Exponential Self Similar Solutions. Equations Invariant Under Combined Translation and Scaling

S.3.4. Exponential Self Similar Solutions. Equations Invariant Under Combined Translation and Scaling

By definition, an exponential self-similar solution is a solution of the form

(13) An exponential self-similar solution exists if the equation under consideration is invariant under

w(x, t) = e αt

V (ξ),

ξ = xe βt .

the transformation t = ¯t + ln C,

C > 0 is an arbitrary constant, (14) for some k and m. Transformation (14) is a combination of a shift in t and scaling in x and w.

Observe that these transformations contain an arbitrary constant

C as a parameter. In practice, the above existence criterion is checked: if a pair of k and m in (14) has been found such that the equation remains the same, then there exists an exponential self-similar solution with the new variables having the form (13), where

(15) These relations follow from the condition that the scaling transformation (14) must preserve the

α = m,

β = −k.

form of the variables of (13):

w=e αt

V (ξ), ξ = xe βt

w=e ¯ =⇒ α ¯t V(¯ ξ), ¯ ξ = ¯xe β ¯t .

Remark. Solutions of the form (13) are sometimes called limit self-similar solutions. Example 5. Let us show that the nonlinear heat equation

∂w =

∂w

∂t

∂x

∂x

Invariant solutions found by using combining translations

1 , and C, C C 2 are arbitrary constants) No.

and scaling (

Form of solution

Invariant transformation

Relations for coefficients

1 w = U (z), z = αx+βy t = ¯t+C 1 , x = ¯x+C 2 α and β are arbitrary constants

2 w=t α U (z), z = xt β

t = C ¯t, x = C k x, w = C ¯ m w ¯

α = m, β = −k

3 w=e αt U (z), z = xe βt

t = ¯t+ln C, x = C k x, w = C ¯ m w ¯

α = m, β = −k

4 w=t α U (z), z = x+β ln t t = C ¯t, x= ¯x+k ln C, w = C m w ¯ α = m, β = −k

admits an exponential self-similar solution. Substituting (14) into (16) yields

m ∂¯ w =

mn+m−2k ∂

Equating the exponents of C, we obtain one linear equation, m = mn + m − 2k. Hence, we have k = 1 2 mn, where m is arbitrary. Further, using formulas (13) and (15) and taking (without loss of generality) m = 2, which is equivalent to scaling of time t, we find the new variables:

Inserting these into (16), we obtain an ordinary differential equation for the function

Example 6. With this method, it can be shown that equation (12) also admits an exponential self-similar solution of the form (17).

Table 16 lists invariant solutions which can be found by combining translation and scaling of the independent variables and scaling of the dependent variable. Apart from traveling-wave (row 1), self-similar (row 2), and exponential self-similar (row 3) solutions considered above, the last row in the table describes another invariant solution. Below we give an example that illustrates the method for the construction of such a solution.

Example 7. Let us show that the nonlinear heat equation (16) admits a solution having the form specified in the fourth row of Table 16 . To that end, we use the transformation

t = C¯t,

x = ¯x + k ln C,

w=C m w ¯

to obtain

m−1 ∂¯ w

C = aC mn+m ∂

Equating the powers of C yields one linear equation, m − 1 = mn + m. Hence, we find that m = −1/n and k may be arbitrary. Therefore (see row 4 in Table 16), equation (16) has a solution of the form

w=t −1 /n U (z),

z = x + β ln t,

where β is arbitrary.

Substituting (18) into (16), we arrive at the autonomous differential equation

a(U n U z ′ ) ′ z − βU z ′ + 1 U = 0. n

The value β = 0 corresponds to an additively separable solution.

The examples considered in Section S.3 show that the construction of exact solutions by means of reducing the dimension of a partial differential equation is possible, provided that the equation in question is invariant under certain transformations (containing one or more arbitrary parameters) or, in other words, the equation possesses a certain symmetry. Below, in Section S.7, we describe a more general approach to the construction of exact solutions. This approach is based on the methods of group-theoretic analysis of differential equations. These methods provide a regular procedure for obtaining invariant solutions of an analogous or more complex structure. ✟✂✠

References for Section S.3: P. W. Bridgman (1931), W. F. Ames (1972), G. W. Bluman and J. D. Cole (1974), G. I. Baren- blatt and Ya. B. Zel’dovich (1972), W. F. Ames, R. J. Lohner, and E. Adams (1981), L. Dresner (1983), G. I. Barenblatt (1989), L. I. Sedov (1993).