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Symbolic computation of solutions for a forced Burgers equation
Applied Mathematics and Computation 216 (2010) 18–26
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Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Symbolic computation of solutions for a forced Burgers equation Alvaro H. Salas Department of Mathematics, Universidad de Caldas, Manizales-Colombia, AA 275, Universidad Nacional de Colombia, Campus la Nubia, Manizales, Colombia
a r t i c l e
i n f o
Keywords: Nonlinear evolution equation pde Burgers equation Reaction–diffusion equation
a b s t r a c t In this paper we give exact solutions for a forced Burgers equation. We make use of the generalized Cole–Hopf transformation and the traveling wave method. Ó 2009 Elsevier Inc. All rights reserved.
1. Introduction The search of exact solutions of nonlinear partial differential equations is of great importance, because these equations appear in complex physics phenomena, mechanics, chemistry, biology and engineering. In this paper we obtain new solutions for the forced Burgers equation [1]
ut þ auux þ buxx ¼ GðtÞx;
ð1:1Þ
Eq. (1.1) is a generalization of the well-known Burgers equation [9]
ut þ uux ¼ muxx :
ð1:2Þ
Owing to the assumptions of the constant coefficients and unforced turbulence, the physical situations in which the classical Burgers equation arises tend to be highly idealized. In practice, the forced of Eq. (1.2) may provide us with more realistic models in many different physical contexts like the long-wave propagation in an inhomogeneous two-layer shallow liquid, directed polymers in a random medium, pinning of vortex lines in superconductors, large eddy simulation, ballistic deposition, passive random walker dynamics on a growing surface, etc. As remarked in Ref. [2], in 1915, Harry Bateman considered a nonlinear equation whose steady solutions were thought to describe certain viscous flows [3]. This equation, modeling a diffusive nonlinear wave, is now widely known as the Burgers equation, and is given by
ut þ uux ¼ muxx ;
ð1:3Þ
where m is a constant measuring the viscosity of the fluid. It is a nonlinear parabolic equation, simply describing a temporal evolution where nonlinear convection and linear diffusion are combined, and it can be derived as a weakly nonlinear approximation to the equations of gas dynamics. Although nonlinear, Eq. (1.3) is very simple, and interest in it was revived in the 1940s, when Dutch physicist Jan Burgers proposed it to describe a mathematical model of turbulence in gas [4]. As a model for gas dynamics, it was then studied extensively by Burgers [5], Eberhard Hopf [6], Julian Cole [7], and others, in particular; after the discovery of a coordinate transformation that maps it to the heat equation. While as a model for gas turbulence the equation was soon rivaled by more complicated models, the linearizing transformation just mentioned added importance to the equation as a mathematical model, which has since been extensively studied. The limit m ! 0 is an hyperbolic equation, called the inviscid Burgers equation: E-mail address: [email protected] 0096-3003/$ - see front matter Ó 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2009.12.008
A.H. Salas / Applied Mathematics and Computation 216 (2010) 18–26
ut þ uux ¼ 0:
19
ð1:4Þ
This limiting equation is important because it provides a simple example of a conservation law, capturing the crucial phenomenon of shock formation. Indeed, it was originally introduced as a model to describe the formation of shock waves in gas dynamics. A first-order partial differential equation for uðx; tÞ is called a conservation law if it can be written in the form ut þ ðf ðuÞÞx ¼ 0. For Eq. (1.3), f ðuÞ ¼ u2 =2. Such conservation laws may exhibit the formation of shocks, which are discontinuities appearing in the solution after a finite time and then propagating in a regular manner. When this phenomenon arises, an initially smooth wave becomes steeper and steeper as time progresses, until it forms a jump discontinuity – the shock. Nowadays, the Burgers equation is used as a simplified model of a kind of hydrodynamic turbulence [8], called Burgers turbulence. Burgers himself wrote a treatise on the equation now known by his name [9], where several variants are proposed to describe this particular kind of turbulence. Eq. (1.2) was originally derived to describe the propagation of nonlinear waves in dissipative media, where mð> 0Þ is the kinematic viscosity, and uðx; tÞ represents the fluid velocity field. It plays an active role in explaining two fundamental effects characteristic of any turbulence: the nonlinear redistribution of energy over the spectrum and the action of viscosity in small scales. Over the decades, the Burgers equation has been widely used to model a large class of physical systems in which the nonlinearity is fairly weak (quadratic) and the dispersion is negligible compared to the linear damping. Hopf [6] and Cole [7] independently discovered a transformation that reduces the Burgers equation (1.2) to a linear diffusion equation. First, we write (1.2) in a form similar to a conservation law
ut þ
@ 1 2 u mux ¼ 0: @x 2
ð1:5Þ
This may be regarded as the compatibility condition for a function W to exist, such that
u ¼ Wx ; 1 mux u2 ¼ Wt : 2
ð1:6Þ ð1:7Þ
We substitute the value of u from (1.6) in (1.7) to obtain
1 2
mWxx W2x ¼ Wt :
ð1:8Þ
Next, we introduce W ¼ 2m log / so that
u ¼ Wx ¼ 2m
/x : /
ð1:9Þ
This is called the Cole–Hopf transformation which, by differentiating, gives
Wxx ¼ 2m
2 /x 2m /xx / /
and Wt ¼ 2m
/t : /
ð1:10Þ
Consequently, (1.8) reduces to the linear heat equation
/t ¼ m/xx :
ð1:11Þ
The relation between the Burgers and the heat equation was already mentioned in an earlier book [10], but the former had not been recognized as physically relevant; hence, the importance of this connection was seemingly not noticed at the time. Using the transformation of Eq. (1.2), known as the Cole–Hopf transformation, it is easy to solve the initial value problem for this equation. Recently, a generalization of the Cole–Hopf transformation has been successfully used to linearize the boundary value problem for the Burgers equation posed on the semiline x > 0 [11]. Many solutions of Eq. (1.11) are well-known in the literature. For a more complete exposition, we exhibit some of them in next sections. To solve Eq. (1.2), we simply substitute the given solution for / in (1.9). Motivated by this idea, we intend to extend the Cole–Hopf transformation to linearize Eq. (1.1). This is our aim in the third section. This paper is organized as follows: In Section 2 we show the general solution to Burgers equation by the traveling wave method. In Section 3 we successfully apply the generalized Cole–Hopf transformation to forced Burgers equation and we reduce the problem of solving this equation to solve the linear heat equation and a special Riccati equation. Section 4 is dedicated to show some exact solutions to linear heat equation by transformation groups. In Section 5 we obtain some exact solutions to Burgers equation. At the end, we give some conclusions. 2. Solutions to Burgers equation by the traveling wave method If GðtÞ 0 in Eq. (1.1), we obtain the generalized Burgers equation
ut þ auux þ buxx ¼ 0:
ð2:1Þ
20
A.H. Salas / Applied Mathematics and Computation 216 (2010) 18–26
Eq. (2.1) may be reduced to classical Burgers equation (1.2). Indeed, let us consider the transformation group given by
x¼
b
am
X;
t¼
b
a2 m
T:
ð2:2Þ
Then
b b u ¼ uðx; tÞ ¼ u X; 2 T ;
am
uT ¼ ux xT þ ut t T ¼ uX ¼ ux xX þ ut t X ¼ uXX ¼
b
am
am
b
a2 m b
am
ut ;
ux ;
ðuxx xX þ uxt t X Þ ¼
b2
a2 m2
uxx ;
from which
uT þ uuX muXX ¼ ut þ auux þ buxx ¼ 0: So it suffices to solve the Eq. (1.2). This means that if u ¼ uðx; tÞ is a solution to Eq. (1.2), then
am a2 m u ¼ u x; t ; b b
ð2:3Þ
is a solution to Eq. (2.1). To illustrate this fact, suppose that the linear heat Eq. (1.11) has a solution of the form
/ðx; tÞ ¼ expðlðx þ ktÞÞ:
ð2:4Þ
Then
/t m/xx ¼ lðk lmÞ expðlðx þ ktÞÞ ¼ 0; so k ¼ lm. Thus, a particular solution to heat Eq. (1.11) is
/ðx; tÞ ¼ expðlðx þ mltÞÞ for any
l:
ð2:5Þ
In virtue of the superposition principle, Eq. (1.11) has also a solution of the form
/ðx; tÞ ¼ 1 þ
n X
ci expðli ðx þ mli tÞÞ;
ð2:6Þ
i¼1
where n is an arbitrary positive integer, ki –0 ðki –kj for i–jÞ and ci are all arbitrary constants. From (1.9) and (2.6), function
uðx; tÞ ¼ 2m
Pn i¼1 c i li expðli ðx þ mli tÞÞ P ; 1 þ ni¼1 ci expðli ðx þ mli tÞÞ
ð2:7Þ
is a solution to Burgers equation (1.2). This solution is called the n-shock-wave solution. For n ¼ 1 solution (2.7) corresponds to the one-shock-wave solution as follows:
uðx; tÞ ¼
2mc1 l1 expðl1 ðx þ ml1 tÞÞ for any c1 and 1 þ c1 expðl1 ðx þ ml1 tÞÞ
l1 :
ð2:8Þ
Solution (2.8) may be written in the form
pffiffiffiffiffi 1 1 uðx; tÞ ¼ ml1 1 þ tanh l1 ðx þ ml1 tÞ þ logð c1 Þ ¼ ml 1 þ tanh lðx mlt þ g0 Þ ; 2 2 where
ð2:9Þ
l and g0 are arbitrary. From (2.3) and (2.9) following function is a solution to generalized Burgers equation (2.1): 1 am a2 m 1 ¼ lm 1 tanh uðx; tÞ ¼ ml 1 þ tanh l x þ ml ðlðamðx þ almtÞ þ n0 ÞÞ : t þ g0 2 2b b b
ð2:10Þ
Similarly, a solution to Eq. (2.1) is
Pn uðx; tÞ ¼ 2m
i¼1 c i
1þ
Pn
li exp abm li ðx þ ali mtÞ
i¼1 c i
: exp abm li ðx þ ali mtÞ
ð2:11Þ
A.H. Salas / Applied Mathematics and Computation 216 (2010) 18–26
21
Solution (2.9) is well-known. A little more general solution to Eq. (1.2) is obtained by the traveling wave method. Indeed, suppose that this admits a solution of the form
u ¼ uðnÞ ¼ uðkðx þ ktÞÞ:
ð2:12Þ
From (1.2) and (2.12), 00
ku0 ðnÞ þ uðnÞu0 ðnÞ mku ðnÞ ¼ 0:
ð2:13Þ
Integrating this equation with respect to n we obtain the famous Riccati equation
u0 ðnÞ ¼
1 2 k C u ðnÞ þ uðnÞ ; 2km km km
ð2:14Þ
where C is the constant of integration. General solution of Eq. (2.14) is
uðnÞ ¼ k
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k2 þ 2C k2 þ 2C tanh ðn þ n0 Þ ; 2km
ð2:15Þ
and then a solution to Burgers equation (1.2) for arbitrary C; k and n0 is
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k2 þ 2C 2 uðx; tÞ ¼ k k þ 2C tanh ðx þ kt þ g0 Þ ; 2m
g0 ¼
n0 : k
ð2:16Þ
Observe that solution (2.9) is covered by solution (2.16). To see this, we take C ¼ 0; k ¼ ml. If k2 þ 2C < 0 we obtain the periodic solution given by
0qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðk2 þ 2CÞ 2 @ uðx; tÞ ¼ k ðk þ 2CÞ tan ðx þ kt þ n0 ÞA: 2m Following is a rational solution to Eq. (1.2) obtained from (2.14) by taking C ¼
uðx; tÞ ¼ k
ð2:17Þ k2 2
2m ; C 1 ¼ const: x þ kt þ C 1
ð2:18Þ
3. Transforming forced Burgers equation via the generalized Cole–Hopf transformation In this section we construct the generalized Cole–Hopf transformation from Eq. (1.2) to the standard heat equation. In order to look for solutions to forced Burgers equation (1.1), we apply the generalized Cole–Hopf transformation given by
uðx; tÞ ¼ f ðx; tÞ
/n ðn; sÞ þ gðx; tÞ; /ðn; sÞ
n ¼ nðx; tÞ; t ¼ tðsÞ;
ð3:1Þ
where f ðx; tÞ–0; gðx; tÞ; nðx; tÞ; /ðn; sÞ and sðtÞ are some functions to be determined later. We have:
nt /s þ st /sn /n nt /n þ st /s / f þ n ft þ g t ; 2 / / / ! 2 nx /nn nx /n / ux ¼ 2 f þ n fx þ g x ; / / / ! ! 2 3 2 2 2 nx /nnn 2nx /n nxx /nn 3n2x /n /nn nxx /n 2nx /nn 2nx /n / fx þ n fxx þ g xx : uxx ¼ þ 2 fþ þ 3 2 2 / / / / / / / / ut ¼
ð3:2Þ ð3:3Þ
ð3:4Þ
We substitute expressions of ut ; ux and uxx from (3.2), (3.3) and (3.4) in (1.1) to obtain following polynomial equation in the variables / and /n :
f nx ð2bnx af Þ/3n þ ðfx ðaf 2bnx Þ f ðagnx þ nt þ bnxx ÞÞ/2n / þ f ðaf nx /nn st /s 3bn2x /nn Þ/n / þ ðagfx þ afg x þ ft þ bfxx Þ/n /2 þ ð/nn ðnx ðafg þ 2bfx Þ þ f nt þ f bnxx Þ þ f st /ns þ bn2x /nnn Þ/2 þ ðg t þ agg x þ bg xx xGðtÞÞ/3 ¼ 0:
ð3:5Þ
22
A.H. Salas / Applied Mathematics and Computation 216 (2010) 18–26
Equating the coefficients of /3n ; /2n /; /n /; /n /2 ; /2 and /3 to zero and taking into account that f –0; n–0 and b–0 yields following system of pde’s
2bnx af ¼ 0;
ð3:6Þ
fx ðaf 2bnx Þ f ðagnx þ nt þ bnxx Þ ¼ 0;
ð3:7Þ
3bn2x /nn
af nx /nn st /s ¼ 0; agfx þ afg x þ ft þ bfxx ¼ 0; /nn ðnx ðafg þ 2bfx Þ þ f nt þ f bnxx Þ þ f st /ns þ bn2x /nnn ¼ 0; g t þ agg x þ bg xx xGðtÞ ¼ 0:
ð3:8Þ ð3:9Þ ð3:10Þ ð3:11Þ
From (3.6) and (3.7),
nx ¼
a 2b
nt ¼
f
a 2b
ð3:12Þ ðafg þ bfx Þ
ð3:13Þ
From (3.12) and (3.13), (3.8) and (3.9),
a2 /nn
f 2; 4b /s 1 g x ¼ ðft þ agfx þ bfxx Þ: af
st ¼
ð3:14Þ ð3:15Þ
From (3.12)–(3.15) and (3.10),
/ns /nnn 4b fx ¼ : /s /nn a f
ð3:16Þ
Observe that if f does not depend on x and /nn ¼ /s , then Eq. (3.16) holds . Thus, we shall assume that
f ¼ f ðtÞ and /s ¼ /nn :
ð3:17Þ
In particular, / satisfies the linear heat equation
/s /nn ¼ 0:
ð3:18Þ
Now, Eqs. (3.11), (3.15) and (3.17) give
8 < gx ¼ 1 a
f 0 ðtÞ ; f ðtÞ
ð3:19Þ
: g ¼ xGðtÞ þ g f 0 ðtÞ : t f ðtÞ Eliminating g from system (3.19) we obtain the Riccati equation
u0 ðtÞ ¼ u2 ðtÞ aGðtÞ where uðtÞ ¼
f 0 ðtÞ : f ðtÞ
ð3:20Þ
Solving this equation and using Eqs. (3.12)–(3.17) we find following particular solutions to system of Eqs. (3.6)–(3.11):
R 8 f ðtÞ ¼ expð uðtÞdtÞ; > > > > gðx; tÞ ¼ 1 xuðtÞ; < a R a x expð uðtÞdtÞ; nðx; tÞ ¼ > 2b > > > R : sðtÞ ¼ a4b2 expð uðtÞdtÞ:
ð3:21Þ
Finally, a solution to forced Burgers equation (1.1) is given by
8 R /n ðn;sÞ x > > > uðx; tÞ ¼ /ðn;sÞ expð uðtÞdtÞ a uðtÞ; > > R R < a x expð uðtÞdtÞ; s ¼ a2 expð uðtÞdtÞ; n ¼ 2b 4b > > where; > > > : /s ¼ /nn and u0 ðtÞ ¼ u2 ðtÞ aGðtÞ;
ð3:22Þ
where / ¼ /ðn; sÞ satisfies the linearheat Eq. (3.18) and uðtÞ is a solution to Riccati Eq. (3.20). As we can see, the problem of finding solutions to forced Burgers equation (1.1) consists on solving the linear heat equation and the Riccati Eq. (3.20). This last equation is solvable exactly for some particular choices of GðtÞ.
A.H. Salas / Applied Mathematics and Computation 216 (2010) 18–26
23
4. Exact solutions to linear heat equation Suppose that / ¼ /ðx; tÞ is a solution to equation
/t ¼ /xx :
ð4:1Þ
Then u ¼ /ðx; mtÞ is a solution to heat equation
/t ¼ m/xx :
ð4:2Þ
Conversely, if u ¼ uðx; tÞ is a solution to (4.2), then / ¼ uðx; t=mÞ is a solution to (4.1). Thus, Eqs. (4.2) and (4.2) are the same. We shall call Eq. (4.1) the normalized heat equation. We already obtained some solutions to Eq. (4.2) in a form of traveling wave. See Eqs. (2.5) and (2.6). In this section we give solutions to this equation by using Lie’s group theory [12]. Definition 4.1. Let I be a system of differential equations. A symmetry group of the system I is a local group of transformations G acting on an open subset M of the space of independent and dependent variables for the system with the property that whenever u ¼ f ðxÞ is a solution of I, and whenever g f is defined for g 2 G, then u ¼ g f ðxÞ is also a solution of the system. (By solution we mean any smooth solution u ¼ f ðxÞ defined on any subdomain X X). Following are symmetry groups of the heat Eq. (4.1):
G1
ðx þ e; t; uÞ;
G2
ðx; t þ e; uÞ;
G3
ðx; t; expðeÞuÞ;
G4
ðexpðeÞx; expð2eÞt; uÞ;
G5
ðx þ 2et; t; u expðex e2 tÞ; pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x t ex2 ; ; ; u 1 4et exp 1 4et 1 4et 1 4et
G6 Ga
ðx; t; u þ eaðx; tÞ:
This means that if / ¼ /ðx; tÞ is a solution of the heat Eq. (4.1), so are the functions
/ð1Þ ¼ /ðx e; tÞ; /ð2Þ ¼ /ðx; t eÞ; /ð3Þ ¼ expðeÞ/ðx; tÞ; /ð4Þ ¼ /ðexpðeÞx; expð2eÞtÞ; /ð5Þ ¼ expðex þ e2 tÞ/ðx 2et; tÞ; 1 ex2 x t / ; /ð6Þ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi exp ; 1 þ 4et 1 þ 4et 1 þ 4et 1 þ 4et /ð7Þ ¼ /ðx; tÞ þ eaðx; tÞ; where e is any real number and aðx; tÞ is any other solution to the heat equation. The symmetry groups G3 and Ga thus reflect the linearity of the heat equation; we can add solutions and multiply them by constants. The groups G1 and G2 demonstrate the time- and space-in variance of the equation, reflecting the fact that the heat equation has constant coefficients. The wellknown scaling symmetry turns up in G4 , while G5 represents a kind of Galilean boost to a moving coordinate frame. The last group G6 is a genuinely local group of transformations. Its appearance is far from obvious from basic physical principles, but it has the following nice consequence. If we let / ¼ c be just a constant solution, then we immediately conclude that the function
c ex2 ; / ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi exp 1 þ 4et 1 þ 4et
ð4:3Þ
pffiffiffiffiffiffiffiffi is also a solution. In particular, if we set c ¼ e=p we obtain the fundamental solution to the heat equation at the point ðx0 ; y0 Þ ¼ ð0; 1=ð4eÞÞ. To obtain the fundamental solution
2 1 x ; / ¼ pffiffiffiffiffiffiffiffi exp 4t 4pt
ð4:4Þ
we need to translate this solution in t using the group G2 (with e replaced by 1=ð4eÞ). It can be shown that the most general solution obtainable from a given solution u ¼ uðx; tÞ by group transformations is of the form
1 /ðx; tÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi exp 1 þ 4e6 t
e3
e5 x þ e6 x2 e5 t 1 þ 4e6 t
ð4:5Þ
24
A.H. Salas / Applied Mathematics and Computation 216 (2010) 18–26
expðe4 Þðx 2e5 tÞ expð2e4 Þt e1 ; e2 þ aðx; tÞ; 1 þ 4e6 t 1 þ 4e6 t
u
ð4:6Þ
where e1 ; . . . ; e6 are real constants and a ¼ aðx; tÞ an arbitrary solution to the heat equation. Previous considerations allow to obtain particular solutions to the heat equation /t ¼ m/xx . Some of them are :
wðxÞ ¼ k ¼ const:;
ð4:7Þ
wðxÞ ¼ Ax þ B;
ð4:8Þ
wðx; tÞ ¼ Aðx2 þ 2mtÞ þ B;
ð4:9Þ
3
wðx; tÞ ¼ Aðx þ 6mtxÞ þ B;
ð4:10Þ
wðx; tÞ ¼ Aðx4 þ 12mtx2 þ 12m2 t2 Þ þ B; n X ð2nÞð2n 1Þ . . . ð2n 2k þ 1Þ ðmtÞk x2n2k ; wðx; tÞ ¼ x2n þ k! k¼1
ð4:11Þ
wðx; tÞ ¼ x2nþ1 þ
n X ð2n þ 1Þð2nÞ . . . ð2n 2k þ 2Þ ðmtÞk x2n2kþ1 ; k! k¼1
wðx; tÞ ¼ A expðml2 t lxÞ þ B; 1 x2 wðx; tÞ ¼ A pffiffi exp þ B; 4mt t
ð4:12Þ ð4:13Þ ð4:14Þ ð4:15Þ
wðx; tÞ ¼ A expðml2 tÞ cosðlx þ BÞ;
ð4:16Þ
wðx; tÞ ¼ A expðml2 tÞ cosðlx þ BÞ þ C;
ð4:17Þ
wðx; tÞ ¼ A expðlxÞ cosðlx 2ml2 t þ BÞ þ K; x wðx; tÞ ¼ A erf pffiffiffiffiffi þ B; 2 mt
ð4:18Þ
where A; B; C, and
ð4:19Þ
l are arbitrary constants, n is a positive integer, Z
2 erfðzÞ pffiffiffiffi
p
z
expðn2 Þ dn;
ð4:20Þ
0
is the error function (probability integral). These solutions are useful in solving forced Burgers equation (1.1) by the formula (3.22). 5. Analytic solutions to forced Burgers equation Suppose we have solved the Riccati Eq. (3.20). Then, making use of (3.22) together with (4.7)–(4.20) (letting replacing x with n and t with s), following are solutions to forced Burgers equation (1.1):
x uðx; tÞ ¼ uðtÞ;
a
uðx; tÞ ¼ uðx; tÞ ¼ uðx; tÞ ¼
A exp An þ B 2An
Z
Aðn2 þ 2sÞ þ B 3Aðn2 þ 2sÞ
ð5:1Þ x
uðtÞdt uðtÞ; a Z
exp
exp Anðn þ 6sÞ þ B 3 4Aðn þ 6nsÞ
Z
2
ð5:2Þ x
uðtÞdt uðtÞ; a
x
uðtÞdt uðtÞ; a Z
4
2
2Þ
ð5:3Þ ð5:4Þ
x uðtÞdt uðtÞ; exp a Aðn Z þ 12n s þ 12s þ B Bl x uðx; tÞ ¼ l uðtÞdt uðtÞ; exp A expð l ð ls þ nÞÞ þ B a Z n x uðx; tÞ ¼ exp uðtÞdt uðtÞ; 2s a Z Al sinðB þ lnÞ x u ðtÞdt uðtÞ; uðx; tÞ ¼ exp A cosðB þ lnÞ þ K expðl2 sÞ a Z AlðsinðB þ lðn 2lsÞÞ þ cosðB þ lðn 2lsÞÞÞ x exp uðtÞdt uðtÞ; uðx; tÞ ¼ A cosðB þ lðn 2lsÞÞ þ K expðlnÞ a n2 Z Ae4s x
exp uðtÞdt uðtÞ: uðx; tÞ ¼ pffiffiffiffiffiffi n ffiffi a p ps Aerf 2 s þ B uðx; tÞ ¼
m ¼ 1 and
ð5:5Þ ð5:6Þ ð5:7Þ ð5:8Þ ð5:9Þ ð5:10Þ
A.H. Salas / Applied Mathematics and Computation 216 (2010) 18–26
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In formulas (5.1)–(5.10),
n¼
a 2b
x exp
Z
uðtÞdt
s¼
and
a2 4b
exp
Z
uðtÞdt :
ð5:11Þ
In particular, if GðtÞ 0, the ’forced’ Burgers equation (1.1) takes the form
ut þ auux þ buxx ¼ 0:
ð5:12Þ
This is the generalized Burgers equation (2.1). We gave its traveling wave solutions in Section 2. We derive here other exact solutions to this equation. Observe that Eq. (3.20) converts to
u0 ðtÞ ¼ u2 ðtÞ;
ð5:13Þ
whose general solution is
uðtÞ ¼
1 ; C ¼ const: Ct
ð5:14Þ
From (3.20), (3.21), (3.22) and (5.14),
8 1 > f ðtÞ ¼ Ct ; > > > x > < gðx; tÞ ¼ aðCtÞ ;
ð5:15Þ
ax ; nðx; tÞ ¼ 2bðCtÞ > > > > > : sðtÞ ¼ a2 : 4bðCtÞ
Making use of (5.1)–(5.10) and taking into account (5.11) and (5.15), following are solutions to Eq. (5.12) :
u1 ðx; tÞ ¼
x
aðC tÞ
ð5:16Þ
;
2bðC tÞðAa BxÞ Aax2 ; aðC tÞðAax þ 2BbðC tÞÞ
x Aa2 6bðt CÞ þ x2 þ 4Bb2 ðC tÞ2
; u3 ðx; tÞ ¼ aðC tÞ Aa2 ð2bðt CÞ þ x2 Þ þ 4Bb2 ðC tÞ2
Aa3 12bx2 ðt CÞ þ 12b2 ðC tÞ2 þ x4 8Bb3 xðC tÞ3
; u4 ðx; tÞ ¼ aðC tÞ Aa3 xð6bðt CÞ þ x2 Þ þ 8Bb3 ðC tÞ3
x Aa4 20bx2 ðt CÞ þ 60b2 ðC tÞ2 þ x4 þ 16Bb4 ðC tÞ4
u5 ðx; tÞ ¼ aðC tÞ Aa4 12bx2 ðt CÞ þ 12b2 ðC tÞ2 þ x4 þ 16Bb4 ðC tÞ4 8 9 1 < Al x=
: u6 ðx; tÞ ¼ C t :A þ B exp alðal2xÞ a; u2 ðx; tÞ ¼
ð5:17Þ
ð5:18Þ
ð5:19Þ
ð5:20Þ
4bðCtÞ
In particular, if B ¼ 0 and A–0,
u7 ðx; tÞ ¼
x al ; aðt CÞ
ð5:21Þ
3=2 a2 Bb bðtCÞ x
qffiffiffiffiffiffiffiffiffiffi ; u8 ðx; tÞ ¼ 1 a2 3 2 a 2A exp 4bðCtÞ x þ B bðtCÞ
2 2 a l al al A exp 4bðCtÞ al sin 2bðCtÞ x þ B þ x cos 2bðCtÞ x þ B þ Cx 2 2
u9 ðx; tÞ ¼ ; a l al cos 2bðCtÞ aðC tÞ A exp 4bðCtÞ xþB þC
8 9 al al 1
; þ u10 ðx; tÞ ¼ al al C t: a; ðx þ alÞ þ B þ K exp 2bðCtÞ x A cos 2bðCtÞ
8 9 al al 1
: u11 ðx; tÞ ¼ þ al al C t: a; ðx þ alÞ þ B þ K exp x A cos 2bðCtÞ
In particular, if K ¼ 0 we get a periodic solution :
2bðCtÞ
ð5:22Þ
ð5:23Þ
ð5:24Þ
ð5:25Þ
26
A.H. Salas / Applied Mathematics and Computation 216 (2010) 18–26
u12 ðx; tÞ ¼
1 xal : x þ al tan B þ aðC tÞ 2bðC tÞ
pffiffiffiffiffiffiffi We obtain from (5.26) another solution by replacing B with B 1 and
1 xa l : u13 ðx; tÞ ¼ x al tanh B þ aðC tÞ 2bðC tÞ 8
9 > > pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > 1 2 > < 2 bðt CÞ exp 4bðCtÞ x = 1 : pffiffiffiffi u14 ðx; tÞ ¼ xþ > aðC tÞ > p > 1 ; : erf pffiffiffiffiffiffiffiffiffiffi x > 2
ð5:26Þ pffiffiffiffiffiffiffi
l with l 1 : ð5:27Þ
ð5:28Þ
bðtCÞ
We may obtain other exact solutions to Burgers equation (1.2) by using (1.9) and 4.7, (4.8)–(4.20). Some other works concerning the problem of finding exact solutions to nonlinear pde’s may be found in Ref. [13–16] 6. Conclusions We successfully applied the generalized Cole–Hopf transformation to forced Burgers equation. As a particular case, we obtained some solutions to Burgers equation. We also derived the general solution of the Burgers equation in the form of a traveling wave. We think that the application of the generalized Cole–Hopf transformation is a useful tool in the search of solutions to other forced equations. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16]
S. Eule, R. Friedrich, A note on the forced Burgers equation, Physics Letters A 351 (2006) 238–241. M. Scott, Encyclopedia of Nonlinear Science, Taylor and Francis, 2005. H. Bateman, Some recent research on the motion of fluids, Monthly Weather Review 43 (1915) 163–170. J. Burgers, Application of a model system to illustrate some points of the statistical theory of free turbulence, Proceedings of the Nederlandse Akademie van Wetenschappen 43 (1940) 2–12. J. Burgers, A mathematical model illustrating the theory of turbulence, Advances in Applied Mechanics 1 (1948) 171–199. E. Hopf, The partial differential equation ut þ uux ¼ luxx , Communications in Pure and Applied Mathematics 3 (1950) 201–230. J. Cole, On a quasilinear parabolic equation occurring in aerodynamics, Quarterly Journal of Applied Mathematics 9 (1951) 225–236. K.M. Case, S.C. Chiu, Burgers turbulence models, Physics of Fluids 12 (1969) 1799–1808. J. Burgers, The Nonlinear Diffusion Equation: Asymptotic Solutions and Statistical Problems, Reidel, Dordrecht and Boston, 1974. A.R. Forsyth, Theory of Differential Equations, Cambridge University Press, Cambridge, 1906. F. Calogero, S. De Lillo, The Burgers equation on the semiline, Inverse Problems 5 (1989) L37. P. Olver, Applications of Lie Groups to Differential Equations, second ed., Springer-Verlag, 1993. A. Salas, Some solutions for a type of generalized Sawada–Kotera equation, Applied Mathematics and Computation 196 (2008) 812–817. A.H. Salas, C.A. Gómez, Computing exact solutions for some fifth KdV equations with forcing term, Applied Mathematics and Computation 204 (2008) 257–260. A.H. Salas, Exact solutions for the general fifth KdV equation by the exp function method, Applied Mathematics and Computation 205 (2008) 291–297. A.H. Salas, C.A. Gómez, Exact solutions for a third KdV equation with variable coefficients and forcing term, Mathematical Problems in Engineering, Hindawi, in press.
Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Symbolic computation of solutions for a forced Burgers equation Alvaro H. Salas Department of Mathematics, Universidad de Caldas, Manizales-Colombia, AA 275, Universidad Nacional de Colombia, Campus la Nubia, Manizales, Colombia
a r t i c l e
i n f o
Keywords: Nonlinear evolution equation pde Burgers equation Reaction–diffusion equation
a b s t r a c t In this paper we give exact solutions for a forced Burgers equation. We make use of the generalized Cole–Hopf transformation and the traveling wave method. Ó 2009 Elsevier Inc. All rights reserved.
1. Introduction The search of exact solutions of nonlinear partial differential equations is of great importance, because these equations appear in complex physics phenomena, mechanics, chemistry, biology and engineering. In this paper we obtain new solutions for the forced Burgers equation [1]
ut þ auux þ buxx ¼ GðtÞx;
ð1:1Þ
Eq. (1.1) is a generalization of the well-known Burgers equation [9]
ut þ uux ¼ muxx :
ð1:2Þ
Owing to the assumptions of the constant coefficients and unforced turbulence, the physical situations in which the classical Burgers equation arises tend to be highly idealized. In practice, the forced of Eq. (1.2) may provide us with more realistic models in many different physical contexts like the long-wave propagation in an inhomogeneous two-layer shallow liquid, directed polymers in a random medium, pinning of vortex lines in superconductors, large eddy simulation, ballistic deposition, passive random walker dynamics on a growing surface, etc. As remarked in Ref. [2], in 1915, Harry Bateman considered a nonlinear equation whose steady solutions were thought to describe certain viscous flows [3]. This equation, modeling a diffusive nonlinear wave, is now widely known as the Burgers equation, and is given by
ut þ uux ¼ muxx ;
ð1:3Þ
where m is a constant measuring the viscosity of the fluid. It is a nonlinear parabolic equation, simply describing a temporal evolution where nonlinear convection and linear diffusion are combined, and it can be derived as a weakly nonlinear approximation to the equations of gas dynamics. Although nonlinear, Eq. (1.3) is very simple, and interest in it was revived in the 1940s, when Dutch physicist Jan Burgers proposed it to describe a mathematical model of turbulence in gas [4]. As a model for gas dynamics, it was then studied extensively by Burgers [5], Eberhard Hopf [6], Julian Cole [7], and others, in particular; after the discovery of a coordinate transformation that maps it to the heat equation. While as a model for gas turbulence the equation was soon rivaled by more complicated models, the linearizing transformation just mentioned added importance to the equation as a mathematical model, which has since been extensively studied. The limit m ! 0 is an hyperbolic equation, called the inviscid Burgers equation: E-mail address: [email protected] 0096-3003/$ - see front matter Ó 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2009.12.008
A.H. Salas / Applied Mathematics and Computation 216 (2010) 18–26
ut þ uux ¼ 0:
19
ð1:4Þ
This limiting equation is important because it provides a simple example of a conservation law, capturing the crucial phenomenon of shock formation. Indeed, it was originally introduced as a model to describe the formation of shock waves in gas dynamics. A first-order partial differential equation for uðx; tÞ is called a conservation law if it can be written in the form ut þ ðf ðuÞÞx ¼ 0. For Eq. (1.3), f ðuÞ ¼ u2 =2. Such conservation laws may exhibit the formation of shocks, which are discontinuities appearing in the solution after a finite time and then propagating in a regular manner. When this phenomenon arises, an initially smooth wave becomes steeper and steeper as time progresses, until it forms a jump discontinuity – the shock. Nowadays, the Burgers equation is used as a simplified model of a kind of hydrodynamic turbulence [8], called Burgers turbulence. Burgers himself wrote a treatise on the equation now known by his name [9], where several variants are proposed to describe this particular kind of turbulence. Eq. (1.2) was originally derived to describe the propagation of nonlinear waves in dissipative media, where mð> 0Þ is the kinematic viscosity, and uðx; tÞ represents the fluid velocity field. It plays an active role in explaining two fundamental effects characteristic of any turbulence: the nonlinear redistribution of energy over the spectrum and the action of viscosity in small scales. Over the decades, the Burgers equation has been widely used to model a large class of physical systems in which the nonlinearity is fairly weak (quadratic) and the dispersion is negligible compared to the linear damping. Hopf [6] and Cole [7] independently discovered a transformation that reduces the Burgers equation (1.2) to a linear diffusion equation. First, we write (1.2) in a form similar to a conservation law
ut þ
@ 1 2 u mux ¼ 0: @x 2
ð1:5Þ
This may be regarded as the compatibility condition for a function W to exist, such that
u ¼ Wx ; 1 mux u2 ¼ Wt : 2
ð1:6Þ ð1:7Þ
We substitute the value of u from (1.6) in (1.7) to obtain
1 2
mWxx W2x ¼ Wt :
ð1:8Þ
Next, we introduce W ¼ 2m log / so that
u ¼ Wx ¼ 2m
/x : /
ð1:9Þ
This is called the Cole–Hopf transformation which, by differentiating, gives
Wxx ¼ 2m
2 /x 2m /xx / /
and Wt ¼ 2m
/t : /
ð1:10Þ
Consequently, (1.8) reduces to the linear heat equation
/t ¼ m/xx :
ð1:11Þ
The relation between the Burgers and the heat equation was already mentioned in an earlier book [10], but the former had not been recognized as physically relevant; hence, the importance of this connection was seemingly not noticed at the time. Using the transformation of Eq. (1.2), known as the Cole–Hopf transformation, it is easy to solve the initial value problem for this equation. Recently, a generalization of the Cole–Hopf transformation has been successfully used to linearize the boundary value problem for the Burgers equation posed on the semiline x > 0 [11]. Many solutions of Eq. (1.11) are well-known in the literature. For a more complete exposition, we exhibit some of them in next sections. To solve Eq. (1.2), we simply substitute the given solution for / in (1.9). Motivated by this idea, we intend to extend the Cole–Hopf transformation to linearize Eq. (1.1). This is our aim in the third section. This paper is organized as follows: In Section 2 we show the general solution to Burgers equation by the traveling wave method. In Section 3 we successfully apply the generalized Cole–Hopf transformation to forced Burgers equation and we reduce the problem of solving this equation to solve the linear heat equation and a special Riccati equation. Section 4 is dedicated to show some exact solutions to linear heat equation by transformation groups. In Section 5 we obtain some exact solutions to Burgers equation. At the end, we give some conclusions. 2. Solutions to Burgers equation by the traveling wave method If GðtÞ 0 in Eq. (1.1), we obtain the generalized Burgers equation
ut þ auux þ buxx ¼ 0:
ð2:1Þ
20
A.H. Salas / Applied Mathematics and Computation 216 (2010) 18–26
Eq. (2.1) may be reduced to classical Burgers equation (1.2). Indeed, let us consider the transformation group given by
x¼
b
am
X;
t¼
b
a2 m
T:
ð2:2Þ
Then
b b u ¼ uðx; tÞ ¼ u X; 2 T ;
am
uT ¼ ux xT þ ut t T ¼ uX ¼ ux xX þ ut t X ¼ uXX ¼
b
am
am
b
a2 m b
am
ut ;
ux ;
ðuxx xX þ uxt t X Þ ¼
b2
a2 m2
uxx ;
from which
uT þ uuX muXX ¼ ut þ auux þ buxx ¼ 0: So it suffices to solve the Eq. (1.2). This means that if u ¼ uðx; tÞ is a solution to Eq. (1.2), then
am a2 m u ¼ u x; t ; b b
ð2:3Þ
is a solution to Eq. (2.1). To illustrate this fact, suppose that the linear heat Eq. (1.11) has a solution of the form
/ðx; tÞ ¼ expðlðx þ ktÞÞ:
ð2:4Þ
Then
/t m/xx ¼ lðk lmÞ expðlðx þ ktÞÞ ¼ 0; so k ¼ lm. Thus, a particular solution to heat Eq. (1.11) is
/ðx; tÞ ¼ expðlðx þ mltÞÞ for any
l:
ð2:5Þ
In virtue of the superposition principle, Eq. (1.11) has also a solution of the form
/ðx; tÞ ¼ 1 þ
n X
ci expðli ðx þ mli tÞÞ;
ð2:6Þ
i¼1
where n is an arbitrary positive integer, ki –0 ðki –kj for i–jÞ and ci are all arbitrary constants. From (1.9) and (2.6), function
uðx; tÞ ¼ 2m
Pn i¼1 c i li expðli ðx þ mli tÞÞ P ; 1 þ ni¼1 ci expðli ðx þ mli tÞÞ
ð2:7Þ
is a solution to Burgers equation (1.2). This solution is called the n-shock-wave solution. For n ¼ 1 solution (2.7) corresponds to the one-shock-wave solution as follows:
uðx; tÞ ¼
2mc1 l1 expðl1 ðx þ ml1 tÞÞ for any c1 and 1 þ c1 expðl1 ðx þ ml1 tÞÞ
l1 :
ð2:8Þ
Solution (2.8) may be written in the form
pffiffiffiffiffi 1 1 uðx; tÞ ¼ ml1 1 þ tanh l1 ðx þ ml1 tÞ þ logð c1 Þ ¼ ml 1 þ tanh lðx mlt þ g0 Þ ; 2 2 where
ð2:9Þ
l and g0 are arbitrary. From (2.3) and (2.9) following function is a solution to generalized Burgers equation (2.1): 1 am a2 m 1 ¼ lm 1 tanh uðx; tÞ ¼ ml 1 þ tanh l x þ ml ðlðamðx þ almtÞ þ n0 ÞÞ : t þ g0 2 2b b b
ð2:10Þ
Similarly, a solution to Eq. (2.1) is
Pn uðx; tÞ ¼ 2m
i¼1 c i
1þ
Pn
li exp abm li ðx þ ali mtÞ
i¼1 c i
: exp abm li ðx þ ali mtÞ
ð2:11Þ
A.H. Salas / Applied Mathematics and Computation 216 (2010) 18–26
21
Solution (2.9) is well-known. A little more general solution to Eq. (1.2) is obtained by the traveling wave method. Indeed, suppose that this admits a solution of the form
u ¼ uðnÞ ¼ uðkðx þ ktÞÞ:
ð2:12Þ
From (1.2) and (2.12), 00
ku0 ðnÞ þ uðnÞu0 ðnÞ mku ðnÞ ¼ 0:
ð2:13Þ
Integrating this equation with respect to n we obtain the famous Riccati equation
u0 ðnÞ ¼
1 2 k C u ðnÞ þ uðnÞ ; 2km km km
ð2:14Þ
where C is the constant of integration. General solution of Eq. (2.14) is
uðnÞ ¼ k
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k2 þ 2C k2 þ 2C tanh ðn þ n0 Þ ; 2km
ð2:15Þ
and then a solution to Burgers equation (1.2) for arbitrary C; k and n0 is
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k2 þ 2C 2 uðx; tÞ ¼ k k þ 2C tanh ðx þ kt þ g0 Þ ; 2m
g0 ¼
n0 : k
ð2:16Þ
Observe that solution (2.9) is covered by solution (2.16). To see this, we take C ¼ 0; k ¼ ml. If k2 þ 2C < 0 we obtain the periodic solution given by
0qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðk2 þ 2CÞ 2 @ uðx; tÞ ¼ k ðk þ 2CÞ tan ðx þ kt þ n0 ÞA: 2m Following is a rational solution to Eq. (1.2) obtained from (2.14) by taking C ¼
uðx; tÞ ¼ k
ð2:17Þ k2 2
2m ; C 1 ¼ const: x þ kt þ C 1
ð2:18Þ
3. Transforming forced Burgers equation via the generalized Cole–Hopf transformation In this section we construct the generalized Cole–Hopf transformation from Eq. (1.2) to the standard heat equation. In order to look for solutions to forced Burgers equation (1.1), we apply the generalized Cole–Hopf transformation given by
uðx; tÞ ¼ f ðx; tÞ
/n ðn; sÞ þ gðx; tÞ; /ðn; sÞ
n ¼ nðx; tÞ; t ¼ tðsÞ;
ð3:1Þ
where f ðx; tÞ–0; gðx; tÞ; nðx; tÞ; /ðn; sÞ and sðtÞ are some functions to be determined later. We have:
nt /s þ st /sn /n nt /n þ st /s / f þ n ft þ g t ; 2 / / / ! 2 nx /nn nx /n / ux ¼ 2 f þ n fx þ g x ; / / / ! ! 2 3 2 2 2 nx /nnn 2nx /n nxx /nn 3n2x /n /nn nxx /n 2nx /nn 2nx /n / fx þ n fxx þ g xx : uxx ¼ þ 2 fþ þ 3 2 2 / / / / / / / / ut ¼
ð3:2Þ ð3:3Þ
ð3:4Þ
We substitute expressions of ut ; ux and uxx from (3.2), (3.3) and (3.4) in (1.1) to obtain following polynomial equation in the variables / and /n :
f nx ð2bnx af Þ/3n þ ðfx ðaf 2bnx Þ f ðagnx þ nt þ bnxx ÞÞ/2n / þ f ðaf nx /nn st /s 3bn2x /nn Þ/n / þ ðagfx þ afg x þ ft þ bfxx Þ/n /2 þ ð/nn ðnx ðafg þ 2bfx Þ þ f nt þ f bnxx Þ þ f st /ns þ bn2x /nnn Þ/2 þ ðg t þ agg x þ bg xx xGðtÞÞ/3 ¼ 0:
ð3:5Þ
22
A.H. Salas / Applied Mathematics and Computation 216 (2010) 18–26
Equating the coefficients of /3n ; /2n /; /n /; /n /2 ; /2 and /3 to zero and taking into account that f –0; n–0 and b–0 yields following system of pde’s
2bnx af ¼ 0;
ð3:6Þ
fx ðaf 2bnx Þ f ðagnx þ nt þ bnxx Þ ¼ 0;
ð3:7Þ
3bn2x /nn
af nx /nn st /s ¼ 0; agfx þ afg x þ ft þ bfxx ¼ 0; /nn ðnx ðafg þ 2bfx Þ þ f nt þ f bnxx Þ þ f st /ns þ bn2x /nnn ¼ 0; g t þ agg x þ bg xx xGðtÞ ¼ 0:
ð3:8Þ ð3:9Þ ð3:10Þ ð3:11Þ
From (3.6) and (3.7),
nx ¼
a 2b
nt ¼
f
a 2b
ð3:12Þ ðafg þ bfx Þ
ð3:13Þ
From (3.12) and (3.13), (3.8) and (3.9),
a2 /nn
f 2; 4b /s 1 g x ¼ ðft þ agfx þ bfxx Þ: af
st ¼
ð3:14Þ ð3:15Þ
From (3.12)–(3.15) and (3.10),
/ns /nnn 4b fx ¼ : /s /nn a f
ð3:16Þ
Observe that if f does not depend on x and /nn ¼ /s , then Eq. (3.16) holds . Thus, we shall assume that
f ¼ f ðtÞ and /s ¼ /nn :
ð3:17Þ
In particular, / satisfies the linear heat equation
/s /nn ¼ 0:
ð3:18Þ
Now, Eqs. (3.11), (3.15) and (3.17) give
8 < gx ¼ 1 a
f 0 ðtÞ ; f ðtÞ
ð3:19Þ
: g ¼ xGðtÞ þ g f 0 ðtÞ : t f ðtÞ Eliminating g from system (3.19) we obtain the Riccati equation
u0 ðtÞ ¼ u2 ðtÞ aGðtÞ where uðtÞ ¼
f 0 ðtÞ : f ðtÞ
ð3:20Þ
Solving this equation and using Eqs. (3.12)–(3.17) we find following particular solutions to system of Eqs. (3.6)–(3.11):
R 8 f ðtÞ ¼ expð uðtÞdtÞ; > > > > gðx; tÞ ¼ 1 xuðtÞ; < a R a x expð uðtÞdtÞ; nðx; tÞ ¼ > 2b > > > R : sðtÞ ¼ a4b2 expð uðtÞdtÞ:
ð3:21Þ
Finally, a solution to forced Burgers equation (1.1) is given by
8 R /n ðn;sÞ x > > > uðx; tÞ ¼ /ðn;sÞ expð uðtÞdtÞ a uðtÞ; > > R R < a x expð uðtÞdtÞ; s ¼ a2 expð uðtÞdtÞ; n ¼ 2b 4b > > where; > > > : /s ¼ /nn and u0 ðtÞ ¼ u2 ðtÞ aGðtÞ;
ð3:22Þ
where / ¼ /ðn; sÞ satisfies the linearheat Eq. (3.18) and uðtÞ is a solution to Riccati Eq. (3.20). As we can see, the problem of finding solutions to forced Burgers equation (1.1) consists on solving the linear heat equation and the Riccati Eq. (3.20). This last equation is solvable exactly for some particular choices of GðtÞ.
A.H. Salas / Applied Mathematics and Computation 216 (2010) 18–26
23
4. Exact solutions to linear heat equation Suppose that / ¼ /ðx; tÞ is a solution to equation
/t ¼ /xx :
ð4:1Þ
Then u ¼ /ðx; mtÞ is a solution to heat equation
/t ¼ m/xx :
ð4:2Þ
Conversely, if u ¼ uðx; tÞ is a solution to (4.2), then / ¼ uðx; t=mÞ is a solution to (4.1). Thus, Eqs. (4.2) and (4.2) are the same. We shall call Eq. (4.1) the normalized heat equation. We already obtained some solutions to Eq. (4.2) in a form of traveling wave. See Eqs. (2.5) and (2.6). In this section we give solutions to this equation by using Lie’s group theory [12]. Definition 4.1. Let I be a system of differential equations. A symmetry group of the system I is a local group of transformations G acting on an open subset M of the space of independent and dependent variables for the system with the property that whenever u ¼ f ðxÞ is a solution of I, and whenever g f is defined for g 2 G, then u ¼ g f ðxÞ is also a solution of the system. (By solution we mean any smooth solution u ¼ f ðxÞ defined on any subdomain X X). Following are symmetry groups of the heat Eq. (4.1):
G1
ðx þ e; t; uÞ;
G2
ðx; t þ e; uÞ;
G3
ðx; t; expðeÞuÞ;
G4
ðexpðeÞx; expð2eÞt; uÞ;
G5
ðx þ 2et; t; u expðex e2 tÞ; pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x t ex2 ; ; ; u 1 4et exp 1 4et 1 4et 1 4et
G6 Ga
ðx; t; u þ eaðx; tÞ:
This means that if / ¼ /ðx; tÞ is a solution of the heat Eq. (4.1), so are the functions
/ð1Þ ¼ /ðx e; tÞ; /ð2Þ ¼ /ðx; t eÞ; /ð3Þ ¼ expðeÞ/ðx; tÞ; /ð4Þ ¼ /ðexpðeÞx; expð2eÞtÞ; /ð5Þ ¼ expðex þ e2 tÞ/ðx 2et; tÞ; 1 ex2 x t / ; /ð6Þ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi exp ; 1 þ 4et 1 þ 4et 1 þ 4et 1 þ 4et /ð7Þ ¼ /ðx; tÞ þ eaðx; tÞ; where e is any real number and aðx; tÞ is any other solution to the heat equation. The symmetry groups G3 and Ga thus reflect the linearity of the heat equation; we can add solutions and multiply them by constants. The groups G1 and G2 demonstrate the time- and space-in variance of the equation, reflecting the fact that the heat equation has constant coefficients. The wellknown scaling symmetry turns up in G4 , while G5 represents a kind of Galilean boost to a moving coordinate frame. The last group G6 is a genuinely local group of transformations. Its appearance is far from obvious from basic physical principles, but it has the following nice consequence. If we let / ¼ c be just a constant solution, then we immediately conclude that the function
c ex2 ; / ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi exp 1 þ 4et 1 þ 4et
ð4:3Þ
pffiffiffiffiffiffiffiffi is also a solution. In particular, if we set c ¼ e=p we obtain the fundamental solution to the heat equation at the point ðx0 ; y0 Þ ¼ ð0; 1=ð4eÞÞ. To obtain the fundamental solution
2 1 x ; / ¼ pffiffiffiffiffiffiffiffi exp 4t 4pt
ð4:4Þ
we need to translate this solution in t using the group G2 (with e replaced by 1=ð4eÞ). It can be shown that the most general solution obtainable from a given solution u ¼ uðx; tÞ by group transformations is of the form
1 /ðx; tÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi exp 1 þ 4e6 t
e3
e5 x þ e6 x2 e5 t 1 þ 4e6 t
ð4:5Þ
24
A.H. Salas / Applied Mathematics and Computation 216 (2010) 18–26
expðe4 Þðx 2e5 tÞ expð2e4 Þt e1 ; e2 þ aðx; tÞ; 1 þ 4e6 t 1 þ 4e6 t
u
ð4:6Þ
where e1 ; . . . ; e6 are real constants and a ¼ aðx; tÞ an arbitrary solution to the heat equation. Previous considerations allow to obtain particular solutions to the heat equation /t ¼ m/xx . Some of them are :
wðxÞ ¼ k ¼ const:;
ð4:7Þ
wðxÞ ¼ Ax þ B;
ð4:8Þ
wðx; tÞ ¼ Aðx2 þ 2mtÞ þ B;
ð4:9Þ
3
wðx; tÞ ¼ Aðx þ 6mtxÞ þ B;
ð4:10Þ
wðx; tÞ ¼ Aðx4 þ 12mtx2 þ 12m2 t2 Þ þ B; n X ð2nÞð2n 1Þ . . . ð2n 2k þ 1Þ ðmtÞk x2n2k ; wðx; tÞ ¼ x2n þ k! k¼1
ð4:11Þ
wðx; tÞ ¼ x2nþ1 þ
n X ð2n þ 1Þð2nÞ . . . ð2n 2k þ 2Þ ðmtÞk x2n2kþ1 ; k! k¼1
wðx; tÞ ¼ A expðml2 t lxÞ þ B; 1 x2 wðx; tÞ ¼ A pffiffi exp þ B; 4mt t
ð4:12Þ ð4:13Þ ð4:14Þ ð4:15Þ
wðx; tÞ ¼ A expðml2 tÞ cosðlx þ BÞ;
ð4:16Þ
wðx; tÞ ¼ A expðml2 tÞ cosðlx þ BÞ þ C;
ð4:17Þ
wðx; tÞ ¼ A expðlxÞ cosðlx 2ml2 t þ BÞ þ K; x wðx; tÞ ¼ A erf pffiffiffiffiffi þ B; 2 mt
ð4:18Þ
where A; B; C, and
ð4:19Þ
l are arbitrary constants, n is a positive integer, Z
2 erfðzÞ pffiffiffiffi
p
z
expðn2 Þ dn;
ð4:20Þ
0
is the error function (probability integral). These solutions are useful in solving forced Burgers equation (1.1) by the formula (3.22). 5. Analytic solutions to forced Burgers equation Suppose we have solved the Riccati Eq. (3.20). Then, making use of (3.22) together with (4.7)–(4.20) (letting replacing x with n and t with s), following are solutions to forced Burgers equation (1.1):
x uðx; tÞ ¼ uðtÞ;
a
uðx; tÞ ¼ uðx; tÞ ¼ uðx; tÞ ¼
A exp An þ B 2An
Z
Aðn2 þ 2sÞ þ B 3Aðn2 þ 2sÞ
ð5:1Þ x
uðtÞdt uðtÞ; a Z
exp
exp Anðn þ 6sÞ þ B 3 4Aðn þ 6nsÞ
Z
2
ð5:2Þ x
uðtÞdt uðtÞ; a
x
uðtÞdt uðtÞ; a Z
4
2
2Þ
ð5:3Þ ð5:4Þ
x uðtÞdt uðtÞ; exp a Aðn Z þ 12n s þ 12s þ B Bl x uðx; tÞ ¼ l uðtÞdt uðtÞ; exp A expð l ð ls þ nÞÞ þ B a Z n x uðx; tÞ ¼ exp uðtÞdt uðtÞ; 2s a Z Al sinðB þ lnÞ x u ðtÞdt uðtÞ; uðx; tÞ ¼ exp A cosðB þ lnÞ þ K expðl2 sÞ a Z AlðsinðB þ lðn 2lsÞÞ þ cosðB þ lðn 2lsÞÞÞ x exp uðtÞdt uðtÞ; uðx; tÞ ¼ A cosðB þ lðn 2lsÞÞ þ K expðlnÞ a n2 Z Ae4s x
exp uðtÞdt uðtÞ: uðx; tÞ ¼ pffiffiffiffiffiffi n ffiffi a p ps Aerf 2 s þ B uðx; tÞ ¼
m ¼ 1 and
ð5:5Þ ð5:6Þ ð5:7Þ ð5:8Þ ð5:9Þ ð5:10Þ
A.H. Salas / Applied Mathematics and Computation 216 (2010) 18–26
25
In formulas (5.1)–(5.10),
n¼
a 2b
x exp
Z
uðtÞdt
s¼
and
a2 4b
exp
Z
uðtÞdt :
ð5:11Þ
In particular, if GðtÞ 0, the ’forced’ Burgers equation (1.1) takes the form
ut þ auux þ buxx ¼ 0:
ð5:12Þ
This is the generalized Burgers equation (2.1). We gave its traveling wave solutions in Section 2. We derive here other exact solutions to this equation. Observe that Eq. (3.20) converts to
u0 ðtÞ ¼ u2 ðtÞ;
ð5:13Þ
whose general solution is
uðtÞ ¼
1 ; C ¼ const: Ct
ð5:14Þ
From (3.20), (3.21), (3.22) and (5.14),
8 1 > f ðtÞ ¼ Ct ; > > > x > < gðx; tÞ ¼ aðCtÞ ;
ð5:15Þ
ax ; nðx; tÞ ¼ 2bðCtÞ > > > > > : sðtÞ ¼ a2 : 4bðCtÞ
Making use of (5.1)–(5.10) and taking into account (5.11) and (5.15), following are solutions to Eq. (5.12) :
u1 ðx; tÞ ¼
x
aðC tÞ
ð5:16Þ
;
2bðC tÞðAa BxÞ Aax2 ; aðC tÞðAax þ 2BbðC tÞÞ
x Aa2 6bðt CÞ þ x2 þ 4Bb2 ðC tÞ2
; u3 ðx; tÞ ¼ aðC tÞ Aa2 ð2bðt CÞ þ x2 Þ þ 4Bb2 ðC tÞ2
Aa3 12bx2 ðt CÞ þ 12b2 ðC tÞ2 þ x4 8Bb3 xðC tÞ3
; u4 ðx; tÞ ¼ aðC tÞ Aa3 xð6bðt CÞ þ x2 Þ þ 8Bb3 ðC tÞ3
x Aa4 20bx2 ðt CÞ þ 60b2 ðC tÞ2 þ x4 þ 16Bb4 ðC tÞ4
u5 ðx; tÞ ¼ aðC tÞ Aa4 12bx2 ðt CÞ þ 12b2 ðC tÞ2 þ x4 þ 16Bb4 ðC tÞ4 8 9 1 < Al x=
: u6 ðx; tÞ ¼ C t :A þ B exp alðal2xÞ a; u2 ðx; tÞ ¼
ð5:17Þ
ð5:18Þ
ð5:19Þ
ð5:20Þ
4bðCtÞ
In particular, if B ¼ 0 and A–0,
u7 ðx; tÞ ¼
x al ; aðt CÞ
ð5:21Þ
3=2 a2 Bb bðtCÞ x
qffiffiffiffiffiffiffiffiffiffi ; u8 ðx; tÞ ¼ 1 a2 3 2 a 2A exp 4bðCtÞ x þ B bðtCÞ
2 2 a l al al A exp 4bðCtÞ al sin 2bðCtÞ x þ B þ x cos 2bðCtÞ x þ B þ Cx 2 2
u9 ðx; tÞ ¼ ; a l al cos 2bðCtÞ aðC tÞ A exp 4bðCtÞ xþB þC
8 9 al al 1
; þ u10 ðx; tÞ ¼ al al C t: a; ðx þ alÞ þ B þ K exp 2bðCtÞ x A cos 2bðCtÞ
8 9 al al 1
: u11 ðx; tÞ ¼ þ al al C t: a; ðx þ alÞ þ B þ K exp x A cos 2bðCtÞ
In particular, if K ¼ 0 we get a periodic solution :
2bðCtÞ
ð5:22Þ
ð5:23Þ
ð5:24Þ
ð5:25Þ
26
A.H. Salas / Applied Mathematics and Computation 216 (2010) 18–26
u12 ðx; tÞ ¼
1 xal : x þ al tan B þ aðC tÞ 2bðC tÞ
pffiffiffiffiffiffiffi We obtain from (5.26) another solution by replacing B with B 1 and
1 xa l : u13 ðx; tÞ ¼ x al tanh B þ aðC tÞ 2bðC tÞ 8
9 > > pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > 1 2 > < 2 bðt CÞ exp 4bðCtÞ x = 1 : pffiffiffiffi u14 ðx; tÞ ¼ xþ > aðC tÞ > p > 1 ; : erf pffiffiffiffiffiffiffiffiffiffi x > 2
ð5:26Þ pffiffiffiffiffiffiffi
l with l 1 : ð5:27Þ
ð5:28Þ
bðtCÞ
We may obtain other exact solutions to Burgers equation (1.2) by using (1.9) and 4.7, (4.8)–(4.20). Some other works concerning the problem of finding exact solutions to nonlinear pde’s may be found in Ref. [13–16] 6. Conclusions We successfully applied the generalized Cole–Hopf transformation to forced Burgers equation. As a particular case, we obtained some solutions to Burgers equation. We also derived the general solution of the Burgers equation in the form of a traveling wave. We think that the application of the generalized Cole–Hopf transformation is a useful tool in the search of solutions to other forced equations. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16]
S. Eule, R. Friedrich, A note on the forced Burgers equation, Physics Letters A 351 (2006) 238–241. M. Scott, Encyclopedia of Nonlinear Science, Taylor and Francis, 2005. H. Bateman, Some recent research on the motion of fluids, Monthly Weather Review 43 (1915) 163–170. J. Burgers, Application of a model system to illustrate some points of the statistical theory of free turbulence, Proceedings of the Nederlandse Akademie van Wetenschappen 43 (1940) 2–12. J. Burgers, A mathematical model illustrating the theory of turbulence, Advances in Applied Mechanics 1 (1948) 171–199. E. Hopf, The partial differential equation ut þ uux ¼ luxx , Communications in Pure and Applied Mathematics 3 (1950) 201–230. J. Cole, On a quasilinear parabolic equation occurring in aerodynamics, Quarterly Journal of Applied Mathematics 9 (1951) 225–236. K.M. Case, S.C. Chiu, Burgers turbulence models, Physics of Fluids 12 (1969) 1799–1808. J. Burgers, The Nonlinear Diffusion Equation: Asymptotic Solutions and Statistical Problems, Reidel, Dordrecht and Boston, 1974. A.R. Forsyth, Theory of Differential Equations, Cambridge University Press, Cambridge, 1906. F. Calogero, S. De Lillo, The Burgers equation on the semiline, Inverse Problems 5 (1989) L37. P. Olver, Applications of Lie Groups to Differential Equations, second ed., Springer-Verlag, 1993. A. Salas, Some solutions for a type of generalized Sawada–Kotera equation, Applied Mathematics and Computation 196 (2008) 812–817. A.H. Salas, C.A. Gómez, Computing exact solutions for some fifth KdV equations with forcing term, Applied Mathematics and Computation 204 (2008) 257–260. A.H. Salas, Exact solutions for the general fifth KdV equation by the exp function method, Applied Mathematics and Computation 205 (2008) 291–297. A.H. Salas, C.A. Gómez, Exact solutions for a third KdV equation with variable coefficients and forcing term, Mathematical Problems in Engineering, Hindawi, in press.