SIAM REV. Vol. 40, No. 1, pp. 96–101, March 1998
c 1998 Society for Industrial and Applied Mathematics
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SIMILARITY TRANSFORMATIONS FOR PARTIAL DIFFERENTIAL EQUATIONS∗ MEHMET PAKDEMIRLI† AND MUHAMMET YURUSOY† Abstract. The importance of similarity transformations and their applications to partial differential equations is discussed. The theory has been presented in a simple manner so that it would be beneficial at the undergraduate level. Special group transformations useful for producing similarity solutions are investigated. Scaling, translation, and the spiral group of transformations are applied to well-known problems in mathematical physics, such as the boundary layer equations, the wave equation, and the heat conduction equation. Finally, a new transformation including the mentioned transformations as its special cases is also proposed. Key words. similarity transformations, partial differential equations, mechanics of solids and fluids AMS subject classifications. 35Q35, 35Q72, 76D10, 35K05, 35L05 PII. S003614459631001X
1. Introduction. In many undergraduate courses on the applications of mathematics, a student is taught how to solve certain well-known problems in mathematical modelling such as the wave equation, the heat equation, etc. Some techniques for finding solutions are presented, usually specific to the problem under consideration. When the student is faced with a different type of problem, he or she realizes that the knowledge of a general treatment of these problems is lacking. When the equations are nonlinear, the student cannot use any of the techniques taught in the courses. General methods for finding exact solutions of linear and nonlinear partial differential equations indeed exist. The classical approach is the Lie group theory, which is discussed in detail in [1, 2]. This method of employing special transformations for finding exact solutions of differential equations was invented by S. Lie approximately one century ago. It is only in the last half of this century, however, that the power of these methods is realized and used widely. An alternative approach using the so-called exterior calculus was proposed in [3] a few decades ago. The detailed presentation of the theory with applications to physical problems can be found in [4]. Although the methods presented in [1–4] are very powerful, they are too complicated to be presented at the undergraduate level. Yet, simplified versions of them may prove to be quite useful and accessible without too much complicated theory. These simplified techniques can be added to certain courses, e.g., differential equations, mechanics. With this understanding, we will present here some simplified techniques which work successfully for a large number of problems. Instead of employing general Lie group transformations, we will use special forms of them such as scaling, translation, and the spiral group. The first two work for most of the equations, and the first one, the scaling transformation, produces nontrivial solutions in the majority of cases. Lie group transformations or their special forms can be used in two different ways: they can be used to produce a new solution from a known solution or they can be ∗ Received by the editors June 22, 1996; accepted for publication (in revised form) August 23, 1996. http://www.siam.org/journals/sirev/40-1/31001.html † Department of Mechanical Engineering, Celal Bayar University, TR-45140, Muradiye, Manisa, Turkey (
[email protected]).
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used to find the so-called similarity solutions (group-invariant solutions). We will concentrate on the latter type of solutions in this article. Similarity transformations are the transformations by which an n-independent variable partial differential system can be converted to a system with n − 1 independent variables. The situation is best when n = 2, since one deals with an ordinary differential equation instead of a partial differential equation. The above mentioned transformations are some of the most effective tools in producing similarity transformations. Of course, these specific transformations will not in general give all of the similarity transformations but may provide a useful partial list. We will apply these special transformations to a couple of basic equations in mathematical physics, namely, the wave equation in mechanics and the boundary layer equation in fluid mechanics. At the end, we will combine all three transformations into one transformation and treat a nonlinear heat equation problem. For further applications of these special transformations, see references [5–9]. 2. Scaling transformation. The scaling transformation is one of the most common. We apply it to the well-known boundary layer equations in fluid mechanics. The equations were derived at the beginning of the century by Prandtl, opening a new era in fluid mechanics. They are asymptotic approximations of the Navier–Stokes equations in the vicinity of the surface. The equations greatly reduce the difficulty of solving the original Navier–Stokes equations. For a two-dimensional incompressible laminar fluid flow, the equations are ∂u ∂v + = 0, ∂x ∂y (1)
u
∂u ∂2u dU ∂u +v = , +U ∂x ∂y ∂y 2 dx
u(x, 0) = 0,
v(x, 0) = 0,
u(x, ∞) = U (x).
Prandtl himself found a similarity transformation for the above equations by employing ad hoc methods. We will show that the equations admit scaling symmetry and reduce the equations to ordinary differential equations in a more systematic way. The scaling transformation for the equations can be written as (2)
x∗ = ea x,
y ∗ = eb y,
u∗ = ec u,
v ∗ = ed v,
U ∗ = ee U.
The above transformation is actually a point transformation from the coordinates (x, y, u, v, U ) to the new coordinates (x∗ , y ∗ , u∗ , v ∗ , U ∗ ) [2]. The transformation parameter is . a, b, c, d, e are arbitrary parameters to be determined by the invariance condition on the equations. Substituting (2) in (1), dividing by the coefficients of the leading terms in each equation, we have the equations in terms of the new variables ∂v ∗ ∂u∗ + e(b+c−a−d) ∗ = 0, ∗ ∂x ∂y (3)
u∗
∗ 2 ∗ ∗ ∂u∗ (b+c−a−d) ∗ ∂u (2b+c−a) ∂ u 2(c−e) ∗ dU + e v = e + e U , ∂x∗ ∂y ∗ ∂y ∗2 dx∗
u∗ (x∗ , 0) = 0,
v ∗ (x∗ , 0) = 0,
u∗ (x∗ , ∞) = e(c−e) U ∗ (x∗ ).
For equations (1), to admit scaling transformation, the transformed equations (3) should have exactly the same form as the original equation. This immediately leads
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to the following invariance conditions: (4)
b + c − a − d = 0,
2b + c − a = 0,
c − e = 0.
Solving (4) in terms of the parameters a and c, we have b = (a − c)/2,
(5)
d = (c − a)/2,
e = c.
Therefore, the scaling transformation admitted by the boundary layer equations has the following form: (6)
x∗ = ea x,
y ∗ = e(a−c)/2 y,
u∗ = ec u,
v ∗ = e(c−a)/2 v,
U ∗ = ec U.
We have two arbitrary parameters a and c, each corresponding to a different transformation. This gives flexibility in assigning specific values to the parameters. Expanding the exponentials in (6) in a Taylor series, keeping terms up to order , denoting the differences in transformed and original variables as differentials and finally solving for and equating each term, we obtain the following equivalent differential system: dy du dv dU dx = = = = . ax by cu dv eU
(7)
Equation (7) will give us the similarity variable and functions. Without loss of generality, we assume that c = ma, where m is another parameter. Using this new parameter m and equations (5), we rewrite equation (7) as follows: (8)
dy du dv dU dx = = = = . x ((1 − m)/2)y mu ((m − 1)/2)v mU
Using the method of characteristics for solving (8), we obtain the similarity variable and functions (9)
ξ = yx(m−1)/2 ,
u = xm f (ξ),
v = x(m−1)/2 g(ξ),
U = kxm ,
where k is a constant. Substituting (9) into the original equations (1), we reduce the partial differential system to an ordinary differential system mf + ((m − 1)/2)ξf 0 + g 0 = 0, (10)
mf 2 + ((m − 1)/2)ξf f 0 + gf 0 = f 00 + mk 2 , f (0) = 0,
g(0) = 0,
f (∞) = k.
If there had been anything wrong in the similarity transformation, the equations could not be expressed completely in terms of the similarity variables, hence providing a check for the calculations. The dependent variable g can be eliminated between the equations, yielding a single equation for f . Solving for f requires numerical techniques, since the equations are highly nonlinear. In this case, we are unable to find closed form solutions. However, a numerical treatment of (10) is easier compared to the original equations (1). We note that most textbooks on fluid mechanics present ad hoc methods for finding the similarity transformations. In contrast, a more systematic way is presented here which can be applied to any problem admitting scaling symmetry.
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3. Translation transformation. Most equations admit translations in one or more of their variables. These transformations, however, usually lead to trivial solutions when used alone. If they are combined with other transformations, useful results can be obtained. In this section, we will apply this transformation to the nondimensional wave equation and produce the well-known moving wave solution. For a single spatial coordinate, the linear wave equation is ∂2u ∂2u = . ∂t2 ∂x2
(11)
The translation transformation for (11) is x∗ = x + a,
(12)
t∗ = t + b,
u∗ = u + c,
where is the transformation parameter and a, b, and c are to be determined from the equation. Substituting (12) into (11), we see that no restrictions are imposed on the parameters a, b, and c. Therefore the equation allows arbitrary translations in all coordinates. For transformation (12), the equivalent differential system is dt du dx = = . a b c
(13)
Choosing c = 0 and a = mb and solving (13) by the method of characteristics, we have the following similarity variable and function: η = x − mt,
(14)
u = F (η).
Substituting into (11), we obtain F 00 (m2 − 1) = 0.
(15)
We now have two choices; either F 00 = 0 or m = ±1. The first choice leads to F = c1 η + c2
(16)
or
u = k1 x + k2 t + k3 ,
which is the trivial solution that can be found directly by examining (11). For m = ±1, we obtain the well-known solution u = F1 (x − t) + F2 (x + t),
(17)
where F1 represents the wave travelling to the right and F2 represents the wave travelling to the left. We didn’t consider any specific boundary conditions for the analysis and hence, the solutions appear in their general form. 4. Spiral group transformation. In this section, we again treat the boundary layer equations given in (1). The spiral group of transformations for these equations are (18)
x∗ = x + a,
y ∗ = eb y,
u∗ = ec u,
v ∗ = ed v,
U ∗ = ee U.
Substituting (18) into (1) and requiring that the equations remain invariant under the transformation yields (19)
b + c − d = 0,
2b + c = 0,
c − e = 0.
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There is no restriction on parameter a. Without loss of generality, we again assume that c = ma. Solving for all parameters in terms of a yields b = −(m/2)a, c = ma, d = (m/2)a, and e = ma. The equations determining the similarity transformations are dy du dv dU dx = = = = . 1 −(m/2)y mu (m/2)v mU
(20)
The similarity variable and functions are (21)
u = emx p(ζ),
ζ = ye(m/2)x ,
v = e(m/2)x q(ζ),
U = kemx .
Substituting (21) into (1), we have mp + (m/2)ζp0 + q 0 = 0, (22)
mp2 + (m/2)ζpp0 + qp0 = p00 + mk 2 , p(0) = 0,
q(0) = 0,
p(∞) = k.
These equations represent another similarity solution for the boundary layer equations. For applications of this transformation to more involved problems, see references [6, 8]. 5. A more general transformation. We propose, in this section, a more general transformation including all the previous ones as special cases. We apply the transformation to the following nonlinear heat conduction equation: ∂ ∂u mu ∂u = e , (23) ∂t ∂x ∂x where u is the temperature distribution and m is a constant. The transformation for the equation is (24)
x∗ = ea x + b,
t∗ = ec t + d,
u∗ = ee u + f.
For b = d = f = 0, the transformation reduces to the scaling transformation; for a = c = e = 0, it reduces to the translation transformation; and finally, for a = d = f = 0, it reduces to the spiral group transformation. Substituting (24) into (23) and requiring the invariance condition yields (25)
e = 0,
2a − c − mf = 0.
Solving for f in terms of the other parameters, we see that the transformation has four arbitrary parameters: (26)
x∗ = ea x + b,
t∗ = ec t + d,
u∗ = u + ((2a − c)/m).
Actually, transformation (26) represents the full group for this heat equation [1]. For a special choice of a = c = m and b = d = 0, we have dt du dx = = . mx mt 1
(27)
Solving (27), we find the following similarity variable and function: (28)
µ = x/t,
u = (1/m) log t + r(µ).
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Substitution of the results into the original equation reduces it to an ordinary differential equation (29)
m(emr r0 )0 + mµr0 = 1.
The above equation is easier to handle compared to the original partial differential equation. Note that we did not consider any boundary conditions for the heat equation. Imposing boundary conditions may further restrict the transformation (26). In all the examples, we treated partial differential equations with two independent variables. For equations having more than two independent variables, similarity transformations can be applied successively to achieve final reductions to ordinary differential equations. An example with three independent variables can be found in [9]. REFERENCES [1] G. W. BLUMAN AND S. KUMEI, Symmetries and Differential Equations, Springer-Verlag, New York, 1989. [2] H. STEPHANI, Differential Equations: Their Solutions Using Symmetries, Cambridge University Press, Cambridge, 1989. [3] B. K. HARRISON AND F. B. ESTABROOK, Geometric approach to invariance groups and solution of partial differential equations, J. Math. Phys., 12 (1971), pp. 653–666. [4] D. G. B. EDELEN, Applied Exterior Calculus, Wiley-Interscience, New York, 1985. [5] A. G. HANSEN AND T. H. NA, Similarity solutions of laminar, incompressible boundary layer equations of non-Newtonian fluids, ASME J. Basic Engrg., 90 (1968), pp. 71–74. [6] M. G. TIMOL AND N. L. KALTHIA, Similarity solutions of three dimensional boundary layer equations of non-Newtonian fluids, Internat. J. Non-Linear Mech., 21 (1986), pp. 475–481. [7] M. PAKDEMIRLI AND E. S. SUHUBI, Similarity solutions of boundary layer equations for second order fluids, Internat. J. Engrg. Sci., 30 (1992), pp. 611–629. [8] M. PAKDEMIRLI, Similarity analysis of boundary layer equations of a class of non-Newtonian fluids, Internat. J. Non-Linear Mech., 29 (1994), pp. 187–196. [9] M. YURUSOY AND M. PAKDEMIRLI, Symmetry reductions of unsteady three-dimensional boundary layers of some non-Newtonian fluids, Internat. J. Engrg. Sci., 35 (1997), pp. 731–740.
