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Fourier Method for Heat Equation

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heat_eqf.mws

1 Example
2 Example
3 Example
4 Example
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6 Example

Fourier method for heat equation

by Aleksas Domarkas

Vilnius University, Faculty of Mathematics and Informatics,

Naugarduko 24, Vilnius, Lithuania

aleksas@ieva.mif.vu.lt

NOTE: In this session we find solutions of boundary value problems for heat equation using Fourier method

1 Example

Problem

`>`	restart;

`>`	*eq:=diff(u(x,t),t)-a^2diff(u(x,t),x,x)=0;#0<x,x<Pi,t>0;**

eq := diff(u(x,t),t)-a^2*diff(u(x,t),`$`(x,2)) = 0

`>`	init_c:=u(x,0)=phi(x);#0<x,x<Pi

`>`	bound_c:=u(0,t)=0,u(Pi,t)=0;

`>`	phi:=x->Pi/2-abs(Pi/2-x);

`>`	'phi(x)'=convert(phi(x),piecewise);

phi(x) = PIECEWISE([x, x < 1/2*Pi],[Pi-x, 1/2*Pi <= x])

`>`	plot(phi,0..Pi,axes=boxed);

[Maple Plot]

>

Eigenvalues and eigenfunctions

`>`	*L:=-diff(y(x),x,x)=lambday(x);**

`>`	s1:=y(0)=0; s2:=y(Pi)=0;

`>`	assume(lambda>0,a>0);

L := -diff(y(x),`$`(x,2)) = lambda*y(x)

`>`	dsolve(L,y(x));

`>`	y :=unapply( rhs(%),x);

`>`	sist:={s1,s2};

$sist := {_C2 = 0, _C1*sin(sqrt(lambda)*Pi)+_C2*cos(sqrt(lambda)*Pi) = 0}$

`>`	linalg[genmatrix](sist,{_C1,_C2});

matrix([[0, 1], [sin(sqrt(lambda)*Pi), cos(sqrt(lambda)*Pi)]])

`>`	chl:=linalg[det](%);

`>`	#chl:=combine(%);

`>`	_EnvAllSolutions := true:

`>`	solve(chl,lambda);

`>`	indets(%) minus {l};

${_Z1}$

`>`	subs(%[1]='k',%%):

`>`	ev:=unapply(%,k);

`>`	y:='y':assume(k,posint);

`>`	subs(lambda=ev(k),L);

-diff(y(x),`$`(x,2)) = k^2*y(x)

`>`	dsolve({%,s1,s2},y(x));

`>`	rhs(%)/sqrt(int(rhs(%)^2,x=0..Pi));

_C1*sin(k*x)*2^(1/2)/(Pi*_C1^2)^(1/2)

`>`	simplify(%,radical,symbolic):

`>`	ef:=unapply(%,(k,x)):

Eigenvalues and eigenfunctions:

`>`	ev(k);ef(k,x);

sin(k*x)*2^(1/2)/Pi^(1/2)

Solving problem

Solution we find in form:

`>`	*spr:=Sum(T[k](t)ef(k,x),k=1..infinity);**

spr := Sum(T[k](t)*sin(k*x)*2^(1/2)/Pi^(1/2),k = 1 .. infinity)

For coefficients T[k](t) we have ODE problem:

`>`	*ode:={diff(u(t),t)+a^2ev(k)u(t)=0,*

`>`	*u(0)=int((phi(x))ef(k,x),x=0..Pi)};**

$ode := {diff(u(t),t)+a^2*k^2*u(t) = 0, u(0) = 2*sin(1/2*Pi*k)*2^(1/2)/Pi^(1/2)/k^2}$

`>`	dsolve(ode,u(t));

u(t) = 2*sin(1/2*Pi*k)*2^(1/2)/Pi^(1/2)/k^2*exp(-a^2*k^2*t)

Solution

`>`	sol:=subs(T[k](t)=rhs(%),spr);

sol := Sum(4*sin(1/2*Pi*k)/Pi/k^2*exp(-a^2*k^2*t)*sin(k*x),k = 1 .. infinity)

`>`	sol_20:=value(subs(infinity=20,sol));

sol_20 := 4/Pi*exp(-a^2*t)*sin(x)-4/9*1/Pi*exp(-9*a^2*t)*sin(3*x)+4/25/Pi*exp(-25*a^2*t)*sin(5*x)-4/49*1/Pi*exp(-49*a^2*t)*sin(7*x)+4/81/Pi*exp(-81*a^2*t)*sin(9*x)-4/121*1/Pi*exp(-121*a^2*t)*sin(11*x)+...

`>`	plot3d(subs(a=1,sol_20),x=0..Pi,t=0..1);

[Maple Plot]

Test

`>`	subs(u(x,t)=sol_20,eq):

`>`	simplify(%);

`>`	plot(subs(t=0,sol_20),x=0..Pi);

[Maple Plot]

`>`	plot(phi,0..Pi);

[Maple Plot]

>

2 Example

Problem

`>`	restart;

`>`	*eq:=diff(u(x,t),t)-a^2diff(u(x,t),x,x)=0;#0<x,x<l,t>0;**

eq := diff(u(x,t),t)-a^2*diff(u(x,t),`$`(x,2)) = 0

`>`	init_c:=u(x,0)=phi(x);#0<x,x<l;

`>`	bound_c:=u(0,t)=0,u(l,t)=0;

`>`	*phi:=x->x(l-x);**

> #l>0;

Eigenvalues and eigenfunctions

`>`	*L:=-diff(y(x),x,x)=lambday(x);**

`>`	s1:=y(0)=0; s2:=y(l)=0;

`>`	assume(lambda>0,l>0,a>0);

L := -diff(y(x),`$`(x,2)) = lambda*y(x)

`>`	dsolve(L,y(x));

`>`	y :=unapply( rhs(%),x);

`>`	sist:={s1,s2};

$sist := {_C1*sin(sqrt(lambda)*l)+_C2*cos(sqrt(lambda)*l) = 0, _C2 = 0}$

`>`	linalg[genmatrix](sist,{_C1,_C2});

matrix([[sin(sqrt(lambda)*l), cos(sqrt(lambda)*l)], [0, 1]])

`>`	chl:=linalg[det](%);

`>`	#chl:=combine(%);

`>`	_EnvAllSolutions := true:

`>`	solve(chl,lambda);

Pi^2*_Z1^2/l^2

`>`	indets(%) minus {l};

${_Z1}$

`>`	subs(%[1]='k',%%):

`>`	ev:=unapply(%,k);

ev := proc (k) options operator, arrow; Pi^2*k^2/l^2 end proc

`>`	y:='y':assume(k,posint);

`>`	subs(lambda=ev(k),L);

-diff(y(x),`$`(x,2)) = Pi^2*k^2/l^2*y(x)

`>`	dsolve({%,s1,s2},y(x));

`>`	rhs(%)/sqrt(int(rhs(%)^2,x=0..l));

_C1*sin(Pi*k/l*x)*2^(1/2)/(_C1^2*l)^(1/2)

`>`	simplify(%,radical,symbolic):

`>`	ef:=unapply(%,(k,x)):

Eigenvalues and eigenfunctions:

`>`	ev(k);ef(k,x);

Pi^2*k^2/l^2

sin(Pi*k/l*x)*2^(1/2)/l^(1/2)

Solving problem

Solution we find in form:

`>`	*spr:=Sum(T[k](t)ef(k,x),k=1..infinity);**

spr := Sum(T[k](t)*sin(Pi*k/l*x)*2^(1/2)/l^(1/2),k = 1 .. infinity)

For coefficients T[k](t) we have ODE problem:

`>`	*ode:={diff(u(t),t)+a^2ev(k)u(t)=0,*

`>`	*u(0)=int((phi(x))ef(k,x),x=0..l)};**

$ode := {u(0) = -2*l^(5/2)*2^(1/2)*((-1)^k-1)/Pi^3/k^3, diff(u(t),t)+a^2*Pi^2*k^2/l^2*u(t) = 0}$

`>`	dsolve(ode,u(t));

u(t) = -2*l^(5/2)*2^(1/2)*((-1)^k-1)/Pi^3/k^3*exp(-a^2*Pi^2*k^2/l^2*t)

Solution

`>`	sol:=subs(T[k](t)=rhs(%),spr);

sol := Sum(-4*l^2*((-1)^k-1)/Pi^3/k^3*exp(-a^2*Pi^2*k^2/l^2*t)*sin(Pi*k/l*x),k = 1 .. infinity)

We convert this solution to other form. Index change to :

`>`	*subs(k=2m+1,op(1,%));assume(m,integer):**

-4*l^2*((-1)^(2*m+1)-1)/Pi^3/(2*m+1)^3*exp(-a^2*Pi^2*(2*m+1)^2/l^2*t)*sin(Pi*(2*m+1)/l*x)

`>`	simplify(%):factor(%);

8*l^2/Pi^3/(2*m+1)^3*exp(-a^2*Pi^2*(2*m+1)^2/l^2*t)*sin(Pi*(2*m+1)/l*x)

`>`	Sum(%,m=0..infinity):

`>`	solution:=subs(a='a',k='k',l='l',m='m',%);

solution := Sum(8*l^2/Pi^3/(2*m+1)^3*exp(-a^2*Pi^2*(2*m+1)^2/l^2*t)*sin(Pi*(2*m+1)/l*x),m = 0 .. infinity)

>

3 Example

Problem

[bic] 694

`>`	restart;

`>`	*eq:=diff(u(x,t),t)+betau(x,t)-a^2diff(u(x,t),x,x)=0;0<x,x<l,t>0;*

eq := diff(u(x,t),t)+beta*u(x,t)-a^2*diff(u(x,t),`$`(x,2)) = 0

`>`	init_c:=u(x,0)=phi(x);

`>`	bound_c:=u(0,t)=0,u(l,t)=0;

> #l>0;

Eigenvalues and eigenfunctions

`>`	*L:=-diff(y(x),x,x)=lambday(x);**

`>`	s1:=y(0)=0; s2:=y(l)=0;

`>`	assume(lambda>0,l>0,a>0);

L := -diff(y(x),`$`(x,2)) = lambda*y(x)

`>`	dsolve(L,y(x));

`>`	y :=unapply( rhs(%),x);

`>`	sist:={s1,s2};

$sist := {_C2 = 0, _C1*sin(sqrt(lambda)*l)+_C2*cos(sqrt(lambda)*l) = 0}$

`>`	linalg[genmatrix](sist,{_C1,_C2});

matrix([[0, 1], [sin(sqrt(lambda)*l), cos(sqrt(lambda)*l)]])

`>`	chl:=linalg[det](%);

`>`	#chl:=combine(%);

`>`	_EnvAllSolutions := true:

`>`	solve(chl,lambda);

Pi^2*_Z1^2/l^2

`>`	indets(%) minus {l};

${_Z1}$

`>`	subs(%[1]='k',%%):

`>`	tr:=unapply(%,k);

tr := proc (k) options operator, arrow; Pi^2*k^2/l^2 end proc

`>`	y:='y':assume(k,posint);

`>`	subs(lambda=tr(k),L);

-diff(y(x),`$`(x,2)) = Pi^2*k^2/l^2*y(x)

`>`	dsolve({%,s1,s2},y(x));

`>`	rhs(%)/sqrt(int(rhs(%)^2,x=0..l));

_C1*sin(Pi*k/l*x)*2^(1/2)/(_C1^2*l)^(1/2)

`>`	simplify(%,radical,symbolic):

`>`	tf:=unapply(%,(k,x)):

Eigenvalues and eigenfunctions:

`>`	tr(k);tf(k,x);

Pi^2*k^2/l^2

sin(Pi*k/l*x)*2^(1/2)/l^(1/2)

Solving problem

Solution we find in form:

`>`	*spr:=Sum(T[k](t)tf(k,x),k=1..infinity);**

spr := Sum(T[k](t)*sin(Pi*k/l*x)*2^(1/2)/l^(1/2),k = 1 .. infinity)

For coefficients T[k](t) we have ODE problem:

`>`	*uz:={diff(u(t),t)+betau(t)+a^2tr(k)u(t)=0,**

`>`	*u(0)=int((phi(xi))tf(k,xi),xi=0..l)};**

$uz := {diff(u(t),t)+beta*u(t)+a^2*Pi^2*k^2/l^2*u(t) = 0, u(0) = int(phi(xi)*sin(Pi*k/l*xi)*2^(1/2)/l^(1/2),xi = 0 .. l)}$

`>`	dsolve(uz,u(t));

u(t) = int(phi(xi)*sin(Pi*k/l*xi)*2^(1/2)/l^(1/2),xi = 0 .. l)*exp((-beta*l^2-k^2*Pi^2*a^2)*t/l^2)

Solution

`>`	sol:=simplify(subs(T[k](t)=rhs(%),spr));

sol := Sum(2/l*int(phi(xi)*sin(Pi*k/l*xi),xi = 0 .. l)*exp(-(beta*l^2+k^2*Pi^2*a^2)*t/l^2)*sin(Pi*k/l*x),k = 1 .. infinity)

`>`	subs(a='a',k='k',l='l',%);

Sum(2/l*int(phi(xi)*sin(Pi*k/l*xi),xi = 0 .. l)*exp(-(beta*l^2+k^2*Pi^2*a^2)*t/l^2)*sin(Pi*k/l*x),k = 1 .. infinity)

>

4 Example

Problem

[komec] 93 p.

`>`	restart;

`>`	*eq:=diff(u(x,t),t)-a^2diff(u(x,t),x,x)=0;0<x,x<l,t>0;**

eq := diff(u(x,t),t)-a^2*diff(u(x,t),`$`(x,2)) = 0

`>`	init_c:=u(x,0)=phi(x);

`>`	bound_c:=u(0,t)=0,D[1](u)(l,t)=0;

`>`	a:=2;l:=3;phi:=x->x;

Eigenvalues and eigenfunctions

`>`	*L:=-diff(y(x),x,x)=lambday(x);**

`>`	s1:=y(0)=0; s2:=D(y)(l)=0;

`>`	assume(lambda>0);

L := -diff(y(x),`$`(x,2)) = lambda*y(x)

`>`	dsolve(L,y(x));

`>`	y :=unapply( rhs(%),x);

`>`	sist:={s1,s2};

$sist := {_C1*cos(3*sqrt(lambda))*sqrt(lambda)-_C2*sin(3*sqrt(lambda))*sqrt(lambda) = 0, _C2 = 0}$

`>`	linalg[genmatrix](sist,{_C1,_C2});

matrix([[cos(3*sqrt(lambda))*sqrt(lambda), -sin(3*sqrt(lambda))*sqrt(lambda)], [0, 1]])

`>`	chl:=linalg[det](%);

`>`	chl:=%/sqrt(lambda);

`>`	_EnvAllSolutions := true:

`>`	solve(chl,lambda);

`>`	indets(%) minus {l};

${_Z1}$

`>`	subs(%[1]='k',%%):

`>`	tr:=unapply(%,k);

`>`	y:='y':assume(k,posint);

`>`	subs(lambda=tr(k),L);

-diff(y(x),`$`(x,2)) = 1/36*Pi^2*(1+2*k)^2*y(x)

`>`	dsolve(%,y(x));

`>`	y:=unapply(rhs(%),x);

`>`	solve(s1,_C2);

`>`	simplify(subs(_C2=%,y(x)));

`>`	select(has,%,x);

`>`	%/sqrt(int(%^2,x=0..l));

`>`	tf:=unapply(%,(k,x));

Eigenvalues and eigenfunctions:

`>`	tr(k);tf(k,x);

Solving problem

Solution we find in form:

`>`	*spr:=Sum(T[k](t)tf(k,x),k=0..infinity);**

spr := Sum(1/3*T[k](t)*sin(1/6*Pi*(1+2*k)*x)*sqrt(6),k = 0 .. infinity)

For coefficients T[k](t) we have ODE problem:

`>`	*uz:={diff(u(t),t)+a^2tr(k)u(t)=0,*

`>`	*u(0)=int((phi(xi))tf(k,xi),xi=0..l)};**

$uz := {u(0) = 12*(-1)^k*6^(1/2)/Pi^2/(1+2*k)^2, diff(u(t),t)+1/9*Pi^2*(1+2*k)^2*u(t) = 0}$

`>`	dsolve(uz,u(t));

u(t) = 12*(-1)^k*6^(1/2)/Pi^2/(1+4*k+4*k^2)*exp(-1/9*Pi^2*(1+2*k)^2*t)

Solution

`>`	simplify(subs(T[k](t)=rhs(%),spr));

Sum(24*(-1)^k/Pi^2/(1+4*k+4*k^2)*exp(-1/9*Pi^2*(1+2*k)^2*t)*sin(1/6*Pi*(1+2*k)*x),k = 0 .. infinity)

Solution

`>`	sol:=subs(k='k',factor(%));

sol := Sum(24*(-1)^k*exp(-1/9*Pi^2*(1+2*k)^2*t)*sin(1/6*Pi*(1+2*k)*x)/Pi^2/(1+2*k)^2,k = 0 .. infinity)

>

5 Example

Problem

[komec] 98 p.

`>`	restart;

`>`	*eq:=diff(u(x,t),t)-a^2diff(u(x,t),x,x)=f(x,t);0<x,x<l,t>0;**

eq := diff(u(x,t),t)-a^2*diff(u(x,t),`$`(x,2)) = f(x,t)

`>`	init_c:=u(x,0)=phi(x);

`>`	bound_c:=D[1](u)(0,t)=0,u(l,t)=0;

`>`	a:=4;l:=7;phi:=x->0;f:=(x,t)->2;

Eigenvalues and eigenfunctions

`>`	*L:=-diff(y(x),x,x)=lambday(x);**

`>`	s1:=D(y)(0)=0; s2:=y(l)=0;

`>`	assume(lambda>0);

L := -diff(y(x),`$`(x,2)) = lambda*y(x)

`>`	dsolve(L,y(x));

`>`	y :=unapply( rhs(%),x);

`>`	sist:={s1,s2};

$sist := {_C1*sqrt(lambda) = 0, _C1*sin(7*sqrt(lambda))+_C2*cos(7*sqrt(lambda)) = 0}$

`>`	linalg[genmatrix](sist,{_C1,_C2});

matrix([[sqrt(lambda), 0], [sin(7*sqrt(lambda)), cos(7*sqrt(lambda))]])

`>`	chl:=linalg[det](%);

`>`	chl:=%/sqrt(lambda);

`>`	_EnvAllSolutions := true:

`>`	solve(chl,lambda);

`>`	indets(%) minus {l};

${_Z1}$

`>`	subs(%[1]='k',%%):

`>`	tr:=unapply(%,k);

`>`	y:='y':assume(k,posint);

`>`	subs(lambda=tr(k),L);

-diff(y(x),`$`(x,2)) = 1/196*Pi^2*(1+2*k)^2*y(x)

`>`	dsolve(%,y(x));

`>`	y:=unapply(rhs(%),x);

`>`	solve(s1,_C1);

`>`	simplify(subs(_C1=%,y(x)));

`>`	select(has,%,x);

`>`	%/sqrt(int(%^2,x=0..l));

`>`	tf:=unapply(%,(k,x));

Eigenvalues and eigenfunctions:

`>`	tr(k);tf(k,x);

`>`	y:='y':

Solving problem

Solution we find in form:

`>`	*spr:=Sum(T[k](t)tf(k,x),k=0..infinity);**

spr := Sum(1/7*T[k](t)*cos(1/14*Pi*(1+2*k)*x)*sqrt(14),k = 0 .. infinity)

For coefficients T[k](t) we have ODE problem:

`>`	*uz:={diff(u(t),t)+a^2tr(k)u(t)=int((f(xi))tf(k,xi),xi=0..l),**

`>`	*u(0)=int((phi(xi))tf(k,xi),xi=0..l)};**

$uz := {u(0) = 0, diff(u(t),t)+4/49*Pi^2*(1+2*k)^2*u(t) = 4*(-1)^k*14^(1/2)/Pi/(1+2*k)}$

`>`	dsolve(uz,u(t));

u(t) = 49*(-1)^k*14^(1/2)/Pi^3/(1+2*k)^3-49*exp(-4/49*Pi^2*(1+2*k)^2*t)*(-1)^k*14^(1/2)/(Pi^3+6*Pi^3*k+12*Pi^3*k^2+8*Pi^3*k^3)

Solution:

`>`	simplify(subs(T[k](t)=rhs(%),spr));

Sum(98*(-1)^(1+k)*cos(1/14*Pi*(1+2*k)*x)*(-1+exp(-4/49*Pi^2*(1+2*k)^2*t))/Pi^3/(1+6*k+12*k^2+8*k^3),k = 0 .. infinity)

Solution

`>`	sol:=subs(k='k',factor(%));

sol := Sum(98*(-1)^(1+k)*cos(1/14*Pi*(1+2*k)*x)*(-1+exp(-4/49*Pi^2*(1+2*k)^2*t))/Pi^3/(1+2*k)^3,k = 0 .. infinity)

Limit of solution, when

`>`	eqn:=subs(u(x,t)=y(x),eq); s1,s2;

eqn := diff(y(x),t)-16*diff(y(x),`$`(x,2)) = 2

`>`	dsolve({eqn,s1,s2},y(x));

`>`	limit(sol,t=infinity);

Sum(-98*(-1)^(1+k)*cos(1/14*Pi*(1+2*k)*x)/Pi^3/(1+2*k)^3,k = 0 .. infinity)

We test this at x=0:

`>`	subs(x=0,%);

Sum(-98*(-1)^(1+k)*cos(0)/Pi^3/(1+2*k)^3,k = 0 .. infinity)

`>`	value(%);

98/Pi^3*hypergeom([1, 1/2, 1/2, 1/2],[3/2, 3/2, 3/2],-1)

`>`	convert(%,StandardFunctions);

98/Pi^3*(-1/2*I*polylog(3,I)+1/2*I*polylog(3,-I))

`>`	simplify(%);

>

6 Example

Problem

`>`	restart;with(linalg,laplacian):

`>`	eq:=diff(u(r,t),t)-a^2*expand(laplacian(u(r,t),[r,alpha],coords=polar))=0; 0<r,r<R,t>0;

eq := diff(u(r,t),t)-a^2*(1/r*diff(u(r,t),r)+diff(u(r,t),`$`(r,2))) = 0

`>`	init_c:=u(r,0)=phi(r);

`>`	bound_c:=u(R,t)=0;

`>`	a:=2;R:=8;phi:=r->64-r^2;

Solution representation formula

u(r,t) = Sum(A[k]*exp(-(a*mu[k])^2*t/(R^2))*BesselJ(0,mu[k]*r/R),k = 1 .. infinity)

where - zeros of Bessel J(0,x) function ,

A[k] = 2*int(r*phi(r)*BesselJ(0,mu[k]*r/R),r = 0 .. R)/(R^2*BesselJ(1,mu[k])^2)

Zeros of Bessel J(0,x) function

`>`	J:=x->BesselJ(0,x);

`>`	plot(J(x),x=0..30); #plot(J(x),x=10..30);

[Maple Plot]

`>`	mu[1]:=fsolve(J(x),x,2..3);

`>`	for k from 1 to Order do

`>`	*mu[k+1]:=fsolve(J(x),x,%..%+2Pi);od;**

Note: see ?BesselJZeros

Solution

`>`	*A:=k->2/(RBesselJ(1,mu[k]))^2int(rphi(r)BesselJ(0,mu[k]/Rr),r=0..R);**

A := proc (k) options operator, arrow; 2/R^2/BesselJ(1,mu[k])^2*int(r*phi(r)*BesselJ(0,mu[k]/R*r),r = 0 .. R) end proc

`>`	k:='k':

`>`	*sol:=sum(A(k)exp(-(amu[k])^2t/(R^2))BesselJ(0,mu[k]r/R),k=1..Order);**

Plots

`>`	plot(simplify(subs(r=0,sol)),t=0..10);

[Maple Plot]

`>`	plots[animate](sol,r=-R..R,t=0..10,frames=20);

[Maple Plot]

3d animation:

`>`	*K := [seq(0.5k,k=0..15)];**

`>`	m:=nops(%);

`>`	f:=x->cat(`t=`,convert(x,symbol));

`>`	tit:=map(f,K);

`>`	*addcoords(z_cylindrical,[z,r,theta],[rcos(theta),rsin(theta),z]):*

`>`	for k to m do Pc\|\|k:=plot3d(subs(t=K[k],sol),r=0..R,theta=0..2*Pi, coords=z_cylindrical, title=tit[k]):end do:

`>`	plots[display3d]([seq(Pc\|\|k,k=1..m)],insequence=true);

[Maple Plot]

>

While every effort has been made to validate the solutions in this worksheet, Waterloo Maple Inc. and the conevibutors are not responsible for any errors contained and are not liable for any damages resulting from the use of this material.

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Fourier Method for Heat Equation

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Maple Add-Ons

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Improving Retention Rates

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Systems Engineering

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Maple T.A. and Möbius

Education

Industries

Automotive and Aerospace

Robotics

Machine Design & Industrial Automation

Other

Application Areas

Product Pricing

Purchasing

Institutional Student Licensing

Maplesoft Elite Maintenance (EMP)

Support

Product Training

Online Product Help

Webinars & Events

Publications

Content Hubs

Examples & Applications

Community

About Maplesoft

Media Center

User Community

Contact

App Preview:

Fourier Method for Heat Equation