Chapter 7
Partial Differential Equations
7.1 Classification of second-order linear equations
The general PDE of order two in two variables can be written as
Auxx + Buxy + Cuyy + Dux + Euy + Fu = G (7.1-1)
Where A, B, C, D, E, F, and G can be functions of x and y. If G = 0, the PDE is homogeneous, examples of homogeneous PDEs are
+ = 0
- c2 + 2B + Au = 0
If G = 0, the PDE is non-homogeneous, examples of non-homogeneous PDEs are
+ = f(x, y)
- c2 + B + Au = 2
The same concepts of linearity and homogeneity apply to the initial and boundary conditions. For example
u(x, 0) + ux(x, 0) = 0 linear, homogeneous
u(x, 0) + ux(x, 0) = 1 linear, non-homogeneous
u(x, 0) + ux2(x, 0) = 1 nonlinear, non-homogeneous
The PDE (7.1-1) might be transformed into two ODEs. Eq. (7.1-1) can be rewritten as
Auxx + Buxy + Cuyy + e = 0 (7.1-2)
where e = Dux + Euy + Fu - G
Let p = ux and q = uy then
dp = dx + dy = uxxdx + uxydy (7.1-3a)
dq = dx + dy = uxydx + uyydy (7.1-3b)
The PDE (7.1-2) in terms of x and y can be transformed into two ODEs in terms of p and q. This can be accomplished by solving for uxx and uyy in terms of dp, dq, dx, dy, and uxy.
uxx = - uxy (7.1-4a)
uyy = - uxy (7.1-4b)
uxx and uyy are then substituted into Eq. (7.1-2) [ Auxx + 2Buxy + Cuyy + e = 0 ]
A{- uxy} + Buxy + C{- uxy} + e = 0 (7.1-5)
Eq. (7.1-5) is then rearranged by collecting the terms with uxy
{- A+ B - C} uxy + A + C+ e = 0 (7.1-6)
The termin Eq. (7.1-6) can be eliminated by multiplying (7.1-6) with - to obtain
{A- B + C} uxy -{ A + C + e} = 0 (7.1-7)
Since Eq. (7.1-7) must be valid for all values of x and y
A- B + C = 0 (7.1-8a)
A + C + e = 0 (7.1-8b)
The solution to the PDE [ Auxx + Buxy + Cuyy + e = 0 ] might be obtained by solving the two ODEs (7.1-8a) and (7.1-8b). is solved first from the quadratic equation (7.1-8a)
= (-B ± {B2 - 4AC}1/2)/(4A)
The curves with slope are called the characteristics of the differential equation. The value of can be real or imaginary depending on the expression D = B2 - 4AC.
If B2 - 4AC < 0, there are no real characteristics and the PDE [ Auxx + Buxy + Cuyy + e = 0 ] is called elliptic.
If B2 - 4AC = 0, there is a single characteristics and the PDE is called parabolic.
If B2 - 4AC > 0, there are two real characteristics and the PDE is called hyperbolic. The solution to the PDE can be obtained by solving for u from
A + C + e = 0 (7.1-8b)
Graphically, the solution to the PDE can be obtained at the intersection of the characteristic curves as shown in Figure 7.1-1.
Figure 7.1-1 Characteristic curves for hyperbolic PDEs.
Examples of different PDEs are
(a) Elliptic
+ = 0 (Laplace’s equation)
+ + e = 0 (Poisson’s equation)
(b) Parabolic
= a; where a is a positive constant
= a(+ ) + e; (Parabolic in time t and elliptic in spatial dimensions x and y)
(c) Hyperbolic
- c2 = 0 (wave’s equation)
The classification of second order linear PDEs might also be explained in terms of one-way and two-way coordinates for a one-dimensional system. In one way coordinate, the conditions at a given location in the coordinate are influenced by changes in conditions on only one side of that location. In two way coordinate, the conditions at a given location in the coordinate are influenced by changes in conditions on either side of that location. Example of one way coordinate is time where yesterday’s events affect today’s happenings, but tomorrow’s conditions have no influence on what happens today. Example of two way coordinate is the one dimensional heat transfer in a rod. Space can also be treated as a one-way coordinate under the action of fluid flow. If there is a strong unidirectional flow in the coordinate direction, then significant influences travel from upstream to downstream. The conditions at a point are then affected largely by the upstream conditions, and very little by the downstream ones.
A situation is parabolic if there exists at least one one-way coordinate; otherwise, it is elliptic. Hyperbolic situation is also a one-way coordinate. However, information is not traveled along coordinate direction but along special curves called characteristics.
Superposition Principle
If u1 and u2 are solutions of a linear homogeneous PDE then
u = C1u1 + C2u2 ; (C1 and C2 are constants)
is also a solution.
Example 7.1-1. Consider the two dimensional Laplace equation
+ = 0
This equation is linear and homogeneous with the following solutions
u1 = x + y; u2 = x2 - y2; u3 = exsin y; …
The following superposition of the three particular solutions is also a solution of the Laplace equation
u = u1 + 2u2 + 3u3 = (x + y) + 2(x2 - y2) + 3exsin y
This fact can be verified as follows
= 1 + 4x + 3exsin y Þ = 4 + 3exsin y
= 1 - 4x + 3excos y Þ = - 4 - 3exsin y
Therefore + = 4 + 3exsin y - 4 - 3exsin y = 0
The superposition principle is also applied to the boundary conditions. If u1 and u2 satisfy a linear homogeneous boundary condition then
u = C1u1 + C2u2
will also satisfy the boundary condition.
7-5