Page 12 - Essentials of applied mathematics for scientists and engineers

book Mobk070 March 22, 2007 11:7

2 ESSENTIALS OF APPLIED MATHEMATICS FOR SCIENTISTS AND ENGINEERS
A partial differential equation expresses a dependent variable, say u, as a function of more
than one independent variable, say x, y,and z. Partial derivatives are normally written as ∂u/∂x.
This is the first-order derivative of the dependent variable u with respect to the independent
variable x. Sometimes we will use the notation u x or when the derivative is an ordinary derivative
2
2
we use u . Higher order derivatives are written as ∂ u/∂x or u xx . The order of the differential

equation now depends on the orders of the derivatives of the dependent variables in terms of
each of the independent variables. For example, it may be of order m for the x variable and of
order n for the y variable. A boundary value problem consists of a partial differential equation
defined on a domain in the space of the independent variables, for example the x, y, z space,
along with conditions on the boundary. Once again, if the partial differential equation and the
boundary conditions contain only terms of first degree in u and its derivatives the problem is
linear. Otherwise it is nonlinear.
A differential equation or a boundary condition is homogeneous if it contains only terms
involving the dependent variable.

Examples
Consider the ordinary differential equation

a(x)u + b(x)u = c (x), 0 < x < A. (1.1)
Two boundary conditions are required because the order of the equation is 2. Suppose

u(0) = 0 and u(A) = 1. (1.2)

The problem is linear. If c (x) is not zero the differential equation is nonhomogeneous. The first
boundary condition is homogeneous, but the second boundary condition is nonhomogeneous.
Next consider the ordinary differential equation

a(u)u + b(x)u = c 0 < x < A (1.3)
Again two boundary conditions are required. Regardless of the forms of the boundary condi-
tions, the problem is nonlinear because the first term in the differential equations is not of first
degree in u and u since the leading coefficient is a function of u. It is homogeneous only if

c = 0.
Now consider the following three partial differential equations:

u x + u xx + u xy = 1 (1.4)
u xx + u yy + u zz = 0 (1.5)

uu x + u yy = 1 (1.6)