Page 573 - Modelling in Transport Phenomena A Conceptual Approach

B.3. SECONDORDER PARTIAL DIFFERENTIAL EQUATIONS 553

It is assumed that the coefficient functions and the given function G are real-valued
and twice continuously differentiable on a region R of the z, y plane.
When G = 0, the equation is homogeneous, otherwise the equation is non-
homogeneous.
The criteria, B2 - AC, that will indicate whether the second-order equation is
a graph of a parabola, ellipse or hyperbola is called the discriminant, A, i.e.,
>O Hyperbolic
= 0 Parabolic
< 0 Elliptic
B.3.2 Orthogonal Functions

Let f(z) and g(z) be real-valued functions defined on the interval a 2 z 5 b. The
inner product of f(z) and g(x) with respect to w(z) is defined by

(B.3-2)

in which the weight function w(z) is considered positive on the interval (a, b).

Example B.10 Find the inner product of f(z) = and g(z) = 1 with respect
to the weight function w(x) = x1I2 on the interval 0 5 x 5 1.

Solution
Application of Eq. (B.3-2) gives the inner product as

2
(f,g) = I'fizdx = 5 x5I2 1 2
0

The inner product has the following properties:
(f, 9) = (9, f) (B.3-3)

(f, 9 + h) = (f, 9) + (f, h) (B.3-4)

(af,g) = a(f,g) a is a scalar (B.3-5)
The inner product of f with respect to itself is

b
(f, f) = J 4.1 f2(4 dz = Ilf(X)1l2 > 0 (B.3-6)

in which nom of f(z) is defined as
llf(4II = m (B.3-7)