Partial Differential Equations with MATLAB

18 An Introduction to Partial Differential Equations with MATLAB©R

2. The heat equation in two space variables,ut=α^2 (uxx+uyy)


3. Laplace’s equation in three space variables,uxx+uyy+uzz=0


4. The wave equation in three space variables,

utt+c^2 (uxx+uyy+uzz)=0

5. Use (1.12) to show that the functionu(x, t)=

∑∞

n=1

cne−n

(^2) t
sinnxis a
solution of the heat equationut=uxx(whenever the series converges,
of course).

6. Show directly that the principle of superposition doesnothold for the
   PDEux+u^2 =0,u=u(x, y), by finding two different solutions, then
   finding a linear combination of them that isnota solution.


7. We may also prove theorems for solutions ofnonhomogeneous PDEs
   that are analogous to those for ODEs. Prove that the general solution
   of the nonhomogeneous PDEL[u]=fisu=uh+up,whereupis any
   one particular solution ofL[u]=f,anduhis the general solution of the
   associated homogeneous PDEL[u] = 0, as follows:

a) First, prove thatuh+upalways is a solution ofL[u]=f.

b)Next,provethat,ifuis any particular solution ofL[u]=f,then

we can always write

u=uh′+up,

whereuh′is a particular case of the solutionuh.

Illustrate the theorem that we proved in Exercise 7 for the nonhomogeneous
PDEs in Exercises 8–12. You may refer to the corresponding exercises in
Section 1.2.

8.uy=2x

9.ux=sinx+cosy

10.uxxy=12x

11.uzz=x+y

12.uxx+ux− 2 u=6,u=u(x, y).

13. a) Ifvis a solution of the PDEL[u]=f,andwis a solution of
    L[u]=g, find a solution of the PDEL[u]=αf+βg,whereαand
    βare any two constants.
    b) Use what you did in part (a) to find a solution of the PDEuxx+
    uyy=3x− 5 y.