Computer Algebra Recipes

264 CHAPTER 6. LINEAR PDE MODELS. PART 2
T (x; 0) = A±(x ¡ a), i.e., is a Dirac delta function 2 of amplitude A located at
x = a>0. The end x = 0 is perfectly insulated so that no heat can °ow across
this end. For no heat °ow to occur, the boundary condition is @T=@xjx=0 =0,
i.e., the gradient of the temperature is zero. The evolution of the temperature
pro¯le is to be animated over the spatial range x = 0 to 20 for the time interval
t = 1 to 200, with a =5,A = 5, and a heat di®usion constant d =1.
In her mathematical physics course, Vectoria has learned that when a gradient
boundary condition is involved for a semi-in¯nite domain problem the
Fourier cosine transform should be used. The Fourier cosine transform F (s) of
a function f(x) forwhichbothf(x) andf 0 (x) ! 0asx !1and its inverse
transform are
r
de¯ned as
Z 1
2
F (s) = f(x) cos(sx) dx; f(x) =
¼
0
r 2
¼
Z 1
F (s) cos(sx) ds:
Guided by her easy conquest of the previous problem, Vectoria realizes that she
will be done by the time Mike arrives to pick her up.
She again loads the plots and integral transform library packages, and enters
the 1-dimensional heat di®usion equation.
> restart: with(plots): with(inttrans):
> DE:=diff(T(x,t),t)=d*diff(T(x,t),x,x);
DE := @ μ
T (x; t) =d @
@t 2
2 T (x; t)
@x
The boundary condition bc and the initial condition ic are speci¯ed. The differential
command D[1](T)(0,t) is used to enter @T(x; t)=@xjx=0, while the
Dirac delta function is entered with the Dirac command.
> bc:=D[1](T)(0,t)=0; ic:=A*Dirac(x-a);
bc := D1(T )(0; t)=0 ic := A Dirac(¡x + a)
The Fourier cosine transforms of the initial condition (assuming that a>0) and
the di®usion equation are calculated. The boundary condition is substituted
into the latter result.
> F0:=fouriercos(ic,x,s) assuming a>0;
F0 := A p 2cos(sa)
p
¼
> FCT:=subs(bc,fouriercos(DE,x,s));
FCT := @
@t fouriercos(T (x; t); x;s)=¡ds2 fouriercos(T (x; t); x;s)
As in the previous recipe, fouriercos(T (x; t); x;s) is replaced with F (t)inFCT .
> FCT:=subs(fouriercos(T(x,t),x,s)=F(t),FCT);
FCT := d
dt F (t) =¡ds2 F (t)
2 The Dirac delta function ±(x ¡ a) is an in¯nitely tall spike located at x = a and is equal
to zero for x 6= a. One of its most important properties is the sifting property, viz., for a
smooth function f(x) (not another delta function) and ²>0, R a+²
f(x) ±(x ¡ a) dx = f(a):
a¡²
0