source term \(R(u)\) significantly changes the structure of the Lie classification problem. In addition to the thermal capacity \(C(u)\) and conductivity \(K(u)\), one must now classify the reaction function \(R(u)\). After introducing the reduced variables \(A=C/K\), \(B=R/K\), and \(G=\left(C'/C-K'/K\right)^{-1}\), the determining equations contain additional contributions involving \(B\) and \(B'\). These terms couple the infinitesimals directly to the reaction source and prevent the reduction of the determining system to a single compatibility equation for \(A\) and \(G\). As a result, classification must be performed simultaneously with respect to the triplet \((C,K,R)\), leading to new compatibility conditions and subclasses that enhance the symmetry of reaction terms that are absent in the source-free setting.

For the algebra \(\mathfrak g^{(2a)}\), the nonzero commutators are \[[\mathcal{X}_{1}^{(2a)},\mathcal{X}_{4}^{(2a)}] = B_0\mathcal{X}_{4}^{(2a)}, \qquad [\mathcal{X}_{2}^{(2a)},\mathcal{X}_{3}^{(2a)}] = (1-\nu)\mathcal{X}_{3}^{(2a)}.\] Thus \(\mathfrak g^{(2a)}\) is the direct sum of two two-dimensional solvable algebras. Using the adjoint action, a one-dimensional optimal system is \[\left\{ \mathcal{X}_{1}^{(2a)}+\alpha\mathcal{X}_{2}^{(2a)}, \mathcal{X}_{1}^{(2a)}+\mathcal{X}_{3}^{(2a)}, \mathcal{X}_{2}^{(2a)}+\mathcal{X}_{4}^{(2a)},\mathcal{X}_{2}^{(2a)}, \mathcal{X}_{3}^{(2a)}, \mathcal{X}_{4}^{(2a)}, \mathcal{X}_{3}^{(2a)}+\mathcal{X}_{4}^{(2a)} \right\}, \; \alpha\in\mathbb{R} .\] For the algebra \(\mathfrak g^{(2b)}\), the nonzero commutators are \[[\mathcal{X}_{1}^{(2b)},\mathcal{X}_{4}^{(2b)}] = \mathcal{X}_{1}^{(2b)}, \qquad [\mathcal{X}_{2}^{(2b)},\mathcal{X}_{3}^{(2b)}] = (1-\nu)\mathcal{X}_{3}^{(2b)}.\] Using the corresponding adjoint action, a one-dimensional optimal system is \[\left\{ \mathcal{X}_{4}^{(2b)}+\alpha\mathcal{X}_{2}^{(2b)}, \mathcal{X}_{4}^{(2b)}+\mathcal{X}_{3}^{(2b)}, \mathcal{X}_{2}^{(2b)}+\mathcal{X}_{1}^{(2b)}, \mathcal{X}_{1}^{(2b)},\mathcal{X}_{2}^{(2b)}, \mathcal{X}_{3}^{(2b)}, \mathcal{X}_{1}^{(2b)}+\mathcal{X}_{3}^{(2b)} \right\}, \; \alpha\in\mathbb{R} .\]

For \(B_0\neq0\), the admitted Lie algebra \[\mathfrak{g}^{(3a)} = \left\langle \mathcal{X}_{1}^{(3a)}, \mathcal{X}_{2}^{(3a)}, \mathcal{X}_{3}^{(3a)}, \mathcal{X}_{4}^{(3a)} \right\rangle\] is generated by \[\begin{align} \begin{cases} \mathcal{X}_{1}^{(3a)} = \partial_t, \\[2mm] \mathcal{X}_{2}^{(3a)} = z\partial_z -2G_0\partial_u, \\[2mm] \mathcal{X}_{3}^{(3a)} = z\ln z\,\partial_z -2G_0(\ln z+1)\partial_u, \\[2mm] \mathcal{X}_{4}^{(3a)} = e^{B_0t}\partial_t + B_0G_0e^{B_0t}\partial_u. \end{cases} \label{alg:3a} \end{align}\tag{31}\]

For \(B_0=0\), the admitted Lie algebra \[\mathfrak{g}^{(3b)} = \left\langle \mathcal{X}_{1}^{(3b)}, \mathcal{X}_{2}^{(3b)}, \mathcal{X}_{3}^{(3b)}, \mathcal{X}_{4}^{(3b)} \right\rangle\] is generated by \[\begin{align} \begin{cases} \mathcal{X}_{1}^{(3b)} = \partial_t, \\[2mm] \mathcal{X}_{2}^{(3b)} = z\partial_z -2G_0\partial_u, \\[2mm] \mathcal{X}_{3}^{(3b)} = z\ln z\,\partial_z -2G_0(\ln z+1)\partial_u, \\[2mm] \mathcal{X}_{4}^{(3b)} = t\partial_t + G_0\partial_u. \end{cases} \label{alg:3b} \end{align}\tag{32}\] The Lie commutation relations of the admitted Lie algebras \(\mathfrak{g}^{(3a)}\) and \(\mathfrak{g}^{(3b)}\) are summarized in Tables 4 and 5, respectively.

The commutation relations listed in Tables 4 and 5 show that both Lie algebras \(\mathfrak g^{(3a)}\) and \(\mathfrak g^{(3b)}\) are decomposable and isomorphic to the direct sum \(A_{2.1}\oplus A_{2.1}\) of two non-Abelian two-dimensional solvable Lie algebras [15]–[19]. Therefore, the classification of one-dimensional subalgebras can be carried out by means of the adjoint representation. Following the standard procedure of Olver [5], equivalent subalgebras are identified under the action of the corresponding inner automorphism group, leading to the optimal systems presented in 6.

The optimal systems for Case (3a) and Case (3b) coincide because both \(\mathfrak g^{(3a)}\) and \(\mathfrak g^{(3b)}\) are isomorphic to \(A_{2.1}\oplus A_{2.1}\).

3 Similarity Reductions and Invariant Solutions↩︎

After identifying the admitted Lie algebras and their associated optimal systems of one-dimensional subalgebras, the subsequent step is to develop group-invariant solutions of the governing equation 2 . For each generator in the optimal system, the corresponding invariants are derived from the characteristic equations: \[\begin{align} \label{rad:17}\frac{dt}{\tau} = \frac{dz}{\xi} = \frac{du}{\eta}. \end{align}\tag{33}\] These invariants produce similarity variables and associated similarity representations of the dependent variable. The substitution of the resultant ansätze into the governing equation transforms the original partial differential equation into an ordinary differential equation, whose solutions yield precise group-invariant solutions. As every one-dimensional subalgebra corresponds to a representative element of the optimal system, it is adequate to do the reduction solely for the generators included within it. Before starting similarity reduction we need to divide 4 both sides by \(K(u)\) and it becomes: \[\begin{align} \label{rad:16} A(u)u_t = u_{zz} +\frac{K'(u)}{K(u)}\,u_z^2 +\frac{\nu}{z}\,u_z +B_0A(u)G(u), \end{align}\tag{34}\] this is done to obtain a pure ODE, because the functions \(A(u), K(u), G(u)\) must be replaced by their invariant functional forms corresponding to the symmetry. In the following, we present similarity reductions and group-invariant solutions associated with selected symmetry generators.

3.1 Case I: \(\mathcal{X}^{(1)}_{1}+\alpha \mathcal{X}^{(1)}_{2}\)↩︎

We consider the one-parameter symmetry group formed by \(\mathcal{X}^{(1)}_{1}+\alpha \mathcal{X}^{(1)}_{2}\) and derive the associated similarity variables and group-invariant solutions. \[\begin{align} \label{rad:22} \mathcal{X}=\,\mathcal{X}^{(1)}_{1}+\alpha \mathcal{X}^{(1)}_{2}=\, \partial_t +\alpha z\,\partial_z -\alpha\left(\frac{b-1}{2}\,u+2G_0\right)\partial_u. \end{align}\tag{35}\] The characteristics system 33 becomes \[\begin{align} \label{rad:18}\frac{dt}{1} = \frac{dz}{\alpha z} = \frac{du}{-\alpha\left(mu+2G_0\right)}, \quad m=\frac{b-1}{2}. \end{align}\tag{36}\] The corresponding invariant form of \(u\) and the similarity variable can be obtained as follows: \[\begin{align} \label{rad:19}u(z,t) = e^{-qt}\phi(\omega) -\frac{4G_0}{b-1}, \quad \omega = z e^{-\alpha t},\qquad q=\,\frac{\alpha\,(b-1)}{2}. \end{align}\tag{37}\] At this stage, we can not substitute the similarity ansätze 37 directly into PDE 34 while leaving \(A(u), K(u), G(u)\) arbitrary, instead we first need to find admissible forms of these functions. Using the notation introduced in Eqs. 23 and 24 , we have \[\begin{align} F(u)=\frac{C'(u)}{C(u)}-\frac{K'(u)}{K(u)}=\frac{A'(u)}{A(u)}, \quad \text{and}\,G(u)=\frac{1}{F(u)} \end{align}\] To reduce 34 into pure ODE we must find the invariant form of \(A(u)\) under symmetry generator 35 . Due to the invariance of PDE 34 the coefficient \(A(u)\) must transform consistently under the prolonged action of symmetry generator 35 , and we find that \[\begin{align} \label{rad:23} \operatorname{pr}^{(1)}\mathcal{X}\!\left(Au_t\right) = \left[ -\alpha\,\left(mu+2G_0\right)A'(u) -\alpha mA(u) \right]u_t. \end{align}\tag{38}\] and \[\begin{align} \label{rad:24} \operatorname{pr}^{(2)}\mathcal{X}\!\left(u_{zz}\right) =-\alpha\,(m+2)\,u_{zz}. \end{align}\tag{39}\] For invariance, the term \(A(u)\,u_{t}\) must scale with same weight as \(u_{zz}\), therefore, the equations 38 and 39 yields \[\begin{align} \label{rad:25} \frac{-\alpha\left[(m\,u+2G_0)A'+m\right]}{A} = -\alpha(m+2) \end{align}\tag{40}\] The solution of 40 will provided he homogeneous transformation requirement for \(A(u)\) \[\begin{align} \label{rad:28}A(u)={A}_0\left(u+\frac{4G_0}{b-1}\right)^{2}. \end{align}\tag{41}\] Define shift variable \[\begin{align} \label{rad:26}v=u+\frac{4G_0}{b-1}. \end{align}\tag{42}\] So that the similarity ansätze 37 becomes: \[\begin{align} \label{rad:27}v=e^{-\alpha m t}\phi(\omega), \qquad \omega=ze^{-\alpha t}. \end{align}\tag{43}\] The coefficient 41 reduce to \[\begin{align} A(u)=A_0 e^{-\alpha (b-1)t}\phi(\omega)^2 \nonumber\\ \label{rad:30} \implies A(u)v_t = - A_0 \alpha e^{-3\alpha m t} \phi^2 \left( m\phi+\omega\phi' \right). \end{align}\tag{44}\] whereas the diffusion term \(v_{zz}\) scale as \[\begin{align} e^{-\alpha(m+2)t} \end{align}\] this gives \[\begin{align} 3\,m=\,m+2 \implies m=1 \implies b=3. \end{align}\] This is the crucial compatibility condition for reduction into a pure ODE under symmetry generator 35 . Now we express \(\frac{K'}{K}\) in invariant form. Let \[P(u)=\frac{K'(u)}{K(u)}.\] Since \(C=AK\), we have \[\frac{C'}{C}=\frac{A'}{A}+\frac{K'}{K}.\] Hence \[\frac{K''}{K}-\frac{C'}{C}\frac{K'}{K} = P'+P^2-\left(\frac{A'}{A}+P\right)P = P'-\frac{A'}{A}P.\] Therefore, the determining equation 25 \[G''+\frac{K'}{K}G' +\left( \frac{K''}{K} -\frac{C'}{C}\frac{K'}{K} \right)G=0\] reduces to \[G''+PG' +\left(P'-\frac{A'}{A}P\right)G=0.\] For the present invariant class, \(b=3\). Thus \[v=u+2G_0,\qquad A=A_0v^2.\] Therefore \[\frac{A'}{A}=\frac{2}{v},\qquad G=\frac{v}{2},\qquad G'=\frac{1}{2},\qquad G''=0.\] Substituting these into the reduced determining equation gives \[\frac{P}{2} + \left(P'-\frac{2}{v}P\right)\frac{v}{2} =0.\] Thus \[\frac{vP'}{2}-\frac{P}{2}=0,\] or \[vP'-P=0.\] Hence \[\frac{P'}{P}=\frac{1}{v},\] which gives \[P=\lambda v.\] Consequently, \[\frac{K'}{K} = \lambda\left(u+2G_0\right), \qquad b=3,\] where \(\lambda\) is an arbitrary constant. Consequently, upon integration \[\begin{align} &K(u) = K_0 \exp\!\left[ \frac{\lambda}{2}\left(u+2G_0\right)^2 \right],\\ &C(u) = A_0K_0\left(u+2G_0\right)^2 \exp\!\left[ \frac{\lambda}{2}\left(u+2G_0\right)^2 \right]. \end{align}\] But for the term \(\frac{K'}{K}u_{z}^{2}\) to scale like the diffusion term we need to set \(\lambda=0\). Finally, substituting the similarity ansätze 37 into the governing equation 34 and collecting the resulting terms, we obtain the reduced equation \[\begin{align} \label{rad:31} \phi''+\frac{\nu}{\omega}\phi' +\alpha A_0\omega\phi^2\phi' +A_0\left(\alpha+\frac{B_0}{2}\right)\phi^3=0. \end{align}\tag{45}\] with the admissible coefficient functions \[\begin{align} &A(u)=A_0\left(u+2G_0\right)^2,\\ & K(u)=K_0, C(u)=C_0\left(u+2G_0\right)^2,\\ &R(u)=\frac{B_0C_0}{2}\left(u+2G_0\right)^3. \end{align}\] The reduced equation 45 is a nonlinear second-order Emden-Fowler-type equation, more precisely a generalized Lane-Emden equation for \(\beta=\,\alpha\,A_{0}\) and \(\gamma=\,A_{0}\left(\alpha+\frac{B_{0}}{2}\right)\) [20] with an additional convective nonlinear term \(\omega\phi^2\phi'\).

Multiply the equation 45 by \(\omega^{\nu}\): \[\omega^{\nu}\phi'' +\nu\omega^{\nu-1}\phi' +\beta\omega^{\nu+1}\phi^{2}\phi' +\gamma\omega^{\nu}\phi^{3} =0.\] The first two terms combine according to \[(\omega^{\nu}\phi')' = \omega^{\nu}\phi'' +\nu\omega^{\nu-1}\phi'.\]

Hence \[(\omega^{\nu}\phi')' +\beta\omega^{\nu+1}\phi^{2}\phi' +\gamma\omega^{\nu}\phi^{3} =0.\]

Now observe that \[\phi^{2}\phi' = \frac{1}{3}(\phi^{3})'.\]

Therefore, \[\beta\omega^{\nu+1}\phi^{2}\phi' = \frac{\beta}{3}\omega^{\nu+1}(\phi^{3})'.\]

Using the product rule, \[\omega^{\nu+1}(\phi^{3})' = (\omega^{\nu+1}\phi^{3})' -(\nu+1)\omega^{\nu}\phi^{3}.\] Substituting this identity produces \[\begin{align} \label{rad:34}(\omega^{\nu}\phi')' +\frac{\beta}{3}(\omega^{\nu+1}\phi^{3})' +\left[ \gamma-\frac{\beta(\nu+1)}{3} \right] \omega^{\nu}\phi^{3} =0. \end{align}\tag{46}\] This equation is exactly solvable if \(\gamma=\frac{\beta(\nu+1)}{3}\). It can be solved easily by direct integration and solution is obtained as follows: \[\begin{align} \label{rad:33}\phi(\omega)=\pm\frac{1}{\sqrt{C_3+\frac{\beta}{3}\omega^2}} \end{align}\tag{47}\] In original notations the compatibility condition must be \[\begin{align} A_0\left(\alpha+\frac{B_0}{2}\right)= \frac{\alpha A_0(\nu+1)}{3}\implies B_{0}=\,\frac{2\,\alpha(\nu-2)}{3} \end{align}\] The original solution is thus can be written as: \[\begin{align} u(z,t) = -2G_0 \pm \frac{1}{\sqrt{ C_3 e^{2\alpha t} +\frac{\alpha A_0}{3}z^2 }}. \end{align}\] This is a rich family of solutions for the governing equation 4 and it is physically relevant with regular bounded profile [21], [22].

3.2 Case II: \(\mathcal{X}^{(1)}_{3}+\beta \mathcal{X}^{(1)}_{2}\)↩︎

The symmetry reduction for the generator \(\mathcal{X}^{(1)}_{3}+\beta \mathcal{X}^{(1)}_{2}\) works similarly to the preceding case. We skip the intermediate calculations and show simply the similarity variables, the reduced ordinary differential equation, and the group-invariant solutions are presented below in this section. The symmetry generator in this case can be written as \[\begin{align} \label{sec95similarity} \mathcal{X} &= e^{B_0 t}\partial_t +\beta z\,\partial_z +\left(B_0 e^{B_0 t}-2\beta\right)\Theta(u)\,\partial_u,\quad \Theta(u)=\frac{b-1}{4}\,u+G_0,. \end{align}\tag{48}\] The similarity transformations associated with this symmetry generator are obtained by solving the characteristic equations 33 , resulting in the following. \[\label{rad:37} \begin{align} u(z,t) &= e^{\frac{(b-1)B_0}{4}t} \exp\!\left( \frac{(b-1)\beta}{2B_0}e^{-B_0 t} \right) \phi(\omega) -\frac{4G_0}{b-1}, \\[2mm] \omega &= z\exp\!\left( \frac{\beta}{B_0}e^{-B_0 t} \right). \end{align}\tag{49}\] and the corresponding reduced equation \[\begin{align} \label{rad:35}\phi'' +\frac{\nu}{\omega}\phi' +\beta A_0\,\phi^{\frac{4}{b-1}} \left( \omega\phi' +\frac{b-1}{2}\phi \right) =0. \end{align}\tag{50}\] The reduced equation 50 is again nonlinear Emden-Fowler/Lane-Emden type ODE with a nonlinear convective term having more general power-law than 45 . It can be solved by first making some parameteric changes, we consider \(n=\frac{4}{b-1}\). Then the reduced ODE \[\phi'' +\frac{\nu}{\omega}\phi' +\beta A_0\phi^{n} \left( \omega\phi' +\frac{b-1}{2}\phi \right)=0\] can be rewritten in divergence form as \[\left(\omega^\nu \phi'\right)' +\frac{\beta A_0}{n+1} \left(\omega^{\nu+1}\phi^{\,n+1}\right)' =0,\] provided that \[\frac{b-1}{2} = \frac{\nu+1}{n+1}.\] Substituting \(n=4/(b-1)\), this condition reduces to \[b=2\nu-1.\] Integrating once yields \[\omega^\nu \phi' +\frac{\beta A_0}{n+1}\, \omega^{\nu+1}\phi^{\,n+1} =C_1,\] where \(C_1\) is an integration constant. For the particular case \(C_1=0\), we obtain the separable equation \[\phi' = -\frac{\beta A_0}{n+1}\, \omega\phi^{\,n+1}.\] Integrating gives \[\phi^{-n} = C_2 +\frac{n\beta A_0}{2(n+1)}\,\omega^2,\] or equivalently \[\phi(\omega) = \left( C_2 +\frac{n\beta A_0}{2(n+1)}\,\omega^2 \right)^{-1/n}.\] Substituting \(n=4/(b-1)\), we arrive at \[\begin{align} \label{rad:36} \phi(\omega) = \left( C_2 +\frac{2\beta A_0}{b+3}\,\omega^2 \right)^{-\frac{b-1}{4}}, \qquad b=2\nu-1. \end{align}\tag{51}\] The original solution of PDE 34 can be obtained by combining 49 with 51 .

3.3 Case III: \(\mathcal{X}_{2}^{(2a)}+\mathcal{X}_{4}^{(2a)}\)↩︎

The reduction corresponding to this symmetry belongs to the branch

\[\begin{align} a=0,\; b=\,\frac{\nu}{2-\nu}, \end{align}\] and for a valid symmetry reduction, we need to take \(K'(u)=0\). In the subcase \(a=0\), one has \[K(u)=K_0, \qquad C(u)=K_0A(u).\] Moreover, since \[\frac{A'}{A}=\frac{1}{G},\] we obtain \[A(u)=A_0G(u)^{1/\alpha}, \qquad \alpha=\frac{b-1}{4}.\] The symmetry generator under consideration is \[\begin{align} \mathcal{X} = \mathcal{X}_{2}^{(2a)} + \mathcal{X}_{4}^{(2a)} = e^{B_0t}\partial_t + z\partial_z + \left(B_0e^{B_0t}-2\right)H(u)\partial_u, \end{align}\] where \[H(u)=\frac{b-1}{4}u+G_0=\alpha u+G_0.\] Solving the characteristic equations 33 gives the and the invariant form and similarity variable \[\begin{align} \label{rad:38}H(u) = \exp\left( \alpha B_0t + \frac{2\alpha}{B_0}e^{-B_0t} \right)\phi(\omega),\quad \omega = \ln z+\frac{e^{-B_0t}}{B_0}, \end{align}\tag{52}\]

Using this ansätze 52 in 34 gives reduces ODE as follows:

\[\begin{align} \label{rad:39}\phi'' + (\nu-1)\phi' + A_0e^{2\omega} \phi^{\frac{2(2-\nu)}{\nu-1}} \left( \phi' + \frac{\nu-1}{2-\nu}\phi \right) =0, \qquad \nu\neq1,2. \end{align}\tag{53}\] The similarity reduction obtained from the generator \(\mathcal{X}_{2}^{(2a)}+\mathcal{X}_{4}^{(2a)}\) is valid only for \(\nu\neq1,2\). From a physical perspective, these values correspond to important radial geometries and are therefore expected to possess distinct symmetry structures. As far as solution of equation 53 is concerned, it is a nonlinear non-autonomous second-order ordinary differential equation containing derivative-dependent power-law nonlinearities. The presence of the explicit factor \(e^{2\omega}\), the mixed nonlinear term \(\phi^{m}\phi'\), and the parameter-dependent exponent \(m\) prevents the reduction to any standard integrable class. As a result of this, the closed-form solutions can be expected only for special parameter values or under additional ansätze.

The equation 53 can be transformed to simpler form by making substitution: \[\begin{align} \label{rad:42}r=\,\frac{\nu-1}{2-\nu} \end{align}\tag{54}\] and it reduced to \[\begin{align} \label{rad:41} \phi''+(\nu-1)\phi'+A_0e^{2\omega}\phi^{2/r}\left(\phi'+r\phi\right) =0. \end{align}\tag{55}\] by making dependent variable transformation; \[\begin{align} \phi=e^{-r\omega}y \end{align}\] it further reduce to \[\begin{align} y''+(\nu-1-2r)y'+\left(r^2-(\nu-1)r\right)y+A_0y^{2/r}y'=0. \end{align}\] Invoking the value of \(r\) from 54 , the most simpler version of 53 can be obtained as follows: \[\begin{align} \label{rad:43}y''-\frac{\nu(\nu-1)}{2-\nu}\,y'+\frac{(\nu-1)^3}{(2-\nu)^2}\,y+A_0y^{\frac{2(2-\nu)}{\nu-1}}y'=0. \end{align}\tag{56}\] It is equivalent to following autonomous Lienard-type equation [20] \[\begin{align} y''+f(y)y'+g(y)=0, \end{align}\] where \[\begin{align} f(y)=\,-\frac{\nu(\nu-1)}{2-\nu}+A_0y^{\frac{2(2-\nu)}{\nu-1}},\;g(y)=\,\frac{(\nu-1)^3}{(2-\nu)^2} \end{align}\] For specific values of the geometric parameter \(\nu\), the reduced equation 56 significantly simplifies, resulting in numerous notable subclasses of nonlinear ordinary differential equations. \[\begin{align} \label{rad:44}y''+A_{0}\,\frac{1}{y^{4}}\,y'-\frac{1}{8}\,y=0, \end{align}\tag{57}\] which is a Lienard equation with inverse-power damping. Second useful subclass can be obtained when power of \(y\) in 56 becomes unity, that is when \(\nu=\frac{5}{3}\), we get \[\begin{align} \label{rad:45}y''+(A_{0}\,y-10)\,y'+8\,y=0. \end{align}\tag{58}\] The third important subclass can obtained, when power of \(y\) becomes 2, that is when \(\nu=\frac{3}{2}\), we get \[\begin{align} \label{rad:46}y''+(A_{0}\,y^{2}-3)\,y'+\frac{1}{2}\,y=0, \end{align}\tag{59}\] which resembles with generalized Rayleigh/Lienard oscillator.

3.4 Case IV: \(\mathcal{X}_{3}^{(3a)}+\mathcal{X}_{4}^{(3a)}\)↩︎

This reduction corresponds to the branch \(\nu=1\) with the admissible coefficient functions \[\begin{align} &G(u)=G_0,\qquad K(u)=K_0,\\ &A(u)=A_0e^{u/G_0}, C(u)=A_0K_0e^{u/G_0}, R(u)=B_0C(u)G_0. \end{align}\] Consider the generator

\[\begin{align} \mathcal{X}= \mathcal{X}_{3}^{(3a)}+\mathcal{X}_{4}^{(3a)} = e^{B_0t}\partial_t + z\ln z\,\partial_z + \left[ B_0G_0e^{B_0t} -2G_0(\ln z+1) \right]\partial_u \end{align}\] The invariant form of \(u\) and the similarity variable is thus obtained as follows: \[\begin{align} \label{rad:40} u(z,t) = \phi(\omega) + G_0\left( B_0t-2\ln z-2\ln(\ln z) \right), \quad \omega= \ln(\ln z)+\frac{e^{-B_0t}}{B_0} \end{align}\tag{60}\] Substituting the similarity ansätz 60 into PDE 34 produces the reduced ODE \[\begin{align} \phi''+\left(A_0e^{\phi/G_0}-1\right)\phi'+2G_0=0. \end{align}\] This equation can be transformed into a simpler form through the substitution \(\phi = G_{0}\ln y\). However, even after this transformation, obtaining a closed-form solution remains unlikely unless additional restrictions or suitable parametric assumptions are imposed.

4 Observation and analysis↩︎

Compared to the conventional symmetry analysis of equations with fixed coefficients, the Lie symmetry classification taken into consideration in this work is significantly more complex. The main challenge is from the existence of arbitrary constitutive functions \(C(u), K(u)\), and \(R(u)\). These functions transforms the determining equations into a coupled functional-differential system. It is typically hard to identify the symmetry generators without concurrently placing limits on the coefficient functions since the infinitesimals are closely associated with the constitutive ratios \(A=\,CK^{-1}\) and \(B=\,RK^{-1}\).

A subsequent difficulty emerges from the radial geometry term \(\nu z^{-1}u_z\), which destroys translational invariance and generates specific geometric branches associated with particular values of \(\nu\). Consequently, the generic classification separates naturally into several exceptional subclasses, such as the logarithmic branch for \(\nu=1\) and the singular branch associated with \(\nu=2\).

Although the introduction of auxiliary variables such as \[A(u)=\frac{C(u)}{K(u)}, \qquad B(u)=\frac{R(u)}{K(u)}, \qquad G(u)=\left(\frac{C'}{C}-\frac{K'}{K}\right)^{-1}\] substantially simplified the determining equations; yet, the resultant system continues to be extremely nonlinear and strongly coupled. Consequently, explicit classification is feasible only upon the imposition of additional compatibility constraints or the consideration of symmetry-enhancing subclasses.

A further limitation of the approach is that the reduced ordinary differential equations such as 53 derived from similarity reductions are, generally, nonlinear non-autonomous equations characterized by derivative-dependent power-law nonlinearities. Since these equations do not fall into the conventional integrable class, exact closed-form invariant solutions are only possible with specific parameter choices or additional assumptions. Despite difficulties, the procedure successfully identifies nontrivial symmetry extensions, admissible constitutive relations, and inequivalent similarity reductions, providing a systematic framework for analyzing generalized radial heat equations with nonlinear reaction sources.

5 Conclusion↩︎

In this paper, Lie symmetry analysis was applied to a generalized nonlinear radial heat equation with an additional nonlinear reaction source. The presence of the arbitrary functions \(C(u), K(u)\), and \(R(u)\) led to a coupled functional-differential classification problem. By introducing reduced constitutive variables, the determining system was simplified and several symmetry-enhancing subclasses were obtained.

The analysis showed that the radial geometry parameter plays an important role in the classification. In particular, special branches arise for exceptional values of the parameter, including logarithmic and singular cases. For each admissible subclass, the corresponding Lie symmetry generators were derived, their commutation relations were studied, and optimal systems of one-dimensional subalgebras were constructed. These optimal systems were then used to obtain similarity variables and reduce the governing equation to ordinary differential equations.

The similarity reductions produced nonlinear Emden-Fowler, Lane-Emden-type, and Lienard-type equations. For selected parameter restrictions, exact invariant solutions were obtained, illustrating the usefulness of the symmetry approach. Overall, the results demonstrate that the nonlinear source term introduces new compatibility conditions and generates richer symmetry structures than those found in the source-free model. The classification presented here provides a systematic basis for constructing exact solutions and for further analytical study of nonlinear heat-transfer models in radial geometries.

Funding↩︎

Not applicable.

Availability of data and material↩︎

Not applicable.

Declaration↩︎

Conflict of interest↩︎

The authors declare that there is no conflict of interest with respect to the publication of this manuscript.

Funding Statement↩︎

This research did not receive external funding.

References↩︎

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1. corresponding author: manjitcsir@gmail.com↩︎