24 An Introduction to Partial Differential Equations with MATLAB©R
- 5ux+4uy− 2 u=0
24.y^2 ux+x^2 uy=025.uxx−uy+u=0- The wave equation,utt−uxx=0
- Laplace’s equation,uxx+uyy=0
28.uxx+2uyy−ux+3uy=0- Laplace’s equation in polar coordinates,urr+^1 rur+r^12 uθθ=0
30.ux+uy−uz=031.ux+uy+uz+u=032.uxx+uy+uz=0- The two-dimensional heat equation,ut=uxx+uyy
- The two-dimensional wave equation,utt=uxx+uyy
- The three-dimensional Laplace equation,uxx+uyy+uzz=0
- Prove that iff(x)=g(y,z) for allx, yandz,thenfandgboth are
constant functions. - One also may try to separate variables in other ways.
a) Find all solutions of the PDEux+uy= 0 of the formu(x, y)=
X(x)+Y(y).
b) Do the same for the eikonal equation,u^2 x+u^2 y=1.- In Section 1.3, we saw that we often are interested in solving a PDE
subject to certain auxiliary conditions, namely, initial and boundary
conditions. In fact, when we solve an initial-boundary-value problem
using separation of variables, we will find it much easier to solve if
we alsoseparate the boundary conditions. For each of the boundary
conditions given below, separate the variables, that is, decide what each
of them tells you about product solutionsu(x, t)=X(x)T(t). (In each
case,ais a constant.)
a)u(a, t) = 0 (the so-calledDirichlet boundary condition)
b) ux(a, t) = 0 (theNeumann condition)
c) αux(a, t)+βu(a, t)=0,whereαandβare constants (theRobin
condition)
d) uxx(a, t)=0
e) uxxx(a, t)=0