Classical and potential symmetries for a generalized Fisher equation

Journal of Computational and Applied Mathematics (

)

–

Contents lists available at ScienceDirect

Journal of Computational and Applied Mathematics journal homepage: www.elsevier.com/locate/cam

Classical and potential symmetries for a generalized Fisher equation M. Rosa ∗ , J.C. Camacho, M.S. Bruzón, M.L. Gandarias Departamento de Matemáticas, Universidad de Cádiz, 11500 Puerto Real, Cádiz, Spain

article

info

Article history: Received 11 June 2016 Received in revised form 25 October 2016 MSC: 76M60 92D25 35Q91 Keywords: Symmetry reductions Potential symmetries Fisher equation Reaction–diffusion Partial differential equations

abstract In this work, we consider a generalized Fisher equation and we have considered this equation from the point of view of the theory of symmetry reductions in partial differential equations. Generalizations of the Fisher equation are needed to more accurately model complex diffusion and reaction effects found in many biological systems. The reductions to ordinary differential equations are derived from the optimal system of subalgebras and new exact solutions are obtained. The potential system has been achieved from the complete list of the conservation laws. Potential symmetries, which are not local symmetries, are carried out for the generalized Fisher equation, these symmetries lead to the linearization of the equation by non-invertible mappings. © 2016 Elsevier B.V. All rights reserved.

Contents 1. 2. 3. 4. 5.

Introduction............................................................................................................................................................................................. Optimal systems and symmetry reductions ......................................................................................................................................... Exact solutions ........................................................................................................................................................................................ Classical potential symmetries............................................................................................................................................................... 4.1. Classical potential symmetries of Eq. (3) .................................................................................................................................. Conclusions.............................................................................................................................................................................................. Acknowledgments .................................................................................................................................................................................. References................................................................................................................................................................................................

1 3 4 5 6 7 7 7

1. Introduction Fisher equations are commonly used in biology for population dynamics models and in bacterial growth problems as well as development and growth of solid tumors. The theory of reaction–diffusion waves begins in the 1930s with the works in population dynamics, combustion theory and chemical kinetics. At the present time, it is a well developed area of research which includes qualitative properties of traveling waves for the scalar reaction–diffusion equation and for system of equations, complex nonlinear dynamics, numerous applications in physics, chemistry, biology and medicine

∗

Corresponding author. E-mail address: [email protected] (M. Rosa).

http://dx.doi.org/10.1016/j.cam.2016.10.028 0377-0427/© 2016 Elsevier B.V. All rights reserved.

2

M. Rosa et al. / Journal of Computational and Applied Mathematics (

)

–

[1–3]. Reaction–diffusion equations are conventionally used in physical chemistry in order to describe concentration and temperature distributions. In this case, heat and mass transfer are described by the diffusion term while the reaction term describes the rate of heat and mass production. The equation analyzed in this paper is a generalized Fisher equation ut = f (u) +

1 c (x)

(c (x)g (u)ux )x ,

(1)

where u(x, t ) denotes the tumor cell density at location x and time t, being x and t the independent variables, g (u) is the diffusion coefficient representing the active motility of cells depending on the variable u, f (u) an arbitrary function and c (x) an arbitrary function depending on the spatial variable x. In the particular case of c (x) = 1 and g (u) = 1, symmetry reductions and exact solutions were obtained using classical and nonclassical symmetries in [4]. Lie symmetry analysis of differential equations provides a powerful and fundamental framework to the exploitation of systematic procedures leading to the integration by quadrature of ordinary differential equations, to the determination of invariant solutions of initial and boundary value problems and to the derivation of conservation laws [5,6]. When c (x) = 1 but g (u) and f (u) are arbitrary functions, equation ut = f (u) + (g (u)ux )x

(2)

is known as the density dependent diffusion–reaction equation which is mentioned by J.D. Murray [2] to model the advance of an advantageous gene through a geographic region. In [7–9], we have derived conservation laws and exact solutions of several interesting particular cases of Eq. (1). It is known that conservation laws play a significant role in the solution process of an equation or a system of differential equations and its applications to PDEs [10–18]. In [19], Anco and Bluman gave a general algorithmic method to find all conservations laws for evolution equations like Eq. (1). Many recent papers using this method have been published [20–22,7,23]. Local symmetries admitted by a nonlinear PDE are also useful to discover whether or not the equation can be linearized by an invertible mapping and construct an explicit linearization when one exists. A nonlinear scalar PDE is linearizable by an invertible contact (point) transformation if and only if it admits an infinite-parameter Lie group of contact transformations satisfying specific criteria [24–26]. An obvious limitation of group-theoretic methods based on local symmetries, in their utility for particular PDEs, is that many of these equations do not have local symmetries. It turns out that PDEs can admit nonlocal symmetries whose infinitesimal generators depend on integrals of the dependent variables in some specific manner. It also happens that if a nonlinear scalar PDE does not admit an infinite-parameter Lie group of contact transformations, it is not linearizable by an invertible contact transformation. However, most of the interesting linearizations involve non-invertible transformations; such linearizations can be found by embedding given nonlinear PDEs in auxiliary systems of PDEs [24]. Krasil’shchik and Vinogradov [27] gave criteria which must be satisfied by nonlocal symmetries of a PDE when realized as local symmetries of a system of PDEs which covers the given PDE. In [24,25], Bluman introduced a method to find a new class of symmetries for a PDE. By writing a given PDE, denoted by R {x, t , u} in a conserved form, a related system denoted by S {x, t , u, v} with potentials as additional dependent variables is obtained. Any Lie group of point transformations admitted by S {x, t , u, v} induces a symmetry for R {x, t , u}; when at least one of the generators of the group depends explicitly on the potential, then the corresponding symmetry is neither a point nor a Lie–Bäcklund symmetry. These symmetries of R {x, t , u} are called potential symmetries. The nature of potential symmetries allows one to extend the uses of point symmetries to such nonlocal symmetries. In particular: (i) Invariant solutions of S {x, t , u, v} yield solutions of R {x, t , u} which are not invariant solutions for any local symmetry admitted by R {x, t , u}. (ii) If R {x, t , u} admits a potential symmetry leading to the linearization of S {x, t , u, v} then R {x, t , u} is linearized by a non-invertible mapping. In [9], we have constructed conservation laws for Eq. (1). These conservation laws were derived by using a conservation theorem due to Ibragimov, as well as the multiplier method of Anco and Bluman [19]. In this work, we consider a particular case of Eq. (1) when c (x) = k1 erx , g (u) = k2 u−2 and f (u) = k3 u with k1 , k2 , k3 and r arbitrary constants ut = k 3 u +

1 k1

erx

k1 erx k2 u−2 ux



 x

.

(3)

By using the symmetry generators previously derived in [9], we perform the optimal system of subalgebras, the similarity reductions, and we find for exact solutions of physical and chemical interest. For Eq. (3) it happens that some of the associated conserved systems admit symmetries that yield to potential symmetries. These symmetries are realized as local symmetries of a related auxiliary system, and lead to the construction of corresponding invariant solutions, as well as to the linearization of the equation by non-invertible mappings.

M. Rosa et al. / Journal of Computational and Applied Mathematics (

)

–

3

Table 1 Functions ci , gi , fi with i = 1 . . . 9 and generators vk with k = 1 . . . 9. i

ci

gi

fi

vk

1 2 3 4

arbitrary k1 xr k1 xr k1 erx

arbitrary k2 uq k2 uq k2 uq

arbitrary k3 up k3 u k3 up

v1 = ∂t v1 , v2 = (q + 1 − p)x∂x + 2(1 − p)t ∂t + 2u∂u v1 , v2 , = qx∂x +2u∂u , v3 = e−k3 qt ∂t +k3 e−k3 qt u∂u v1 , v4 = ∂x

5

k1 erx

k2 u2

k3 u

v1 , v3 , v4 , v5 =

rx

v1 , v4 , v6 =

p

e−rx r

∂x + e−rx u∂u

(−x+k2 rt )(p−1)

6

k1 e

k2

k3 u

7

k1 erx

k2 equ

k3 epu

v1 , v4 (p ̸= q)

8

k1 erx

k2 equ

2k2 equ r 2 9q

v1 , v4 , v7 =

9

k1 erx

k2

k3 epu

v1 , v4 , v9 =

−3qe

2

∂x − (p − 1)t ∂t + u∂u

−rx 3

∂ +e

x 2r p(−x+k2 rt ) x 2

−rx 3

∂u , v8 = −qt ∂t +∂u

∂ − pt ∂t + ∂u

Table 2 Commutator table.

[vi , vj ]

v1

v3

v4

v5

v1 v3 v4 v5

0 k3 q v3 0 0

−k3 q v3

0 0 0 0

0 0

0 0 r v5

−r v5 0

2. Optimal systems and symmetry reductions In [9], we have applied the classical Lie method to Eq. (1). To apply the classical method to Eq. (1), one looks for infinitesimal generators of the form v = ξ (x, t , u)

∂ ∂ ∂ + τ (x, t , u) + φ(x, t , u) , ∂x ∂t ∂u

(4)

that leave invariant this equation. The functional forms of c (x), f (u) and g (u) as well as the corresponding generators are given in Table 1. We focus our attention on Eq. (3) corresponding to case 5 in Table 1. We seek solutions of the invariant equation ut = k3 u +

1 k1

erx

k1 erx k2 u−2 ux



 x

,

(5)

the symmetries of which are given by v1 = ∂t ,

v3 = e−k3 qt ∂t + k3 e−k3 qt u∂u ,

v4 = ∂x ,

v5 =

e−rx r

∂x + e−rx u∂u .

The optimal system of one-dimensional subalgebras [28] is a systematic procedure from which all the possible invariant solutions are obtained. We follow the approach given in [28] to construct the optimal system, we construct the commutator tables and the adjoint tables which shows the separate adjoint actions of each element in vi , i = 1 . . . 5, as it acts on all other elements. This construction is done easily by summing the Lie series (see Tables 2 and 3). The commutator operation is vi , vj = vi (vj ) − vj (vi ) i, j = 1, 3, 4, 5.





The adjoint representation is Ad(exp(ϵ vi ))vj = vj − ϵ vi , vj +





 ϵ 3     ϵ2   vi , vi , vj − vi , vi , vi , vj + · · · , ϵ ∈ R. 2! 3!

The corresponding generators of the optimal system of subalgebras are

⟨v1 + av4 , v3 + bv5 , v1 + bv5 , v3 + av4 , v5 ⟩ where a, b ∈ R are arbitrary constants. In the following, similarity solutions and reductions of Eq. (3) to ODE’s are obtained using the generators of the optimal system (see Tables 4 and 5).

4

M. Rosa et al. / Journal of Computational and Applied Mathematics (

)

–

Table 3 Adjoint table. v1

Ad v1 v3 v4 v5

v3 k3 q ε

v1 v1 − ε k3 q v3 v1 v1

e v3 v3 v3

v3

v4

v5

v4 v4 v4 v4 − ε r v5

v5 v5 er ε v 5 v5

Table 4 Similarity solutions. i

Generator

zi

ui

1

v1 + av4

x − at

h (z )

2 3

v3 + bv5 v1 + bv5

er x + b e2 k 3 er x − b t

4 5

v3 + av4 v5

x+ t

− 2 k3 t

2 h(z ) k3 er x+k3 t h(z ) er x

a e − 2 k3 t 2 k3

h(z ) ek3 t h(z ) er x

Table 5 Reduced ODEs. i

ODEsi

1

hz z + hz r + hz z +

2

h3 k3 k2 r 2 a h 2 hz k2 r h4 k3

3 4 5

h3 k 3

+ a hk2hz − 2 (hhz ) = 0

k2 4 b h 2 hz k 3 2 k2 r 2 b h 2 hz k2 r 2

+

2

2

−

2 (hz )2 h

+ hz z −

=0

2 ( hz ) 2 h

=0

+ hzr z − 2 (hhzr ) + hz = 0 − h3 hz = 0 2

3. Exact solutions In this section we give exact solutions for some reduced equations. ODE1 : We realize that the ODE1 admits the group corresponding to the generator w1 = ∂z . Taking into account the invariants of its first prolongation and using the new variables h = ζ,

h′ = v(ζ ),

h′′ = v(ζ )

dv dζ

,

(6)

we obtain that ODE1 can be reduced to the first order ODE

v vξ −

2 v2

ξ

+rv+

ξ2 a v k2

+

ξ 3 k3 k2

.

(7)

ODE2 : We realize that the ODE2 admits the group corresponding to the generators w1 = ∂z and w2 = −2z ∂z + h∂h . For generator w1 and taking into account the invariants of its first prolongation and using the variables (6), we obtain that ODE2 can be reduced to the first order ODE

v vξ −

2 v2

ξ

+

4 ξ 2 b k3 v k2 r 2

,

(8)

whose solution is:

v = ξ2

 C−

4 ξ b k3



k2 r 2

,

(9)

where C is arbitrary constant. ODE3 : We realize that the ODE3 admits the group corresponding to the generator w1 = ∂z . Taking into account the invariants of its first prolongation and using the variables (6), we obtain that ODE3 can be reduced to the first order ODE

v vξ −

2 v2

ξ

+

ξ2 b v k2

r2

+

ξ 3 k3 k2 r 2

.

(10)

M. Rosa et al. / Journal of Computational and Applied Mathematics (

)

–

5

ODE4 : We obtain this implicit solution: ln a (h)2 − k2 r − hC1 k2



−

2





− k2 C1 arctanh 



2ah − k2 C1 2

4k2 ra + k2 C1

2

1



4k2 ra + k2 2 C1 2

+ ln (h) − zr − rC2 = 0,

where C1 and C2 are arbitrary constants. ODE5 : We obtain this solution: h = C1 ek3 z .

(11)

4. Classical potential symmetries In [19] Anco and Bluman gave a general treatment of a direct conservation law method for partial differential equations expressed in a standard Cauchy–Kovalevskaya form, in particular for evolution equations ut = G(x, u, ux , uxx , . . . , unx ). The nontrivial conservation laws are characterized by a multiplier λ with no dependence on ut satisfying Eˆ [u] (Λut − ΛG(x, u, ux , uxx , . . . , unx )) = 0. Here Eˆ [u] :=

∂ ∂ ∂ ∂ − Dt − Dx + D2x + ··· . ∂u ∂ ut ∂ ux ∂ uxx

The conserved current must satisfy

Λ = Eˆ [u]Φ t and the flux Φ x is given by [29]

∂Φt Φ = −Dx (ΛG) − G + GDx ∂ ux −1

x



∂Φt ∂ uxx



+ ··· .

The conservation law will be written Dt (Φ t ) + Dx (Φ x ) = 0. For Eq. (3) by using Maple software, we got the following multipliers. Each multiplier determines a corresponding conserved density and flux. For f (u) = u, g (u) = uq , c (x) = ex , we get that multiplier is

Λ = k1 ex−t + k2 e−t and we obtain the corresponding conserved density and flux:

Λ = e x −t φ t = e x −t u

Λ = e −t φ t = e −t u

φ x = −ex−t uq ux

φ x = −e−t uq ux −

e − t uq + 1 q+1

.

In [30], Bluman introduced a method to find a new class of symmetries for a PDE. Suppose a given scalar PDE of second order F (x, t , u, ux , ut , uxx , uxt , utt ) = 0,

(12)

where the subscripts denote the partial derivatives of u, can be written as a conservation law D Dt

f (x, t , u, ux , ut ) −

D Dx

g (x, t , u, ux , ut ) = 0,

for some functions f and g of the indicated arguments. Here

∂ ∂ ∂ ∂ + ux + uxx + uxt + ··· , Dx ∂x ∂u ∂ ux ∂ ut D ∂ ∂ ∂ ∂ = + ut + uxt + utt + ··· . Dt ∂t ∂u ∂ ux ∂ ut D

=

(13) D Dx

and

D Dt

are total derivative operators defined by

6

M. Rosa et al. / Journal of Computational and Applied Mathematics (

)

–

Through the conservation law (13) one can introduce an auxiliary potential variable v and form an auxiliary potential system

vx = f (x, t , u, ux , ut ), vt = g (x, t , u, ux , ut ).

(14)

Any Lie group of point transformations XS = ξ (x, t , u, v)

∂ ∂ ∂ ∂ + τ (x, t , u, v) + φ(x, t , u, v) + ψ(x, t , u, v) , ∂x ∂t ∂u ∂v

(15)

admitted by (14) yields a nonlocal symmetry potential symmetry of the given PDE (13) if and only if the following condition is satisfied

ξv2 + τv2 + φv2 ̸= 0.

(16)

4.1. Classical potential symmetries of Eq. (3) In order to find potential symmetries of Eq. (3) we write the equation in a conserved form and the associated auxiliary system is given by



vx = e−t u, vt = −e−t u−2 ux + e−t u−1 .

(17)

A Lie point symmetry characterized by an infinitesimal transformation of the form x∗ = x + εξ (x, t , u, v) + O (ε 2 ), t ∗ = t + ετ (x, t , u, v) + O (ε 2 ),

(18)

u∗ = u + εφ(x, t , u, v) + O (ε 2 ),

v ∗ = v + εφ(x, t , u, v) + O (ε 2 ) is admitted by system (17). In the present work, we will present the point symmetries of (17) and we will study which of these symmetries induce potential symmetries of Eq. (3). These symmetries are such that the condition (16) is satisfied. If the above relation does not hold, then the point symmetries of (17) project into point symmetries of (3). System (17) admit Lie symmetries if and only if pr (1) X (vx − e−t u) = 0,

pr (1) X (vt + e−t u−2 ux − e−t u−1 ) = 0, where pr (1) X is the first extended generator of XS . We get

η = −g e−x − τ=

k4 e2 t 2

−

k5 v 2

+

2 k5 e−2 t 2

k2 v 2

−

k5 e−2 t 2

+

k6 2

+ k1 ,

+ k6 ,

φ = −g u e−x + gv u2 e−x−t + k5 e−t u2 v − ψ = k5 e−2 t v − k6 v −

k2 e−2 t 2

k 2 e − t u2 2

+

k4 e2 t u 2

+

k5 e−2 t u 2

,

+ k3 ,

where g = g (t , v) must satisfy gt e2 t + gv v = 0. The generators corresponding to k2 , k5 and g (t , v) correspond to potential symmetries, these are

∂ ∂ ∂ − e − t u2 − e−t , ∂x ∂u ∂v ∂ ∂ −2t ∂ −2t ∂ v2 = −e −e + e−t (2u2 v + u) + e−2t v , ∂x ∂u  ∂t  ∂v ∂ g (t , v) 2 −x−t ∂ −x ∂ −x v∞ = −ge − g (t , v)ue − u e . ∂x ∂v ∂u v1 = v

(19)

M. Rosa et al. / Journal of Computational and Applied Mathematics (

)

–

7

The nonlinear equation (3) does not admit an infinite-parameter Lie group of contact transformations; however, its associated auxiliary system (17) admits an infinite-parameter Lie group of point transformations with infinitesimal generator v∞ , where g (t , v) is an arbitrary function satisfying Eq. (19). One can obtain the invertible mapping z1 = v

z2 = e−t

w1 = ex

w2 =

−e x u

which transforms any solution (w1 (z1 , z2 ), w2 (z1 , z2 )) of the linear system

∂w1 w2 =− ∂ z1 z2 ∂w1 ∂w2 = ∂ z1 ∂ z2 to a solution (u(x, t ), v(x, t )) of the nonlinear system (17) and hence to a solution u(x, t ) of Eq. (3). 5. Conclusions In this paper, by using Lie symmetries of the generalized Fisher equation (1) derived in [9], we have obtained the optimal system of one-dimensional subalgebras of Eq. (3). We have obtained reductions to ODE’s and some exact solutions. The potential system has been achieved from the complete list of the conservation laws. We have constructed nonlocal symmetries (potential symmetries) which are realized as local symmetries of a related auxiliary system of differential equations, by using potential symmetries we have also linearized (3) by an explicit non-invertible mapping. We have used Maxima and Maple software. Acknowledgments The support of DGICYT project MTM2009-11875, UCA and Junta de Andalucía FQM-201, is gratefully acknowledged. References [1] J. Belmonte-Beitia, G. Calvo, V. Pérez-García, Effective particle methods for the Fisher-Kolmogorov equations: Theory and applications to brain tumor dynamics, Commun. Nonlinear Sci. Numer. Simul. 19 (2014) 3267–3283. [2] J. Murray, Mathematical Biology, third ed., Springer-Verlag, New York, Berlin, Heidelberg, 2002. [3] K.R. Swanson, C. Bridgea, J. Murray, C. Ellsworth, J. Alvord, Virtual and real brain tumors: using mathematical modeling to quantify glioma growth and invasion, J. Neurol. Sci. 216 (2003) 1–10. [4] P.A. Clarkson, E.L. Mansfield, Symmetry reductions and exact solutions of a class of nonlinear heat equations, Physica D 70 (1993) 250–288. [5] M. Abd-el-Malek, A.M. Amin, Lie group analysis of nonlinear inviscid flows with a free surface under gravity, J. Comput. Appl. Math. 258 (2014) 17–29. [6] R. de la Rosa, M. Bruzón, On the classical and nonclassical symmetries of a generalized gardner equation, Appl. Math. Nonlinear Sci. 1 (2015) 263–272. http://dx.doi.org/10.21042/AMNS.2016.1.00021. [7] M. Rosa, M. Gandarias, Multiplier method and exact solutions for a density dependent reaction–diffusion equation, Appl. Math. Nonlinear Sci. 1 (2) (2016) 311–320. http://dx.doi.org/10.21042/AMNS.2016.2.00026. [8] M. Rosa, M. Bruzón, M. Gandarias, Symmetry analysis and exact solutions for a generalized Fisher equation in cylindrical coordinates, Commun. Nonlinear Sci. Numer. Simul. 25 (2015) 74–83. [9] M. Rosa, M. Bruzón, M. Gandarias, Lie symmetry analysis and conservation laws for a Fisher equation with variable coefficients, Appl. Math. Inf. Sci. 9 (2015) 1–11. [10] M. Gandarias, M. Bruzón, M. Rosa, Nonlinear self-adjointness and conservation laws for a generalized Fisher equation, Commun. Nonlinear Sci. Numer. Simul. 18 (2013) 1600–1606. [11] R. Tracinà, M.S. Bruzón, M.L. Gandarias, On the nonlinear self-adjointness of a class of fourth-order evolution equations, Appl. Math. Comput. 275 (2016) 299–304. [12] R. Tracinà, I.L. Freire, M. Torrisi, Nonlinear self-adjointness of a class of third order nonlinear dispersive equations, Commun. Nonlinear Sci. Numer. Simul. 32 (2016) 225–233. [13] R. de la Rosa, M. Gandarias, M. Bruzón, Equivalence transformations and conservation laws for a generalized variable-coefficient gardner equation, Commun. Nonlinear Sci. Numer. Simul. 40 (2016) 71–79. http://dx.doi.org/10.1016/j.cnsns.2016.04.009. [14] N. Ibragimov, The answer to the question put to me by l.v. ovsyannikov 33 years ago, Arch. ALGA 3 (2006) 80. [15] M. Gandarias, Weak self-adjoint differential equations, J. Phys. A 44 (2011) 262001. http://dx.doi.org/10.1088/1751-8113/44/26/262001. [16] N. Ibragimov, Nonlinear self-adjointness and conservation laws, J. Phys. A 44 (2011) 432002. [17] M. Gandarias, Nonlinear self-adjointness through differential substitutions, Commun. Nonlinear Sci. Numer. Simul. 19 (2014) 3523–3528. [18] M. Rosa, M. Bruzón, M. Gandarias, A conservation law for a generalized chemical Fisher equation, J. Math. Chem. 53 (2015) 941–948. [19] S.C. Anco, G. Bluman, Direct construction method for conservation laws for partial differential equations part ii: General treatment, Eur. J. Appl. Math. 41 (2002) 567–585. [20] M. Gandarias, C. Khalique, Symmetries solutions and conservation laws of a class of nonlinear dispersive wave equations, Commun. Nonlinear Sci. Numer. Simul. 32 (2016) 114–121. [21] K. Adem, C. Khalique, Symmetry analysis and conservation laws of a generalized two-dimensional nonlinear kp-mew equation, Math. Probl. Eng. 2015 (2015) 805763. http://dx.doi.org/10.1155/2015/805763. [22] M. Gandarias, M. Bruzón, M. Rosa, Symmetries and conservation laws for some compacton equation, Math. Probl. Eng. 2015 (2015) 430823. http://dx.doi.org/10.1155/2015/430823. [23] M. Gandarias, M. Rosa, On double reductions from symmetries and conservation laws for a damped Boussinesq equation, Chaos, Solitons and Fractals: Interdiscip. J. Nonlinear Sci. Nonequilibrium Complex Phenom. 89 (2016) 560–565. http://dx.doi.org/10.1016/j.chaos.2016.03.030. [24] G.W. Bluman, S. Kumei, Symmetries and Differential Equations, Springer, Berlin, 1989.

8

M. Rosa et al. / Journal of Computational and Applied Mathematics (

)

–

[25] G.W. Bluman, S. Kumei, Symmetry-based algorithms to relate partial differential equations: I. local symmetries, European J. Appl. Math. 1 (1990) 189–216. [26] G. Bluman, J. Cole, Similarity Methods for Differential Equations, Springer, Berlin, 1974. [27] I. Krasil’shchik, A. Vinogradov, Symmetry and integrability byquadratures of ordinary differential equations, Acta Appl. Math. 15 (1989) 161–209. [28] P. Olver, Applications of Lie Groups to Differential Equations, Springer, Berlin, 1986. [29] N. Euler, M. Euler, On nonlocal symmetries, nonlocal conservation laws and nonlocal transformations of evolution equations: two linearisable hierachies, J. Nonlinear Math. Phys. 6 (2009) 489–504. [30] G. Bluman, G. Reid, S. Kumei, New classes of symmetries for partial differential equations, J. Math. Phys. 29 (1988) 806–811.