Page 322 - Schaum's Outline of Differential Equations

CHAP. 31] AN INTRODUCTION TO PARTIAL DIFFERENTIAL EQUATIONS 305

and Laplace's equation (named in honor of P. S. Laplace (1749-1827), a French mathematician and
scientist)

These equations are widely used as models dealing with heat flow, civil engineering, and acoustics to name but
three areas. Note that k is a positive constant in Eqs. (31.3) and (31.4).

SOLUTIONS AND SOLUTION TECHNIQUES
If a function, u(x, y,z, ...), is sufficiently differentiable - which we assume throughout this chapter for all
functions - we can verify whether it is a solution simply by differentiating u the appropriate number of times
with respect to the appropriate variables; we then substitute these expressions into the PDE. If an identity is
obtained, then u solves the PDE. (See Problems 31.1 through 31.4.)
We will introduce two solution techniques: basic integration and separation of variables.
Regarding the technique of separation of variables, we will assume that the/orw? of the solution of the PDE
can be "split off or "separated" into a product of Junctions of each independent variable. (See Problems 31.4
and 31.11). Note that this method should not be confused with the ODE method of "separation of variables"
which was discussed in Chapter 4.

Solved Problems

31.1. Verify that u(x, t) = sin x cos kt satisfies the wave equation (31.4).
Taking derivatives of u leads us to u x = cos x cos kt, u xx = - sin x cos kt, u, = - k sin x sin kt, and u tt = -k 2
sin x cos kt. Therefore u = —-u implies -sin x cos kt = —^ (— k sin x cos fcf) = - sin x cos fcf; hence, u indeed
is a solution.

31.2. Verify that any function of the form F(x + kt) satisfies the wave equation, (31.4).
Let u = x + kt; then by using the chain rule for partial derivatives, we have F x = F uu x = F u(l) = F u;
2
2
F xx = F uuu x = F xx(l) = F^, F, = F uu, = F u (k); F tt = kF uuu, = k F uu. Hence, F a = F aa = -^F tt = -^(k F aa) = F aa, so we
K K
have verified that any sufficiently differentiable function of the form F(x + kt) satisfies the wave equation. We note
1
that this means that functions such as -Jx + kt, tan" ^ + kt) and In (x + kt) all satisfy the wave equation.
kt
31.3. Verify u (x, t) = e~ sin x satisfies the heat equation (31.3).
h
Differentiation implies u x = e~ kt cos x, u xx = -e~ kt sin x, u t = -ke~ sin x. Substituting u^. and u, in (31.3)
kt
clearly yields an identity, thus proving that u(x, t) = e~ sin x indeed satisfies the heat equation.
3 2
2 2
9 6
31.4. Verify u(x, t) = (5x - 6X 5 + x )t satisfies the PDE X t u xtt - 9x t u tt = tu xxt + 4u xx.
We note that u(x, t) has a specific form; i.e., it can be "separated" or "split up" into two functions: a function
of x times a function of t. This will be discussed further in Problem 31.11. Differentiation of u(x, t) leads to:
1
5
9
5
6
7
3
4
4
3
8
««< = (5 - 30x + 9x )(30t ), u^ = (-12CU + 12x )(t ), u^, = (-120.x + 12x )(6t ), and u tt = (5x -6x + x )(30?).