6.2 Significance of Second order PDE

Many physical problems from fluid mechanics, heat transfer, rigid body dynamics and elasticity are modelled by second order PDE's. In some problems fourth order PDE's do arise, however, as we split higher order ordinary differential equations into system of first order equations, it is also a common practice to split a 4th order PDE into two second order PDE 's along with the necessary boundary and initial conditions and solve them together.

Hence in solution of PDE's, understanding the methods to solve second order PDE is very important to solve real world problems.

Classification of Second order PDE : Consider the following second order equation in two independent variables

.................(6.2.1)

where A, B, C, D, E & F are all functions of x, y, u, and only so that the equation is atmost quasi-linear. Then the equation (6.2.1) is elliptic, parabolic or hyperbolic depending on the discriminant, B² - 4AC which is negative, zero or positive, respectively.

Canonical forms: Consider the general transformation of the independent variables x and y to x and h where

x = x(x,y) and h = h(x,y)

.................(6.2.2)

such that x and h are continuosly differentiable with the Jacobian

.................(6.2.3)

  in some domain .

Now using the definition of partial derivatives one can write

.................(6.2.4)

substituting 6.2.4 in 6.2.1 gives

.................(6.2.3)

  Canonical form for hyperbolic equations : Consider

Since the Jacobian

is positive and also B² - 4AC for hyperbolic equations, is positive, that shows that canonical forms of PDE retain their type.

If

If we choose

and

Now the condition B² - 4AC > 0 implies that the slope l of the curve. x(x,y) = C₁ and h(x,y) = C₂ are real. That is, at any point (x,y) there exists two real directions given by the two roots along which the PDE takes the canonical form. These directions are called characteristic equations or directions.