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# Verification of Solutions to PDEs

In this section we will verify that certain functions are solutions to first order PDEs. Consider the first order wave equation,
 (1.2)

where is a constant called the wave speed. We now verify that the following are all solutions to Equation (1.2).

Example 1. .5

1. The solution is shown in Figure 1.1 at various values of . It clearly demonstrates that the initial shape of the function is maintained and simply propagates to the right.

2. Hence, the equation is satisfied. Note that is continuous but the partial derivatives are not continuous. The solution is shown in Figure 1.2 for various values of .

3. where is an arbitrary function. It is easier to understand what is happening if we define , so that . Hence, the partial derivatives, on using the chain rule, are

Therefore,

for any functional form for . This illustrates an important point. Ordinary differential equations have arbitrary constants but Partial differential equations have arbitrary functions. The function is determined by an initial condition.

4. Assume that the initial condition, at , is , where is a prescribed function. Now if

Hence, the unknown function, is the prescribed function, but the argument is replaced by . Hence, the solution is

As an illustration, if , then

satisfies Equation (1.2) and the initial condition.
The next example will verify the solution to other equations, in terms of an arbitrary function with a specific argument.

Example 1. .6

1. Consider the linear, equation

The solution is , where is an arbitrary function.

Hence,

and the equation is satisfied.

2. Consider the linear equation without non-constant coefficients,

The solution is given by , where is again an arbitrary function but this time .

and so the equation is clearly satisfied.
How do we decide on the argument of the arbitrary functions?

Next: Method of Characteristics Up: Partial Differential Equations of Previous: Revision of Partial Differentiation
Prof. Alan Hood
2000-10-30