Classical Lie point symmetry analysis of nonlinear diffusion equations describing thermal energy storage

Applied Mathematics and Computation 216 (2010) 251–260

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Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc

Classical Lie point symmetry analysis of nonlinear diffusion equations describing thermal energy storage R.J. Moitsheki a,*, O.D. Makinde b a

Center for Differential Equations, Continuum Mechanics and Applications, School of Computational and Applied Mathematics, University of the Witwatersrand, Private Bag 3, Wits 2050, South Africa b Faculty of Engineering, Cape Peninsula University of Technology, P.O. Box 1906, Bellville 7535, South Africa

a r t i c l e

i n f o

Keywords: Unsteady nonlinear heat diffusion Thermal energy storage Classical Lie point symmetries Group invariant solutions

a b s t r a c t In this paper, we employed the linear transformation group approach to time dependent nonlinear diffusion equations describing thermal energy storage problem. Symmetry analysis of the governing equation resulted in admitted large Lie symmetry algebras for some special cases of the arbitrary constants and the source term. Some transformations that lead to equations with fewer arbitrary parameters are applied and classical Lie point symmetry methods are employed to analyze the transformed equations. Some symmetry reductions are performed and wherever possible the reduced ordinary differential equations are completely solved subject to realistic boundary conditions. Ó 2010 Elsevier Inc. All rights reserved.

1. Introduction The operation of hot water storage tanks for thermal energy storage is classified as static mode and dynamic mode [1–3]. The static mode is further classified into fully charged and partially charged mode. Dynamic mode of operation includes charging and discharging cycle. In static fully charged mode, the storage tank is initially completely filled with hot water at a constant temperature and is subjected to convective heat loss from the tank walls to the ambient [4]. In static partially charged mode, the storage tank is initially charged with different levels of hot and cold water separated by thermocline [2]. The heat loss from the stored fluid to the ambient decreases the temperature of the fluid near to the tank wall, thereby increasing its density. Therefore, accurate and efficient design of solar heating or cooling thermal energy storage tanks generally requires adequate theoretical framework from which insight might be gleaned into an inherently complex physical process [4]. The solution of the transient nonlinear heat conduction for thermal energy storage applicable in rectangular, cylindrical and spherical coordinates remains an extremely important problem of practical relevance in the engineering sciences [5,6]. The mathematical technique employed in the present investigation is the parameter-group transformation [7]. The foundation of the group-theoretical method is contained in the general theory of continuous transformation groups that were introduced and treated extensively by Lie (see e.g. [8]). The group methods, as a class of methods that lead to a reduction of the number of independent variables, were utilized in [9]. Several other authors such as [6–13] have applied group method analysis to investigate many problems in physical sciences and engineering. Group-theoretical methods provide a powerful tool for analyzing nonlinear differential models, since they are not based on linear operators, superposition, or any other aspects of linear solution techniques.

* Corresponding author. E-mail addresses: [email protected] (R.J. Moitsheki), [email protected] (O.D. Makinde). 0096-3003/$ - see front matter Ó 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2010.01.046

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2. Mathematical formulation The transient heat conduction equation with heat source term modeling the thermal storage problem in a rectangular, cylindrical or spherical coordinate system is given as [2,6];

qCðTÞ

  @T 1 @ @T ¼ m KðTÞr m þ SðTÞ; @t r @r @r

ð1Þ

with the initial condition

Tðr; tÞ ¼ T 0

at t ¼ 0;

ð2Þ

where T is the temperature, q is the density, r is the space variable, t is time, T 0 is the initial temperature of the body and S(T) is the temperature-dependent heat source term. Following [6], the power law temperature-dependent thermal conductivity and heat capacity are taken as

KðTÞ ¼ K 0



T  T1 T0  T1

n

and CðTÞ ¼ C 0



T  T1 T0  T1

b ;

where K 0 ; C 0 ; n and b are constants, T 0 is the initial temperature of the body, T 1 is the ambient temperature of the surrounding environment. The geometry of the body is specified by m ¼ 0; 1; 2 representing rectangular, cylindrical and spherical coordinates respectively. Eqs. (1) and (2) are made dimensionless by introducing the following quantities:

r r ¼ ; a

t ¼ K 0 t ; qC 0 a2

T¼

T  T1 ; T0  T1

S¼

S

qC 0

:

Neglecting the bar symbol for clarity, the dimensionless governing equation then become

Tb

  @T 1 @ @T þ SðTÞ: ¼ m T n rm @t r @r @r

ð3Þ

We will refer to Eq. (3) as the governing equation. We aim to determine cases or forms of S(T) for which extra classical Lie point symmetries are admitted by (3), with both both b and n nonzero. Note that if these two parameters are equal to zero then both heat capacity and heat conductivity are merely constants. 3. Symmetry techniques for differential equations The theory and applications of continuous symmetry groups were founded by Lie in the 19th century [14]. The modern accounts of this theory may be found in excellent text such as those of [10,12]. We restrict our discussion to classical Lie point symmetries, since we will only use such symmetries. The reader is referred to [10,12] for more details on this theory. Given a continuous one parameter symmetry group, it is possible to reduce the number of independent variables by one. Lie’s fundamental result is that the whole of one parameter group can be determined from the transformation laws up to the first degree of the parameter , i.e. determination of symmetry groups involve finding transformation of the form

xj ¼ xj þ X j ðx; uÞ þ Oð2 Þ; u ¼ u þ Uðx; uÞ þ Oð2 Þ; that leave the mth order governing partial differential equation,

Dðx; u; uð1Þ ; . . . ; uðmÞ Þ ¼ 0;

ð4Þ ðkÞ

invariant. Here, x is set of n independent variables ðx1 ; x2 ; . . . ; xn Þ and u denotes a set of coordinates corresponding to all the kth order partial derivatives of u with respect to x1 ; x2 ; . . . ; xn . The coefficients X j and U are the components of the infinitesimal symmetry generator, which is one of the vector fields

C ¼ X j ðx; uÞ

@ @ þ Uðx; uÞ ; @xj @u

ð5Þ

which span the associated Lie algebra. Here, we sum over repeated index (see e.g. [10]). The infinitesimal criterion for invariance of a P.D.E. such as Eq. (4) is given by

CðmÞ DjD¼0 ¼ 0; ðmÞ

ð6Þ

where C is the mth extension or prolongation of the infinitesimal generator C in (5). The invariance condition (6) results in an overdetermined linear system of determining equations for the coefficients X j ðx; uÞ and Uðx; uÞ. Manipulation of these determining equations to find their solutions is very long and tedious. However, we use the freely available program Dimsym [15], which is written as a subprogram for the computer algebra package Reduce [16] to construct the admitted symmetries.

R.J. Moitsheki, O.D. Makinde / Applied Mathematics and Computation 216 (2010) 251–260

253

If a differential equation is invariant under some point symmetry, one can often construct similarity solutions which are invariant under some subgroup of the full group admitted by the equation in question. These solutions result form solving a reduced equation in fewer variables. More often, differential equations arising in real world problem involve one or more functions depending on either the independent variable or on the dependent variables. It is possible by symmetry techniques to determine the cases which allow the equation in question to admit extra symmetries. The exercise of searching for the forms of arbitrary functions that extend the principal Lie algebra is called group classification. The problem of group classification was introduced by Lie [17] and recent accounts on this topic may be found for example in [11,18–21]. We adopt methods in [11] (which exclude explicit equivalence transformation analysis) to perform group classification of Eq. (3). 4. Symmetry analysis of Eq. (3) In the initial symmetry analysis of Eq. (3), where S(T) remained arbitrary we obtained nothing beyond the translation in t. However, Dimsym [15] flagged a number of possibilities which may lead to extra symmetries for Eq. (3). For m = 0 (the rectangular dimensions) with S(T) remaining arbitrary Eq. (3) admits translations in t and r, also captured in [22]. In fact, the case m = 0 has been extensively studied in [22], also appearing in [23]. The case S = 0, m = 0 was studied in [7,18], we therefore, omit these cases in our analysis. In [24] the authors considered one case namely SðTÞ ¼ S0 T nþ1 , with S0 being a constant, m and n are arbitrary constants, and the work is extended here to include as many as possible cases for which the governing equation admit extra symmetries. 4.1. Case: b = n Eq. (3) with b = n, admits extra point symmetries for the following cases: 4.1.1. Case a: S(T) = 0, and arbitrary m and n In this case, the admitted Lie algebra is finite four dimensional and spanned by the base vectors



 9 r2 þ 2ðm þ 1Þt @ @ @ > > T þ rt þ t2 ; = 4ðn þ 1Þ @T @r @t @ @ @ n @ > > ; C2 ¼ r þ 2t ; C3 ¼ ; C4 ¼  T @r @t @t n þ 1 @T

C1 ¼ 

ð7Þ

and the infinite symmetry generator

C5 ¼ Hðr; tÞT n

@ @T

with H being the arbitrary solution of equation

@H @ 2 H m @H ¼ 2 þ : @t @r r @r For n ¼ 1, the symmetry generators C1 and C4 in (7) are replaced by

C1 ¼ ðr 2 þ 2ðm þ 1ÞtÞT log T

@ @ @ þ 4rt þ 4t2 ; @T @t @t

C4 ¼ T logðTÞ

@ : @T

For m = 0, b = n = 1 we obtain extra symmetries

C5 ¼

@ ; @r

C6 ¼ rTlog T

@ @ þ 2t @T @r

and the infinite symmetry generator

C7 ¼ Hðr; tÞT

@ ; @T

with H satisfying the linear heat equation, Ht ¼ Hrr . 4.1.2. Case b: S(T) = 0, m = 2 and arbitrary n The obtained extra symmetries are given by

9  r 2 þ 6t @ @ @ > > T þ rt þ t 2 ; > > > 4ðn þ 1Þ @T @r @t > > = @ @ @ n @ C2 ¼ r þ 2t ; C3 ¼ ; C4 ¼  T ; > @r @t @t n þ 1 @T > >  2  > 1 @ @ r þ 2t @ @ > > ; T C5 ¼  þ ; C6 ¼  þt > ðn þ 1Þr @T @r 2ðn þ 1Þr @T @r 

C1 ¼ 

and the infinite symmetry generator,

ð8Þ

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C7 ¼ Hðr; tÞT n

@ ; @T

where H satisfies the equation

@H 2 @H @ 2 H ¼ þ 2: @t r @r @r 4.1.3. Case c: SðTÞ ¼ S0 T nþ1 , m, n arbitrary The Lie algebra is spanned by the base vectors;

9 1 @ @ @ > ð4S0 ðn þ 1Þt 2  r 2  2ðm þ 1ÞtÞT þ rt þ t 2 ; > = 4ðn þ 1Þ @T @r @t > @ @ @ @ n @ > ; C2 ¼ 2S0 tT þ r þ 2t ; C3 ¼ ; C4 ¼  T @T @r @t @t n þ 1 @T

C1 ¼

ð9Þ

and the infinite symmetry generator

C5 ¼ Hðr; tÞT n

@ @T

with H being the arbitrary solution of equation

@H @ 2 H m @H ¼ 2 þ þ S0 ðn þ 1ÞH: @t @r r @r 4.1.4. Case d: SðTÞ ¼ S0 T nþ1 ; m ¼ 2, n arbitrary In this case, symmetry analysis yields



 9 r 2  6t @ @ @ @ @ @ > T þ rt þ t 2 ; C2 ¼ 2S0 tT þ r þ 2t ; > > 4ðn þ 1Þ @T @r @t @T @r @t > > > > = @ n @ T @ @ ; C5 ¼  þ ; C3 ¼ ; C4 ¼  > @t n þ 1 @T ðn þ 1Þr @T @r > > > > > r 2 þ 2t @ @ @ n > ; C6 ¼  T þ t ; C7 ¼ Hðr; tÞT ; 2ðn þ 1Þr @T @r @T

C1 ¼ S0 t2 

ð10Þ

where H is an arbitrary solution of a P.D.E.

S0 ðn þ 1ÞrH þ r

@2H @H @H þ2 ¼ : @r 2 @r @t

4.1.5. Case e: SðTÞ ¼ expðS0 T nþ1 Þ; m ¼ 2, n arbitrary Beyond translation in t, Eq. (3) admits

C2 ¼ 

1 @ r @ @ þ þt : @t S0 ðn þ 1ÞT n @T 2 @r

4.1.6. Case f: SðTÞ ¼ S0 T n ; m and n are arbitrary Beyond translation in t, Eq. (3) admits

C2 ¼ 2T

@ @ @ þ r þ 2t : @T @r @t

This algebra is further extended by a translation in r when m = 0. 4.1.7. Case g: SðTÞ ¼ a log T; m and n arbitrary In this case, we only obtained the translation of t. However, the algebra becomes larger for special choices of the constants m and n. For arbitrary m and n = 1 we have the admitted algebra L4  L1 given by

9

C1 ¼ ðr 2 þ 4at2  2ðm þ 1ÞtÞT log T @T@ þ 4rt @r@ þ 4t 2 @t@ ; > = C2 ¼ 2atT log T @T@ þ r @r@ þ 2t @t@ ; C3 ¼ @t@ ; C4 ¼ T log T @T@ ; > ; C5 ¼ Hðr; tÞT @T@ ; where H satisfies the P.D.E.

ð11Þ

R.J. Moitsheki, O.D. Makinde / Applied Mathematics and Computation 216 (2010) 251–260

aH þ

255

@ 2 H m @H @H þ ¼ : @r 2 r @r @t

For m = 2 (the spherical dimension) we obtain other than the infinite symmetry generator, the finite six dimensional Lie algebra namely,

9

C1 ¼ ðr 2 þ 4at 2  2ðm þ 1ÞtÞT log T @T@ þ 4rt @r@ þ 4t 2 @t@ ; > > = C2 ¼ 2atT log T @T@ þ r @r@ þ 2t @t@ ; C3 ¼ @t@ ; C4 ¼ T log T @T@ ; 

C5 ¼ 

r 2 þ2t r



@ þ @r@ ; T log T @T

C6 ¼ 



r 2 þ2t 2r



> > @ þ t @t@ ; T log T @T

and the infinite symmetry generator

C7 ¼ Hðr; tÞT

@ ; @T

where H satisfies

@2H @H @H þ2 ¼r : @r 2 @r @t

arH þ r

For m = 0 (the rectangular shape) then C5 and C6 change to

C5 ¼

@ @r

and C6 ¼ rT log T

@ @ þ 2t : @T @r

4.2. Case: b – n For b – n; i.e. b, n and m are arbitrary, the principal Lie algebra extends as follows: 4.2.1. Case h: S(T) = 0 Beyond translation in t, Eq. (3) admits

C2 ¼ r

@ @ þ 2t ; @r @t

C3 ¼

  1 @ @ : 2T þ ðn  bÞr 2ðn  bÞ @T @r

4.2.2. Case i: SðTÞ ¼ S0 – 0 Beyond translation in t, Eq. (3) admits

C2 ¼

  1 @ @ @ : 2T þ ðn þ 1Þr þ 2ðb þ 1Þt 2ðb þ 1Þ @T @r @t

For m = 2, b = 1 we obtain extra symmetries

C2 ¼ 

  eðnþ1ÞS0 t @ @ S0 T þ ; @T @t ðn þ 1ÞS0

C3 ¼ 

  2 @ @ 2T þ ðn þ 1Þr : nþ1 @T @r

4.2.3. Case j: SðTÞ ¼ S0 T p ; p 2 R Beyond translation in t, Eq. (3) admits

C2 ¼

  1 @ @ @ 2T þ ðn  p þ 1Þr þ 2ðb  p þ 1Þt : 2ðb  p þ 1Þ @T @r @t

For p = 1, m = 1, n = 1 we obtain



 @ 1 @ þ ; @T S0 @t @ @ C4 ¼ 2T  r : @T @r

C2 ¼ eS0 t T

C3 ¼ 2ðlog r þ 1ÞT

@ @ þ r log r ; @T @r

4.2.4. Case k: SðTÞ ¼ S0 T bþ1 Beyond translation in t, Eq. (3) admits

C2 ¼

  1 @ @ ; 2T þ ðn  bÞr 2ðb  nÞ @T @r

C3 ¼ 

The fourth admitted symmetry when m ¼ 3nþ4þb is nþ2þb

  eðbnÞS0 t @ @ : þ S0 T @T @t S0 ðb  nÞ

ð12Þ

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C4 ¼

2ðnþ1Þ   r bþnþ2 @ @ þ ½ðb þ nÞ2 þ 4ðb þ n þ 1Þr : ð2b þ 2n þ 4ÞT @T @r 2ðb  nÞðn þ 1Þ

Furthermore, with b ¼ ð4 þ 3nÞ; m ¼ 0 we obtain

  @ 1 @ @ ; C4 ¼ T þ 2ðn þ 1Þr ; @r 2ðn þ 1Þ @T @r   r @ @ C5 ¼ T þ ðn þ 1Þr : nþ1 @T @r

C2 ¼

Dimsym [15] flagged the case b = 1 which may lead to extra symmetries being admitted. We treat this case separately in the following paragraph. 4.3. Case: b ¼ 1; n –  1 In this case, the generic symmetry is a translation in time. 4.3.1. Case l: SðTÞ ¼ S0 T p ; p 2 R Other than the generic symmetry , we obtain

C2 ¼

  1 @ @ @ : 2T þ ðp  n  1Þr þ ð2p  1Þt 2p @T @r @t

4.3.2. Case m: SðTÞ ¼ S0 ; S0 2 R The principal Lie algebra is extended by the base vectors

C2 ¼ 

  eS0 ðnþ1Þt @ 1 @ þ ; T @T S0 @t nþ1

C3 ¼

  m @ @ 2T  ðn þ 1Þr : ðm  1Þðn þ 1Þ @T @r

For m = 3 we obtain the fourth symmetry

C4 ¼

  1 @ @ 2T þ ðn þ 1Þr : 4ðn þ 1Þr2 @T @r

4.3.3. Case n: S(T) = 0 Other than the generic symmetry, the extra admitted symmetries are given by

C2 ¼ r

@ @ þ 2t ; @r @t

C3 ¼

  1 @ @ ; 2T þ ðn þ 1Þr 2ðn þ 1Þ @T @r

and for m = 3 we obtain the fourth admitted symmetry

C4 ¼

  1 @ @ : 2T  ðn þ 1Þr 4ðn þ 1Þr2 @T @r

5. Mapping to equations with less number of arbitrary parameters One may adopt a new dependent variable

u¼

Z

T b dT;

ð13Þ

so that it may be assumed that the heat capacity is unity (or simply allowing b to vanish), and assuming that heat capacity be raised to b = 1 in Eq. (3). Although the physics behind the original problem may now have been changed, it is worth noting that the problem becomes mathematically easier since the arbitrary parameters are reduced. The transformation (13) and appropriate scalling reduce Eq. (3) to

  @u 1 @ @u ¼ m rm u0 uN þ SðuÞ; @t r @r @r

b– 1

ð14Þ

and

  @u 1 @ @u þ SðuÞ; ¼ m rm enu @t r @r @r

b ¼ 1;

ð15Þ

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R.J. Moitsheki, O.D. Makinde / Applied Mathematics and Computation 216 (2010) 251–260

nb  ¼ n þ 1. If we allow n to take same values as b (i.e. b = n), then this will render Eq. (14) where N ¼ bþ1 ; u0 ¼ ðb þ 1ÞN and n linear. Note that a linear Eq. (14) is equivalent to a nonlinear Eq. (3). It is well known that one of the generic symmetry admitted by the linear equation is an infinite generator; not surprisingly Eq. (3) admits infinite symmetry generator. The zero source term case for both Eqs. (14) and (15) have been extensively studied in [25], wherein m ¼ 1  N; N – 1. We will herein omit this case. Eqs. (14) and (15) with m = 0 are the subclasses of a class considered in [22]. Eq. (14) with a source term proportional to u has been studied in [26]. In the initial symmetry analysis, both Eqs. (14) and (15) admit the time translation. Furthermore, for m = 0, one obtains a translation in r. Extra symmetries admitted by (14) and (15) are listed in Tables 1–3, respectively.

6. Some symmetry reductions and invariant solutions: illustrative examples The main purpose for determining symmetries is to obtain invariant (similarity) solutions or generate new solutions from trivial ones. The procedure for finding such solutions is fully discussed in text such as ([8,10–13]). It is possible to find solutions using any linear combination of the admitted symmetries. In order to ensure a minimal set of inequivalent reductions, an optimal system may be constructed. However, we will make use of some of the obtained symmetries to find invariant solutions. In general if a differential equation such as (4) admits the symmetry generator (5) one may obtain the invariants and hence the functional form of the solutions by solving the system

du dx1 dx2 ¼ ¼  ¼ U X1 X2

Example 1. Given S ¼ S0 T n with arbitrary m and n as in Case f, the C2 yields the functional form of the invariant solution

r T ¼ tGðcÞ with c ¼ pffiffi where G satisfies the O:D:E: t c 0 m 0 1 0 2 G  G ¼ nG ðG Þ þ G þ G00 þ S0 ; 2 c which is highly nonlinear and difficulty to solve exactly.

Table 1 Extensions of the principal Lie symmetry algebra admitted by (14) with N ¼ 0. S(r, t)

Parameters

Symmetries

S¼0

m arbitrary

@ @ @ C2 ¼ ðr2 þ 2ðm þ 1ÞtÞu @u þ 4rt @r þ 4t2 @t @ @ @ C3 ¼ r @r þ 2t @t ; C4 ¼ u @u @ C5 ¼ Hðr; tÞ @u where H satisfies

Ht ¼ Hrr þ mr Hr m¼2

@ @ C5 ¼ 1r ðr @r  u @u Þ 1 @ @ C6 ¼ 2r ½ðr2 þ tÞu @u þ 2rt @r  @ C7 ¼ Hðr; tÞ @u where H satisfies

Ht ¼ Hrr þ 2r Hr p

S¼u

m arbitrary

1 @ @ @ C2 ¼ 2ðp1Þ ð@u þ ðp  1Þr @r þ 2ðp  1Þt @t Þ

S¼u

m arbitrary

@ @ @ C2 ¼ ½r2 þ 4t2  2ðm þ 1Þtu @u þ 4rt @r þ 4t 2 @t @ @ @ C3 ¼ 2ut @u þ r @r þ 2t @t @ @ C4 ¼ u @u ; C5 ¼ Hðr; tÞ @u where

m¼2

2

¼ H þ @@rH2 þ mr @H @r

@ @ C5 ¼ 1r u @u þ r @r @H @t

1 @ @ C6 ¼ 2r ½ðr2 þ tÞu @u þ 2rtu @r  @ C7 ¼ Hðr; tÞ @u where

@H @t

2

¼ H þ @@rH2 þ mr @H @r .

S ¼ epu ; p 2 R

m arbitrary

1 @ @ @ C2 ¼ 2p ð2 @u þ pr @r þ 2t @t Þ

S ¼ u1=ðnþ1Þ

m arbitrary

@ @ C2 ¼ ð4t2  r 2  2ðm þ 1ÞtÞu @u þ 4rt @r @ 4t 2 @t @ @ @ C3 ¼ 2tu @u þ r @r þ 2t @t ;

@ C4 u @u 2

@ @H C5 ¼ Hðr; tÞ @u ; @t ¼ H þ @@rH2 þ mr @H @r

m¼2

@ @ C5 ¼ 1r ðr @r  u @u Þ 1 @ C6 ¼ 2r ð2rt @r  ðr 2 þ 2tÞÞ 2

@ @H C7 ¼ Hðr; tÞ @u ; @t ¼ H þ @@rH2 þ 2r @H @r

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Table 2 Extensions of the principal Lie symmetry algebra admitted by (14) with N – 0. S(r,t)

Parameters

Symmetries

S ¼ S0 uq ; S0 ; q 2 R

m; N arbitrary

1 @ @ @ C2 ¼ 2ðq1Þ ð2u @u þ ðq  N  1Þr @r þ 2ðq  1Þt @t Þ,

m; N arbitrary

@ @ C2 ¼  eNS0 0 ðS0 u @u þ @t Þ

q; S0 – 0 S ¼ S0 u; S0 2 R

S Nt

1 @ @ C3 ¼ 2n ð2u @u þ Nr @r Þ

m ¼ 3Nþ4 Nþ2

S¼q–0

2ðNþ1Þ

r Nþ2 @ @ C3 ¼ 2Nð2Nþ4Þ ðð2N þ 4Þu @u  ðN2 þ 4N þ 4Þr @r Þ

m ¼ 0; N ¼ 4=3

@ C4 ¼ @r ;

m ¼ 1; N ¼ 1

@ @ C4 ¼ 2ðlog r þ 1Þu @u þ r log r @r @ @ @ C2 ¼ 2u @u þ ðN þ 1Þr @r þ 2t @t

m; N arbitrary

@ @ C5 ¼ rð3u @u þ r @r Þ

Table 3 Extensions of the principal Lie symmetry algebra admitted by (15). S(r, t)

Parameters

Symmetries

S ¼ S0 equ ; S0 ; q 2 R

 arbitrary m; n

1 @ @ @  Þr @r C2 ¼ 2q ð2 @u þ ðq  n þ 2qt @t Þ

q; S0 –0 S ¼ S0 enu þ S1 ;

C2 ¼ 

eS1 nt  S1 n

@ @ ðS1 @u þ @t Þ

S0 ; S1 2 R S¼q–0

m arbitrary

q nt

@ @ C2 ¼  e nq ðq @u þ @t Þ m @ @  r @r C3 ¼  nðm1Þ ð2 @u þn Þ

S¼0

m¼3

@ @  r @r C4 ¼ 4n1r2 ð2 @u n Þ

m arbitrary

@ C2 ¼ 2t @t@ þ r @r ; @ @ C2 ¼ r @r þ 2t @t ; @ @ C4 ¼ 2n @u þ r @r

m¼3

@ @ C3 ¼ 2n @u þ r @r @ @  r @r C3 ¼ 4n1r2 ð2 @u n Þ

Example 2. Suppose we impose the realistic boundary conditions (see e.g. [6,26])

@T @T ¼ 0; r ¼ 0 and T n ¼ Bi gðtÞ T; r ¼ 1; @r @r where Bi is the Biot number and g(t) is the heat transfer coefficient, then given case j where S ¼ S0 T bþ1 and using C3 we obtain the functional form

T ¼ eS0 t f ðrÞ;

ð16Þ

wherein f is a function of the invariant r, and satisfies the O.D.E.

n 0 2 m 0 ðf Þ þ f þ f 00 ¼ 0; f r

ð17Þ

and the boundary conditions transforms to

f 0 ð0Þ ¼ 0;

f 0 ð1Þ ¼ Bif nS0 t

1n

ð1Þ

wherein g(t) is given by e . One may observe that the functional form (16) is a consequence of separation of variables. However, this form has been exposed by Lie symmetry methods especially with the given nontrivial source term. Four cases arise for the solution to Eq. (17) (see also [27]) case (i)

n ¼ 1; m – 1,

f ¼ c2 expðc1 r 1m Þ: case (ii)

n = 1, m = 1,

f ¼ c 2 r c1 : case (iii)

n –  1; m ¼ 1.

f ¼ ½c1 ln r þ c2 ðnþ1Þ :

259

R.J. Moitsheki, O.D. Makinde / Applied Mathematics and Computation 216 (2010) 251–260 Table 4 Explicit correspondence between symmetries in Section 4 and Tables 1–3. Table

Section

Table 1 First case Second case Third case Fourth case Fifth case

Section 4.1 Cases (a) and (b) Case (g) Case (f) Case (e) Cases (c) and (d)

Table 2 First case Second case Third case

Section 4.2 Case (k) Case (j) Case (i)

Table 3 First and second case Third case Fourth case

Section 4.3 Case (l) Case (m) Case (n)

case (iv)

n –  1; m – 1.

1 n h c ionþ1 1 f ¼  ðn þ 1Þ : r 1m þ c2 m1

For case (iv) and in terms of original variables, the invariant solution satisfying the physical boundary conditions is given

T¼



1 ðn þ 1ÞBi n S0 t 1m e r nþ1 ; m1

m – 1; n –  1:

7. Conclusions We have obtained forms of the source term for which the principal Lie algebra is extended. It is possible to classify group invariant solutions according to the symmetries admitted by the governing equation for each of the determined cases of the source term. In general any linear combination of the admitted symmetries may lead to exotic similarity (invariant) solutions. In fact one may construct a set of elements, known as the optimal system, that yield inequivalent reductions. Although in some cases, as in example 1, the reduced equation may be difficult to solve analytically; numerical solutions may be sought. The power of Lie symmetry methods to solve differential equations especially nonlinear P.D.E.s, sometimes with realistic boundary conditions, has been displayed in Example 2 of Section 6. Eq. (3) admits the symmetries corresponding to the ones listed in Table 1 (cf. Sections 4.1), Table 2 (cf. Sections 4.2) and Table 3 (cf. Section 4.3). The calculations have been simplified by transforming Eq. (3) to Eqs. (14) and (15), since parameters were reduced. The explicit one to one correspondence of the symmetries in Section 4 and Tables 1–3 is given in Table 4. Acknowledgements The authors wish to thank the National Research Foundation of South Africa under Thuthuka program, for the generous financial support, and we thank the anonymous referees for their useful comments. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13]

J.A. Duffie, W.A. Becham, Solar Engineering of Thermal Processes, Wiley, 1980. p. 485. Y. Jahiria, S.K. Gupta, Decay of thermal stratification in a water body for solar energy storage, Solar Energy 28 (2) (1982) 137. M.N.A. Hawlader, B.J. Brinkworth, An analysis of the non-convective solar pond, Solar Energy 27 (3) (1981) 195. M.S. Shin, H.S. Kim, D.S. Jang, S.N. Lee, H.G. Yoon, Numerical and experimental study on the design of a stratified thermal storage system, Appl. Thermal Eng. 24 (2004) 17–27. T.W. Davies, Transient conduction in a plate with counteracting convection and thermal radiation at the boundary, Appl. Math. Model. 9 (1985) 337– 340. M.B. Abd-el-Malek, M.M. Helal, Group method solution for solving nonlinear heat diffusion problems, Appl. Math. Model. 30 (9) (2006) 930–940. L.V. Ovsiannikov, Group Analysis of Differential Equations, Springer, New York, 1982. N.H. Ibragimov, Elementary Lie Group Analysis and Ordinary Differential Equations, Wiley, New York, 1999. M.J. Moran, R.A. Gaggioli, Reduction of the number of variables in system of partial differential equations with auxiliary conditions, SIAM J. Appl. Math. 16 (1968) 202–215. G.W. Bluman, S.C. Anco, Symmetry and Integration Methods for Differential Equations, Springer-Verlag, New York, USA, 2002. G.W. Bluman, S. Kumei, Symmetry and Differential Equations, Springer-Verlag, New York, USA, 1989. P.J. Olver, Applications of Lie Groups of Differential Equations, Springer-Verlag, New York, 1986. p. 495. H. Stephani, Differential Equations – Their Solutions Using Symmetries, Cambridge University Press, Cambridge, 1989.

260 [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27]

R.J. Moitsheki, O.D. Makinde / Applied Mathematics and Computation 216 (2010) 251–260 S. Lie, Theorie der transformationsgruppenm, Math. Ann. 16 (1880) 441–528. J. Sherring, DIMSYM Symmetry Determination and Linear Differential Package, Latrobe University, 1993. A.C. Hearn, Reduce User’s Manual Version 3.4, Santa Monica, California, Rand Publication CP78, The Rand Cooporation, 1985. S. Lie, On integration of a class of linear partial differential equations by means of definite integral, Arch. Math. VI (3) (1881) 328–368. L.V. Ovsiannikov, Group relations of the equation of non-linear heat conductivity, Dokl. Akad. Nauk SSSR. 125 (1959) 492–495. N.M. Ivanova, C. Sophocleous, On the group classification of variable nonlinear diffusion–convection equations, J. Comput. Appl. Math. 197 (2) (2006) 322–344. C. Sophocleous, Symmetries and form-preserving transformation of generalised inhomogeneous nonlinear diffusion equations, Physica A 324 (2003) 509–529. O.O. Vaneeva, A.G. Johnpillai, R.O. Popovych, C. Sophocleous, Enhanced group analysis and conservation laws of variable coefficient reaction-diffusion equations with power nonlinearities, J. Math. Anal. Appl. 330 (2007) 1363–1386. V.A. Dorodnitsyn, Group Properties and Invariant Solutions of a Nonlinear Heat Equation With a Source or a Sink, Preprint Number 57, Keldysh Institute of Applied Mathematics of Academy of Sciences, USSR, Moscow, 1979, 2007. N.H. Ibragimov, CRC handbook of Lie group analysis of differential equations, Symmetries, Exact Solutions and Conservation Laws, vol. 1, CRC Press, Boca Raton, Florida, USA, 1994. R.J. Moitsheki, O.D. Makinde, Application of Lie point symmetry and Adomain decomposition techniques to thermal-storage nonlinear diffusion models, J. Phys: Conf. Ser. 128 (2008) 012058. 10pp.. C. Sophocleous, On symmetries of radially symmetric nonlinear diffusion equations, J. Math. Phys. 33 (11) (1992) 3687–3693. R.J. Moitsheki, Transient heat diffusion with temperature dependent conductivity and time dependent heat transfer coefficient, Mathematical Problems in Engineering, vol. 2008, 9pp., doi:10.1155/2008/347568. A.D. Polyanin, V.F. Zaitsev, Handbook of Exact Solutions for Ordinary Differential Equations, CRC Press, Boca Raton, Florida, USA, 1995.