Symmetry solutions for reaction-diffusion equations with spatially dependent diffusivity

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1 Symmetry solutions for reaction-diffusion equations with spatially dependent diffusivity This is the author submitted original manuscript (pre-print) version of a published work that appeared in final form in: Bradshaw-Hajek, Bronwyn & Moitsheki, RJ 215 'Symmetry solutions for reaction-diffusion equations with spatially dependent diffusivity' Applied mathematics and computation, vol. 254, pp Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms are not reflected in this document and it does not replicate the final published authoritative version for which the publisher owns copyright. It is not the copy of record. This output may be used for non-commercial purposes. The final definitive published version (version of record) is available at: Persistent link to the Research Outputs Repository record: General Rights: Copyright and moral rights for the publications made accessible in the Research Outputs Repository are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognize and abide by the legal requirements associated with these rights. Users may download and print one copy for the purpose of private study or research. You may not further distribute the material or use it for any profit-making activity or commercial gain You may freely distribute the persistent link identifying the publication in the Research Outputs Repository If you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately and investigate your claim.

2 The Library Educating Professionals, Creating and Applying Knowledge, Engaging our Communities NOTICE: this is the author s version of a work that was accepted for publication in Applied Mathematics and Computation. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in Applied Mathematics and Computation, Volume 254, 1 March 215, Pages 3 38 DOI# 1.116/j.amc

3 Symmetry solutions for reaction-diffusion equations with spatially dependent diffusivity B.H. Bradshaw-Hajek a,, R.J. Moitsheki b a Phenomics and Bioinformatics Research Centre, School of Information Technology and Mathematical Sciences, University of South Australia, Mawson Lakes, SA 582, Australia b Center for Differential Equations, Continuum Mechanics and Applications, School of Computational and Applied Mathematics, University of the Witwatersrand, Private Bag 3, Wits 25, South Africa Abstract Nonclassical and classical symmetry techniques are employed to analyse a reactiondiffusion equation with a cubic source term. Here, the diffusivity(diffusion term) is assumed to be an arbitrary function of the spatial variable. Classification using Lie point and nonclassical symmetries is performed. It turns out that the diffusivity needs to be given as a quadratic function of the spatial variable for the given governing equation to admit nonclassical symmetries. Both nonclassical and classical symmetries are used to construct some group-invariant (eact) solutions. The results are applied to models arising in population dynamics. Keywords: Classical Lie point symmetries, Nonclassical symmetries, Eact solutions, Reaction-diffusion equations, Spatially dependent diffusion Introduction Classical Lie point and nonclassical symmetry analysis are useful techniques when searching for analytic solutions to differential equations. The main difference underlying the determination of classical Lie point and nonclassical symmetries (also known as conditional symmetries, Q-conditional symmetries or reduction operators [1]) is in the condition imposed in the infinitesimal criterion for invariance. For classical symmetries it is required that the differential equation in question be invariant on solutions of the equation itself, however for the determination of nonclassical symmetries, an etra condition is that the equation is invariant both on its solutions and its invariant surface condition (see for eample [2]). Bluman and Cole [2] pioneered the idea of nonclassical symmetries. Nonclassical symmetry methods may give rise to eact solutions which cannot be obtained by classical methods (see for eample [3]). Corresponding author addresses: Bronwyn.Hajek@unisa.edu.au (B.H. Bradshaw-Hajek), Raseelo.Moitsheki@wits.ac.za (R.J. Moitsheki) Preprint submitted to Elsevier August 23, 214

4 Nonclassical symmetries methods have been applied to reaction-diffusion equations or the nonlinear diffusion equation with a source term (among others see e.g. [4, 5, 6, 7, 8, 9, 1, 11, 12]). Serov [4], and later Arrigo et al. [5] and Clarkson and Mansfield [6], found that the quasilinear reaction-diffusion equation u t = [D u ] +Q(u), with constant diffusivity, D, admits strictly nonclassical symmetries only if Q(u) is a cubic polynomial. Nucci and Clarkson [13] also found solutions for particular forms of cubic Q(u). Vaneeva and Zhalij [14] performed the group classification via equivalence transformations of the reaction-diffusion (generalized Huley) equation u t = u +g()q(u), while the nonclassical symmetry analysis of this equation with Q(u) cubic was carried out in [8] and with Q(u) = u 2 (1 u) in [15]. Hashemi and Nucci [9] applied nonclassical symmetry analysis to a related class of reaction-diffusion equations. In this paper, we use assume that the diffusivity depends on the space variable so that the equation in question is given by F = u t (k()u ) Q(u) =, (1) where Q(u) is a factorisable cubic. We refer to equation (1) as the governing equation. Conditions under which a general PDE may be mapped to another are discussedin[16]. Asmentionedabove, forequation(1), inthecasewherek() = k constant and Q(u) is cubic, nonclassical solutions have been found [4, 5, 6, 1, 13]. Here, we employ the classical and nonclassical symmetry techniques to construct eact (group-invariant) solutions for equation (1) when k() constant. Equations of this type (1) can be used to model problems arising in heat conduction [17], biology (population dynamics [7, 18, 19], transmission of nerve signals [2]), and combustion theory [21]. Eact solutions play an important role in identifying interesting behaviours in nonlinear systems. This paper is organized as follows; in Section 2 we discuss the classical and nonclassical symmetry techniques in brief. In Section 3, we provide analysis of the governing equation using nonclassical symmetry techniques to construct some eact (group-invariant) solutions. In Section 4, solutions are found using classical Lie point symmetries. In Section 5 we discuss a possible application of the constructed solutions to a problem in population dynamics. Lastly, we provide some final remarks in Section Classical and nonclassical symmetry analysis In brief, a Lie point symmetry of a differential equation is an invertible transformation of the independent and dependent variables that preserves the governing equation and depends smoothly on a continuous parameter. The literature in this area is sizable (see for eample [22, 23, 24, 25, 26]) and will 2

5 only be described in brief here. To find symmetries of a second order partial differential equation, one seeks the transformations generated by the base vector = G(t,,u,u t,u,u tt,u t,u ) =, t = t+εξ 1 (t,,u)+o(ε 2 ), = +εξ 2 (t,,u)+o(ε 2 ), u = u+εη(t,,u)+o(ε 2 ), Γ = ξ 1 (t,,u) t +ξ2 (t,,u) +η(t,,u) u, which leave the governing equation invariant. The operator Γ is a classical Lie point symmetry of the governing equation provided the invariance criteria Γ [2] ( ) = = (2) holds, where Γ [2] is the second prolongation of Γ (defined in [25], for eample). Equation (2) results in an overdetermined system of linear homogeneous partial differential equations, known as the determining equations. Albeit tedious, these equations can be algorithmically solved. Calculations can be done by hand or facilitated by interactive computer software such as Maple [27] or Reduce [28]. The nonclassical symmetry techniques developed by Bluman and Cole [2] generalise Lie s classical symmetry method for obtaining analytic solutions to partial differential equations. We seek invariance of the governing equation subject to the constraint of the invariant surface condition (ISC) ξ 1 u t +ξ 2 u = η, which can sometimes lead to additional reductions which are not obtainable using the classical method. In terms of the second prolongation of the nonclassical symmetry generator, Γ [2], the determining relations for parabolic PDEs are given by Γ [2] ( ) =, ISC =. (3) Unlike in the case of searching for classical Lie point symmetries, the determining equations resulting from (3) are a set of overdetermined nonlinear equations which must be solved to find ξ 1 (,t,u), ξ 2 (,t,u) and η(,t,u). For 1+1 dimensional evolution equations, the case where ξ 1 = is known to be unfeasible [29] since finding any nonclassical operators is equivalent to solving the original equation. Therefore, we restrict our study to the case of regular nonclassical operators and without loss of generality set ξ 1 = 1 (see also [3]). Solution of the determining equations may require specific restrictions to be placed on arbitrary functions in the case of equation (1), k(). Again, calculations may be facilitated by an interactive code in Maple. 3

6 Results using nonclassical symmetry analysis 3.1. Results for u t = (k()u ) a 2 u(u r 1 )(u r 2 ) In this section, we assume that Q(u) = a 2 u(u r 1 )(u r 2 ). The more general case(whereallthreerootsofthecubicaredistinctandnon-zero)canbeobtained by a simple translation in u, i.e. u = ū r 3. Upon manipulation of the determining equations, we find we must restrict the form of k() in order to solve (3). The Lie point symmetries are also recovered, however only the strictly nonclassical symmetry reductions are discussed here. From eamination of the determining equations, we find a nonclassical symmetry may eist if k() satisfies the fourth order differential equation 12k 3 k (iv) +6k(k ) 2 k 8a 2 (r 2 1 +r 2 2 r 1 r 2 )k 2 k +4a 2 (r 2 1 +r 2 2 r 1 r 2 )k(k ) 2 3(k ) 4 =. 93 Although we cannot solve this equation completely, we have found that k() = (α+β) 2 94 is a special solution. In this case, equation (1) can be written as u t = ( (α+β) 2 u ) a2 u(u r 1 )(u r 2 ). (4) Using the transformations = (α + β)/β and t = β 2 t, setting ā = a/β and dropping the bars for convenience, we can rewrite this equation as u t = ( 2 u ) a2 u(u r 1 )(u r 2 ). (5) Equation (4) admits the nonclassical symmetry generator Γ = [ ] a 2 t + (3u (r 1 +r 2 )) 1 [ ] 3a 2 2 u(u r 1)(u r 2 ) u. (6) The associated invariant surface condition is a 2 u t +[ (3u (r 1 +r 2 )) 1 ]u = 3a2 2 u(u r 1)(u r 2 ). (7) 99 1 Using the governing equation (4) and the ISC (6) we can eliminate u t. After rearranging, we obtain an ordinary differential equation (ODE) for u(, t), a 2 2 u +[ (3u (r 1 +r 2 ))+1 ]u + a2 2 u(u r 1)(u r 2 ) = Via the transformation y = ln, we can rewrite this equation as u yy + a 2 (3u (r 1 +r 2 ))u y + a2 2 u(u r 1)(u r 2 ) =, 4

7 which is the second member of the Riccati chain [31]. The Hopf-Cole transfor- 2 v y mation u = [5] transforms the above equation into a v which has the solution ) a 2 v(y,t) = c 1 (t)ep( r 1y v yyy a 2 (r 1 +r 2 )v yy + a2 2 r 1r 2 v y =, ) a 2 +c 2 (t)ep( r 2y +c 3 (t). After working back through the transformations, we find that u(,t) = c 1(t)r 1 mr1 +c 2 (t)r 2 mr2 c 1 (t) mr1 +c 2 (t) mr2 +c 3 (t), where m = a/ 2. The functions c i (t) can be found by substituting this form for u(, t) into the ISC, so that the nonclassical symmetry solution to equation (4) corresponding to the nonclassical symmetry generator (5) is given by u(,t) = c 1 r 1 e mr1(mr1+1)t mr1 +c 2 r 2 e mr2(mr2+1)t mr2 c 1 e mr1(mr1+1)t mr1 +c 2 e mr2(mr2+1)t mr2 +e 2m2 r 1r 2t, (8) where m = a/ 2. Solution (7) seems, as far as we know, not to be recorded in the literature. Figure 1 shows this solution for various values of the parameters, and for the more general case where k() = (α + β) 2. The figures show u(,t) taking on negative values whilst these cases may not have any physical meaning (depending on the problem under consideration), the range of possible behaviours is interesting Results for u t = (k()u ) a 2 u 2 (u r 1 ) In this section, we assume that Q(u) = a 2 u 2 (u r 1 ). Once again, the more general case (where the two roots of the cubic are distinct and non-zero) can be obtained by a simple translation in u. As before, in addition to finding a nonclassical symmetry, we recover the Lie point symmetries (dicussed in Section 4). We discuss only the strictly nonclassical symmetries here. Upon manipulation of the determining equations we find that a nonclassical symmetry eists if k() satisfies 12k 3 k (iv) +6k(k ) 2 k 8a 2 (r 1 r 2 ) 2 k 2 k +4a 2 (r 1 r 2 ) 2 k(k ) 2 3(k ) 4 =. 125 Although we cannot completely solve this equation, we find that one solution is k() = (α+β) As in Section 3.1, by using scale transformations and a translation in, we can simplify this so that the equation under investigation can be written u t = ( 2 u ) a2 u 2 (u r 1 ). (9) 5

8 (a) u, t.5 1 K.5 increasing time u(,t) t 1 1 (c) 1. u, t.5 increasing time 7.5 u(,t) t Figure 1: Solution (7) at times t =,1,...5 for various values of the parameters, demonstrating the different possible behaviours. The more general case is plotted, where k() = (α+β) 2. (a) a = 1, k() = 2, r 1 = 1, r 2 = 1, c 1 = 1, c 2 = 2, c 3 = 4, (b) a = 1, k() = 2, r 1 = 4, r 2 = 1, c 1 = 1, c 2 = 2, c 3 = 1.5, (c) a = 1, k() = (2 + ) 2, r 2 = 1, r 3 = 2, c 1 = 1, c 2 = 2, c 3 = 4, (d) a = 1, k() = 2, r 1 = 6, r 2 = 11, c 1 = 1, c 2 = 4, c 3 = The nonclassical generator is given by Γ = [ ] a 2 t + (3u r 1 ) 1 [ ] 3a 2 2 u2 (u r 1 ) u, (1) and the associated ISC is a 2 u t +[ (3u r 1) 1 ]u = 3a2 2 u2 (u r 1 ) Following the same proceedure as that outline in Section 3.1, we find the solution to equation (1) corresponding to the nonclassical symmetry generator (11) is 6

9 c 1m +c 3 r 1 e mr1(mr1+1)t mr1 u(,t) = (11) c 2 +(1 2mr 1 )c 1 t+c 1 ln+c 3 e mr1(mr1+1)t mr1 Solution (12) seems, as far as we know, not to be recorded in the literature. Figure 2 plots solution (12) for various values of the parameters. (a) 4 3 increasing time (b) K1 K u, t 2 u, t K3 increasing time 1 K K5 Figure 2: Solution (12) for various values of the parameters, demonstrating the different possible behaviours. (a) a = 1, k() = 2, r 1 = 4, r 2 = 1, c 1 = 1, c 2 = 2, c 3 = 4, at times t =,1,...5, (b) a = 1, k() = 2, r 1 = 4, r 2 = 1, c 1 = 1, c 2 = 2, c 3 = 4 at times t =,.2, Results for u t = (k()u ) a 2 u 3 In this section, we assume that Q(u) = a 2 u 3. As before, the more general case (where the repeated root is non-zero) can be obtained by a translation in u. This equation, with a = 2, was eamined in [17]. For general a, the details in addition to finding a nonclassical symmetry, the Lie point symmetries are 142 recovered, however these are discussed in Section Upon manipulation of the determining equations, we find that nonclassical 144 symmetries may eist if k() satisfies 4k 3 k (iv) +2k(k )2k (k ) 4 =. (12) 145 Although we cannot solve equation (13) completely, we find that k() = (α+β) 6/5 and k() = (α+β) are solutions. In the case where k() = (α+β) 6/5, reductions using the nonclassical symmetry generator lead to an equation which could not be integrated. In the case where k() = (α + β) 2, we may once again simplify the problem 7

10 using scale transformations and a translation. We therefore provide the solution for the case where k() = 2. In this case, the governing equation is given by u t = ( 2 u ) a2 u 3. (13) This equation admits the nonclassical symmetry generator Γ = [ ] 3a t + u 1 2 3a2 2 u3 u, (14) which has the associated ISC [ 3a u t + u 1 ]u = 3a2 2 2 u3. (15) Following the procedure outlined in Section 3.1, we find that the solution to the governing equation (14) corresponding to the nonclassical symmetry (15) is 2ln+2t+c 1 6 u(,t) = m(ln) 2 +m(2t+c 1 6)ln+m(t 2 +c 1 t c 2 ) (16) This solution was given in [17] for the case when a = 2. For general a, solution 155 (17) seems not to be recorded in the literature. Figure 3 3 plots solution (17) for various values of the parameters. (a).5 (b) K increasing time u, t K.5 increasing time u, t K.2 K.3 K1. K.4 K.5 Figure 3: Solution (17) for various values of the parameters, at times t =,.2,...1, demonstrating the different possible behaviours. (a) a = 1, k() = (1 + ) 2, r 1 = 1, r 2 = 1, c 1 = 1, c 2 = 2, c 3 = 4, (b) a = 1, k() = (1+) 2, r 1 = 1, c 1 = 1, c 2 = Results using classical Lie point symmetry analysis In order to compare the solutions obtained by nonclassical symmetry analysis with those obtained by Lie point symmetry analysis, we have eamined u t = [ (α+β) 2 u ] +Q(u), (17) 8

11 where Q(u) is a factorisable cubic. The symmetries admitted by equation (18) are listed in Table 1. Q(u) admitted symmetry generators Q(u) = a 2 (u r 1 )(u r 2 )(u r 3 ) Γ 1 = t Q(u) = a 2 (u r 1 ) 2 (u r 2 ) Q(u) = a 2 (u r 1 ) 3 Γ 2 = (α+β) Γ 1 = t Γ 2 = (α+β) Γ 3 = 1 2 (u r 1) u + 1 2β [(α+β)ln(α+β) β 2 t(α+β) ] +t t Table 1: Lie point symmetries admitted by equation (1) for different types of factorisable cubic Q(u) Reduction for Q(u) = a 2 (u r 1 )(u r 2 )(u r 3 ) Taking a linear combination of the two admitted symmetries, Γ 1 +γγ 2, we obtain the invariants z = (α+β) 1/β e γt and u = F(z). 166 We can rewrite the governing equation in terms of these invariants to obtain z 2 F zz +(1+β γ)zf z a 2 (F r 1 )(F r 2 )(F r 3 ) = Introducing y = lnz, we can rewrite this as F yy +(β γ)f y a 2 (F r 1 )(F r 2 )(F r 3 ) =. Via the transformation w = df, we obtain a first order nonlinear ODE dy w dw df +(β γ)w a2 (F r 1 )(F r 2 )(F r 3 ) =. (18) Equation (19) is an Abel equation of the second kind [32] whose solution depends on the parameters β, γ, r 1, r 2 and r 3. For eample, (i) if γ = β, equation (19) is separable and the solution to equation (4) can be written in terms of integrals over Jacobi elliptic integrals [33]. (ii) if a 2 = 2, r 1 =, r 2 = 1, and r 3 = 1 2 (1 + β γ), then w = F F2 is a particular solution [34]. Working back through the transformations, we obtain [ u(,t) = 1+c(α+β) 1/β e γt] 1, (19) where c is a constant. Solution (2) is therefore a solution to equation (4) and seems, as far as we know, not to be recorded in the literature. 9

12 Reduction for Q(u) = a 2 (u r 1 ) 2 (u r 2 ) Once again, we take a linear combination of the two admitted symmetries, Γ 1 + γγ 2. Via a set of transformations similar to those described above, we obtain the first order nonlinear ODE w dw df +(β γ)w a2 (F r 1 ) 2 (F r 2 ) =. (2) Once again, this is an Abel equation of the second kind [32] whose solution depends on the parameters β, γ, r 1 and r 2. For eample, (i) if γ = β, once again, equation (21) is separable, and the solution to equation (1) can be written in terms of integrals over Jacobi elliptic integrals [33]. (ii) if a 2 = 2, r 1 =, and r 2 = β γ we find that w(f) = F 2 + (β γ)f is a particular solution to equation (21). Working back through the transformations, we find that u(,t) = (β γ) [1+c ] 1 (α+β) 1 γ 1 β e γ(β γ)t (21) isasolutiontoequation(1)andseemsnottoberecordedintheliterature. (iii) if a 2 = 2, r 1 = (β γ), and r 2 = we find that w(f) = F 2 + (β γ)f is a particular solution to equation (21). Working back through the transformations, we find that u(,t) = (β γ) [1+c 1 (α+β) γ β 1 e γ(β γ)t] 1 (22) is a solution to equation (1) and seems, as far as we are aware, not to be recorded in the literature Reduction for Q(u) = a 2 (u r 1 ) 3 In this case, the governing equation admits the same two generators, plus an additional one. By taking a linear combination of the first two symmetries, Γ 1 +γγ 2, and using a similar set of transformations to those in Section 4.1, we obtain the first order nonlinear ODE w dw df +(β γ)w a2 (F r 1 ) 3 =. (23) This is an Abel equation of the second kind whose solution depends on the parameters β, γ and r 1. For eample, (i) if γ = β, equation (24) can be integrated to obtain w(f) = a 2 (F r 1 ) 2. Working back through the transformations, we find that u(,t) = r 1 β 2 a is a solution to equation (14). [ ln(α+β)+β 2 t+c 1 ] 1 (24) 1

13 (ii) if r 1 =, the classical Lie point symmetry analysis for equation (14) is given in [17], and herein omitted. In the case of the third admitted generator, Γ 3, we find the invariants z = 1 t ln(α+β)+β 2 t and u = 1 t F(z)+r The governing equation can then be rewritten as β 2 F zz zf z F a2 F 3 =. 21 Eact analytic solutions to this ODE could not be found Applications to population genetics As discussed briefly in Section 1, equations of the form (1) can be used to model problems in population genetics. In particular, we eamine the problem originally described by Fisher[35], where a new advantageous but recessive allele is introduced to a population which is living in a one dimensional habitat such as along a river or a shoreline (for eample, in the occurrence of a mutation). If this new allele becomes established, it will spread through the population. To describe this spread in a region where diffusion is spatially dependent, we first write equations which describe the density of each genotype ρ 11 = ( k() ρ ) 11 µρ 11 +γ 11 p 2 ρ, t ρ 12 = ( k() ρ ) 12 µρ 12 +2γ 12 p(1 p)ρ, (25) t ρ 22 = ( k() ρ ) 22 µρ 22 +γ 11 (1 p) 2 ρ, t where γ ij is the reproductive success rate of genotype A i A j, µ is the common death rate, ρ(,t) is the total population density, ρ(,t) = ρ 11 (,t)+ρ 12 (,t)+ ρ 22 (,t) and p(,t) is the frequency of allele A 1. The frequency of allele A 2 is given by 1 p(,t). The spatially dependent diffusion is assumed to be the same for each genotype. The frequency of allele A 1 is can be written p = 2ρ 11 +ρ 12 2ρ. (26) By differentiating equation (27) with respect to t, we find that equations (26) remarkably collapse into a single equation that describes the change in frequency of the new gene, A 1, p t = ( k() p ) + 2k() ρ p ρ +p(1 p)(g 1 g 2 p), (27) 11

14 where g 1 = γ 12 γ 22 and g 2 = γ 11 +2γ 12 γ 22. Equation (28) with k() = was first presented in [19] and subsequently derived, with some eact solutions presented in [7, 18]. In the case where k() =, nonclassical symmetries of a related class of equations, u t = u +f(u)u +Q(u) with Q(u) cubic, have been eamined [?? ]. In the case where the total population density is constant (ρ = ), equation (28) reduces precisely to equation (1), where Q(u) is a factorisable cubic. The solutions derived within this paper are then directly applicable here. Figure 4 shows a plot of the spatial dependence of the diffusion coefficient, and solution (7) for γ 11 = 2, γ 12 = 3, γ 22 = 1, (ie r 1 =, r 2 = 1, r 3 = 2/3). In this case, the genotype with one copy of each allele is the most advantageous (γ 12 > γ 11, γ 22 ). Also, α =, β = 1, and c 1 = 1, c 2 = 2, c 3 = 4. Figure 4(a) shows that the spatially dependent diffusion is higher in the right of the range than in the left of the range. Figure 4(b) shows how the frequency of the advantageous gene, p(, t), changes with time. Initially, the new gene is more common in the right of the range and, over time, it spreads across the range, taking over the whole area. (a) (b) k p, t.5 increasing time Figure 4: Nonclassical symmetry solution as applied to population genetics. (a) shows the spatially dependent diffusion coefficient, (b) shows the change in frequency of advantageous gene Final remarks In this paper, the nonclassical and classical Lie point symmetry techniques have been successfully employed to analyse a reaction-diffusion equation with an arbitrary spatially dependent diffusion coefficient and a cubic source term. This equation admits nonclassical symmetries when the diffusivity is given as a quadratic function of the space variable. New eact solutions (equations (7), (12), (17), (2), (22), (23), (25)) are constructed using both nonclassical and 12

15 classical symmetries. An application of one of the solutions to a problem in population genetics is discussed. Bibliography [1] W.I. Fushchych, I.M. Tsyfra, On a reduction and solutions of the nonlinear wave equations with broken symmetry, J Phys A: Math Gen, 2 (1987) L45-L48. [2] G.W. Bluman and J.D. Cole, The general similarity solution of the heat equation, J Math Mech, 18 (1969) [3] M.C. Nucci, Nonclassical symmetries as special solutions of heir-equations, J Math Anal Appl, 279 (23) [4] N.I. Serov, Conditional invariance and eact solutions of the nonlinear equation, Ukrainian Mathematical Journal, 42 (199) 1, [5] D. Arrigo, J.M. Hill and P. Broadbridge, Nonclassical symmetry reductions of the nonlinear diffusion equation with a nonlinear source, IMA J Appl Math, 52 (1994) 1-24 [6] P.A. Clarkson and E.L. Mansfield, Symmetry reductions and eact solutions of a class of nonlinear heat equations, Physica D, 7 (1994) [7] B.H. Bradshaw-Hajek, P. Broadbridge, A robust cubic reaction-diffusion system for gene propagation, Math Comput Model, 39 (24) [8] B.H. Bradshaw-Hajek, M.P. Edwards, P. Broadbridge, G.H. Williams, Nonclassical symmetry solutions for reaction-diffusion equations with eplicit spatial dependence, Nonlinear Analysis, 67 (27) [9] M.S. Hashemi and M.C. Nucci, Nonclassical symmetries for a class of reaction diffusion equations: the method of Heir-equations, J Nonlinear Math Phys, 2(1) (213), [1] O.O. Vaneeva, R.O. Popovych, C. Sophocleous, Enhanced group analysis and eact solutions of variable coefficient semilinear diffusion equations with a power source, Acta Appl Math, 16 (29) [11] O.O. Vaneeva, R.O. Popovych, C. Sophocleous, Reduction operators of variable coefficient semilinear diffusion equations with a power source, Proceedings of 4 th International Workshop Group Analysis of Differential Equations and Integrable Systems (October 26-3, 28, Protaras, Cyprus) (29) , arxiv: [12] O.O. Vaneeva, R.O. Popovych, C. Sophocleous, Reduction operators of variable coefficient semilinear diffusion equations with an eponential source, Proceedings of 5 th International Workshop Group Analysis of Differential Equations and Integrable Systems (June 6-1, 21, Protaras, Cypris) (211) , arxiv:

16 [13] M.C. Nucci and P.A. Clarkson, The nonclassical method is more general than the direct method for symmetry reductions: An eample of Fitzhugh- Nagumo equation, Phys Lett A, 164 (1992) [14] O. Vaneeva and A. Zhalij, Group classification of variable coefficient quasilinear reaction-diffusion equations, Publications de l Institut Mathematique- Beograd, 94(18) (213) [15] N.M. Ivanova and C. Sophocleous, On nonclassical symmetries of generalized Huley equations, Proceedings of 5 th International Workshop Group Analysis of Differential Equations and Integrable Systems (June 6-1, 21, Protaras, Cyprus) (211) 91-98, arxiv: v1 [16] G.W Bluman and S. Kumei, Symmetry-based algorithms to relate partial differential equations: I. Local Symmetries, Euro J Appl Math (199) [17] R.J. Moitsheki and B.H. Bradshaw-Hajek, Symmetry analysis of a heat conduction model for heat transfer in a longitudinal rectangular fin of a heterogeneous material, Commun Nonlinear Sci, 18 (213) [18] P. Broadbridge, B.H. Bradshaw, G.R. Fulford, G.K. Aldis, Huley and Fisher equations for gene propagation: an eact solution, ANZIAM J. 44 (22) 11-2 [19] A.D Bazykin, Hypothetical mechanism of speciation, Evolution, 23 (1969) [2] A.C. Scott, The electrophysics of a nerve fiber, Rev Mod Phys, 47 (1975) [21] Y.B. Zeldovich and D.A. Frank-Kamenetsky, Acta Physicochim URS, 9 (1938) 341 [22] G.W. Bluman and S. Kumei, Symmetries and Differential Equations, AMS No. 81, Springer, New York, 1989 [23] G.W. Bluman and S.C. Anco, Symmetry and Integration Methods for Differential Equations, Applied Mathematical Sciences, 22 [24] G.W. Bluman, A.F. Cheviakov and S.C. Anco, Applications of symmetry methods to partial differential equations, AMS No 168, Springer, New York, 1989 [25] P.J. Olver, Applications of Lie group to differential equations, AMS No 17, Springer, New York, 1982 [26] N.H. Ibragimov, Elementary Lie group analysis and ordinary differential equations, John Wiley and Sons, New York,

17 [27] MAPLE 15 User Manual, Toronto: Maplesoft a division of Waterloo Maple Inc., [28] A.C. Hearn, Reduce user manual version 3.4. Santa Monica, California: Rand Publication CP78, Rand Corporation, [29] R.Z. Zhdanov, V.I. Lahno, Conditional symmetry of a porous medium equation, Physica D, 122 (1998) [3] M. Kunzinger, R.O. Popovych, Singular reduction operators in two dimensions, J Phys A, 41 (28) 5521, 24pp [31] W.F. Ames, Nonlinear Ordinary Differential Equations in Transport Processes. Academic Press, New York, 1968 [32] A.D. Polyanin and V.F. Zaitsev, Handbook of Eact Solutions for Ordinary Differential Equations, Chapman and Hall/CRC, Boca/Raton, Florida, 23 [33] J. Spanier and K.B. Oldham, An Atlas of Functions, Hemisphere Publishing, 1987 [34] B.H. Bradshaw-Hajek, Reaction-diffusion equations for population genetics, PhD thesis, 24 [35] R.A. Fisher, The wave of advance of advantageous genes, Ann Eugenic, 7 (1937) [36] R. Cherniha, M. Serov, Symmetries, ansaetze and eact solutions of nonlinear second-order evolution equations with convection terms, Eur J Appl Math, 9 (1998) [37] R. Cherniha, New Q-conditional symmetries and eact solutions of some reaction-diffusion-convection equations arising in mathematical biology, J Math Anal Appl, 326 (27)