A Solution to the Convection Equation 

>

restart:

 

>	with(plots):

 

>	F:=x->exp(-x^2);

 

>	plot(F(x),x=-2..2);

 

>	u:=F(x-c*t);

 

>	ut:=diff(u,t);ux:=diff(u,x);

 

>	PDE:=ut+c*ux;simplify(PDE);

 

>

c:=1;u;

 

>	u1:=subs(t=0,u);u2:=subs(t=1,u);u3:=subs(t=2,u);

 

>	plot({u1,u2,u3},x=-5..5,color=[red,blue,green]);

 

>	plot3d(u,x=-6..6,t=0..2,axes=boxed);

 

>	animate(u,x=-6..6,t=0..2);

 

A Solution to Laplace's Equation 

>

restart:

 

>	with(plots):

 

>	Phi:=exp(-x)*cos(y);

 

>	Phi_xx:=diff(Phi,x$2);Phi_yy:=diff(Phi,y$2);

 

>	PDE:=Phi_xx+Phi_yy;

 

>	plot3d(Phi,x=0..1,y=0..2*Pi,axes=frame);

 

A Fourier Series 

>

restart:

 

>	with(plots):

 

>	h:=N->2*sum((-1)^(m+1)*sin(m*x)/m,m=1..N);

 

>	h2:=h(2);h4:=h(4);h8:=h(8);h16:=h(16);h32:=h(32);

 

>	plot({x,h2,h4,h8,h16,h32},x=0..Pi);

 

>