Hodograph Transformation

S.2.2. Hodograph Transformation

In some cases, nonlinear equations and systems of partial differential equations can be simplified by means of the hodograph transformation.

1 ◦ . For an equation with two independent variables x, t and an unknown function w = w(x, t), the hodograph transformation consists in representing the solution in implicit form

(4) or t = t(x, w). Thus, t and w are treated as independent variables, while x is taken to be the dependent

x = x(t, w)

variable. The hodograph transformation (4) does not change the order of the equation and belongs to the class of point transformations (equivalently, it can be represented as x=e w, t = e t, w = e x).

2 ◦ . For a system of two equations with two independent variables x, y and two dependent variables w = w(x, y), v = v(x, y), the hodograph transformation implies that w, v are treated as the independent variables and x, y as the dependent variables. In other words, one looks for a solution in the form

(5) The hodograph transformation is used in gas dynamics and the theory of jets for the linearization

x = x(w, v),

y = y(w, v).

of equations and finding solutions of certain boundary value problems. Below we consider some applications of the hodograph transformation to solving specific equations of mathematical physics.

Example 3. Consider the nonlinear second-order equation

Let us seek its solution in implicit form. Differentiating relation (4) with respect to both variables as an implicit function and taking into account that w = w(x, t), we get

1= x w w x

(differentiation in x),

0= x w w t + x t

(differentiation in t),

x 2 ww w x + x w w xx (double differentiation in x),

where the subscripts indicate the corresponding partial derivatives. We solve these relations to express the “old” derivatives through the “new” ones,

w t =− t ,

w w xx =− x x ww =− x ww .

Substituting these expressions into (6), we obtain the following second-order linear equation:

Example 4. Let us represent the equation

as the following system of equations:

We now take advantage of the hodograph transformation (5), which amounts to taking w, v as the independent variables and x, y as dependent variables. Differentiating each relation in (5) with respect to x and y (as composite functions) and eliminating the partial derivatives x w , x v , y w , y v from the resulting relations, we obtain

∂w , where J= ∂w ∂v − ∂w ∂v .

Using (9) to eliminate the derivatives w x , w y , v x , v y from (8), we arrive at the system

∂y = ∂x , −

∂x

∂y

f (w)

∂v

∂w

∂v

∂w ∂w

Similarly, from system (10), we obtain another linear equation for the function y = y(w, v),

Given a particular solution x = x(w, v) of equation (11), we substitute this solution into system (10) and find y = y(w, v) by straightforward integration. Eliminating v from (5), we obtain an exact solution w = w(x, y) of the nonlinear equation (7).

1 ◦ . Equation (11) with an arbitrary

f (w) admits a simple particular solution, namely,

x=C 1 wv + C 2 w+C 3 v+C 4 ,

where C 1 , ...,C 4 are arbitrary constants. Substituting this solution into system (10), we obtain

= −( C 1 w+C 3 ) f (w).

Integrating the first equation in (14) yields

2 C 1 v 2 + C 2 v + ϕ(w). Substituting this solution into the second equation in (14), we find the function ϕ(w), and consequently

Formulas (13) and (15) define an exact solution of equation (7) in parametric form ( v is the parameter). 2 ◦ . In a similar way, one can construct a more complex solution of equation (7) in parametric form,

3 ◦ . Using a particular solution of equation (12), we obtain another exact solution of equation (7):

1 2 Z x=− 2 C 1 v − C 2 v+C 1 F (w) dw + C 3 w+C 4 ,

f (w) dw. See also 5.4.4.8 for a more general equation and some other solutions.

Example 5. Consider the system of gas dynamic type equations

Treating w, v as the independent variables and x, y as the dependent ones, we arrive at the following system of linear equations (the calculations are similar to those of Example 4):

References for Subsection S.2.2: N. E. Kochin, I. A. Kibel’, and N. V. Roze (1963), B. L. Rozhdestvenskii and N. N. Yanenko (1983), A. M. Siddiqui, P. N. Kaloni, and O. P. Chandna (1985), G. G. Chernyi (1988), R. Courant and D. Hilbert (1989), P. A. Clarkson, A. S. Fokas, and M. J. Ablowitz (1989), V. F. Zaitsev and A. D. Polyanin (2001 b).