Introduction 5
Exercises 1.
- Find the order of each PDE:
a) Theconvectionoradvection equation,ut+cux=
b) Thewave equation,utt=c^2 uxx
c) Theeikonal equation,u^2 x+u^2 y=
d) TheEuler–Bernoulli beam equation,utt+α^4 uxxxx=
e) uxxyyz−u^8 +u^6 xx=- Show that each given function is a solution of the corresponding PDE:
a)u=x^2 y,xux− 2 yuy=
b) u=xsiny,uxx−uyy=u
c) u=yf(x),uyy=0(wherefis any function with continuous second
derivative)
d) u=ex+2y+ex−^2 y, 4 uxx−uyy=
e) u=excosy+ax+by, Laplace’s equation in two dimensions in
rectangular coordinatesuxx+uyy=0(whereaandbare any
constants)
f)u=xyz, 2 xux−yuy−zuz=
g)u=x^2 y^3 z^2 −xz^3 , 3 x^2 uxx+2yuy+2xy^3 z^2 uzzz=- Consider theconvection equationut+cux=0,wherecis a constant.
a) Show thatu= sin(x−ct),u=cos(x−ct)andu=5(x−ct)^2 are
solutions.
b) Show thatu=7sin(x−ct),u=3cos(x−ct)andu=7sin(x−
ct)−3cos(x−ct) also are solutions.
c) Show thatu=f(x−ct) is a solution for “any” functionf.
d) Why is “any” in quotation marks in part (c)?- Consider theone-dimensional wave equationutt=c^2 uxx,wherecis a
constant.
a) Show that all of the functions in Exercise 3 are solutions of this
equation, as well.
b) Show thatu=g(x+ct) also satisfies the wave equation for “any”
functiong.- Consider theeikonal equationu^2 x+u^2 y=1.
a) Show thatu=xandu=yare solutions.
b) Areu=3xandu=− 4 ysolutions?