Application Center - Maplesoft

Contact Maplesoft Request Quote

Submit your work

Application Center Applications Cauchy problems for wave equations (updated to Maple 7) Preview

App Preview:

Cauchy problems for wave equations (updated to Maple 7)

You can switch back to the summary page by clicking here.

Learn about Maple

Download Application

wave_eqc.mws

Cauchy problem for 1d wave equation (1)
Cauchy problem for 1d wave equation (2)
Cauchy problem for 2d wave equation
Cauchy problem for 3d wave equation
Non-homogenous wave equation
Decomposition method
Radial-symmetric solution

Solving Cauchy problems for wave equations

by Aleksas Domarkas

Vilnius University, Faculty of Mathematics and Informatics,

Naugarduko 24, Vilnius, Lithuania

aleksas@ieva.mif.vu.lt

NOTE: In this session we solve Cauchy problems for wave equations.

Cauchy problem for 1d wave equation (1)

Problem

from M.Kawski

ftp://math.la.asu.edu/pub/kawski/MAPLE/362/dAlembert2hats.mws

`>`	restart;

`>`	*eq:=diff(u(x,t),t,t)=4diff(u(x,t),x,x);#x>-infinity,x<infinity,t>0;**

`>`	ic:=u(x,0)=u0(x);ict:=D[2](u)(x,0)=u1(x);#x>-infinity,x<infinity;

eq := diff(u(x,t),`$`(t,2)) = 4*diff(u(x,t),`$`(x,2))

`>`	hat2:=x->1-abs(abs(x)-2);

`>`	ramp:=x->(x+abs(x))/2;

`>`	u0:=ramp@hat2;

`>`	u1:=0;

`>`	plot(u0,-4..4,-1..2,scaling=constrained,axes=boxed);

[Maple Plot]

`>`	convert(u0(x),piecewise);

PIECEWISE([0, x <= -3],[x+3, x <= -2],[-x-1, x < -1],[0, x <= 1],[x-1, x < 2],[-x+3, x < 3],[0, 3 <= x])

`>`	u0:=unapply(convert(u0(x),Heaviside),x);

>

Solving(1 method)

We use d'Alembert's formula for solution:

u(x,t) = (u0(x-a*t)+u0(x+a*t))/2+int(u1(xi),xi = x-a*t .. x+a*t)/(2*a)

> a:=2:

`>`	*sol1:=(u0(x-at)+u0(x+at))/2;*

>

Solving(2 method)

`>`	assume(t>0);

`>`	with(inttrans):

`>`	fourier(eq,x,w);

diff(fourier(u(x,t),x,w),`$`(t,2)) = -4*w^2*fourier(u(x,t),x,w)

`>`	ode:=subs(fourier(u(x,t),x,w)=s(t),%);

ode := diff(s(t),`$`(t,2)) = -4*w^2*s(t)

`>`	dsolve({ode,s(0)=fourier(u0(x),x,w), D(s)(0)=fourier(u1(x),x,w)},s(t));

`>`	sol2:=simplify(invfourier(rhs(%),w,x));

>

Checking the Solution

sol1=sol2 ???

`>`	simplify(sol1-sol2);

`>`	u:=unapply(sol1,(x,t)):

`>`	simplify(rhs(eq)-lhs(eq));

`>`	simplify(rhs(ic)-lhs(ic));

`>`	simplify(rhs(ict)-lhs(ict));

Plots

`>`	plot3d(sol1,x=-10..10,t=0..5,orientation=[-60,20],axes=framed);

[Maple Plot]

`>`	with(plots):with(plottools):

Warning, the name changecoords has been redefined

Warning, the name arrow has been redefined

Animation, created by M.Kawski:

> display([
seq(
display([
textplot([1.4,1.4,cat(`t = `,convert(tt,string))],
'align=RIGHT',font=[TIMES,ROMAN,18]),
plot(subs(t=tt,sol1),x=-6..6,-1..2,
thickness=3)
]),
tt=[seq(evalf(k/20,3),k=0..100)])
],insequence=true,
titlefont=[TIMES,BOLD,16],
title=`Animation: d'Alembert's solution`);

[Maple Plot]

>

Cauchy problem for 1d wave equation (2)

Problem

`>`	restart;

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

`>`	ic:=u(x,0)=u0(x);ict:=D[2](u)(x,0)=u1(x);

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

`>`	u0:=unapply(randpoly([x]),x);

`>`	u1:=unapply(randpoly([x]),x);

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

Solving(1 method)

We use d'Alembert's formula for solution:

u(x,t) = (u0(x-a*t)+u0(x+a*t))/2+int(u1(xi),xi = x-a*t .. x+a*t)/(2*a)

`>`	*(u0(x-at)+u0(x+at))/2+int(u1(xi),xi = x-at .. x+at)/(2a);**

`>`	sol1:=simplify(%);

>

Solving(2 method)

`>`	L:=f->diff(f,x,x);

L := proc (f) options operator, arrow; diff(f,x,x) end proc

`>`	d:=max(degree(u0(x)),degree(u1(x)));

`>`	Sum(a^(2k)t^(2k)/(2k)!(L@@k)(u0(x))+ a^(2k)t^(2k+1)/(2k+1)!(L@@k)(u1(x)),k=0..d);

Sum(a^(2*k)*t^(2*k)/(2*k)!*`@@`(L,k)(-85*x^5-55*x^4-37*x^3-35*x^2+97*x+50)+a^(2*k)*t^(2*k+1)/(2*k+1)!*`@@`(L,k)(79*x^5+56*x^4+49*x^3+63*x^2+57*x-59),k = 0 .. 5)

`>`	sol2:=simplify(value(%));

`>`	is(sol1=sol2);

>

Solving(3 method)

`>`	with(inttrans):

`>`	fourier(eq,x,w);

diff(fourier(u(x,t),x,w),`$`(t,2)) = -a^2*w^2*fourier(u(x,t),x,w)

`>`	ode:=subs(fourier(u(x,t),x,w)=s(t),%);

ode := diff(s(t),`$`(t,2)) = -a^2*w^2*s(t)

`>`	dsolve({ode,s(0)=fourier(u0(x),x,w), D(s)(0)=fourier(u1(x),x,w)},s(t));

`>`	sol3:=invfourier(rhs(%),w,x);

`>`	is(sol1=sol3);

Checking the Solution

`>`	u:=unapply(sol1,(x,t)):

`>`	simplify(rhs(eq)-lhs(eq));

`>`	simplify(rhs(ic)-lhs(ic));

`>`	simplify(rhs(ict)-lhs(ict));

>

Cauchy problem for 2d wave equation

Problem( 2d wave equation)

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

`>`	*eq:=diff(u(x,y,t),t,t)=a^2laplacian(u(x,y,t),[x,y]);**

`>`	ic:=u(x,y,0)=u0(x,y);ict:=D[3](u)(x,y,0)=u1(x,y);

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

`>`	u0:=unapply(randpoly([x,y]),x,y);

`>`	u1:=unapply(randpoly([x,y]),x,y);

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

Solving(1 method)

We use Poisson formula for solution:

u(x,y,t) = Int(Int(u1(xi)/sqrt(a^2*t^2-r^2),xi[1] = B .. ``),xi[2] = `` .. ``)/(2*Pi*a)+Diff(Int(Int(u0(xi)/sqrt(a^2*t^2-r^2),xi[1] = B .. ``),xi[2] = `` .. ``),t)/(2*Pi*a)

where B={( ): (xi[1]-x)^2+(xi[2]-y)^2 <= (a*t)^2 }.

For computing integrals we use polar coordinates:

`>`	*T:={xi=x+rcos(phi),eta=y+rsin(phi)};*

`>`	subs(T,1/(2Pia)* Doubleint(u1(xi,eta)/sqrt((at)^2-r^2)r, phi=0..2Pi,r=0..at)+1/(2Pia)Diff( Doubleint(u0(xi,eta)/sqrt((at)^2-r^2)r, phi=0..2Pi,r=0..a*t),t));

1/2/Pi/a*Int(Int((54-5*y-5*r*sin(phi)+99*(x+r*cos(phi))^3-61*(x+r*cos(phi))^2*(y+r*sin(phi))-50*(x+r*cos(phi))^3*(y+r*sin(phi))-12*(x+r*cos(phi))^5)/(a^2*t^2-r^2)^(1/2)*r,phi = 0 .. 2*Pi),r = 0 .. a*t)...

`>`	sol1:=simplify(value(%));

Solving(2 method)

`>`	L:=f->laplacian(f,[x,y]);

`>`	d:=max(degree(u0(x,y)),degree(u1(x,y)));

`>`	Sum(a^(2k)t^(2k)/(2k)!(L@@k)(u0(x,y))+ a^(2k)t^(2k+1)/(2k+1)!(L@@k)(u1(x,y)),k=0..d);

Sum(a^(2*k)*t^(2*k)/(2*k)!*`@@`(L,k)(79*y+56*x*y+49*x^3+63*x^3*y^2+57*x^2*y^3-59*y^5)+a^(2*k)*t^(2*k+1)/(2*k+1)!*`@@`(L,k)(54-5*y+99*x^3-61*x^2*y-50*x^3*y-12*x^5),k = 0 .. 5)

`>`	sol2:=simplify(value(%));

`>`	is(sol1=sol2);

>

Solving(3 method)

`>`	with(inttrans);

`>`	fourier(eq,x,w1);

diff(fourier(u(x,y,t),x,w1),`$`(t,2)) = a^2*(-w1^2*fourier(u(x,y,t),x,w1)+diff(fourier(u(x,y,t),x,w1),`$`(y,2)))

`>`	fourier(%,y,w2);

diff(fourier(fourier(u(x,y,t),x,w1),y,w2),`$`(t,2)) = a^2*(-w1^2*fourier(fourier(u(x,y,t),x,w1),y,w2)-w2^2*fourier(fourier(u(x,y,t),x,w1),y,w2))

`>`	subs(fourier(fourier(u(x,y,t),x,w1),y,w2)=s(t),%);

diff(s(t),`$`(t,2)) = a^2*(-w1^2*s(t)-w2^2*s(t))

`>`	dsolve({%,s(0)=fourier(fourier(u0(x,y),x,w1),y,w2), D(s)(0)=fourier(fourier(u1(x,y),x,w1),y,w2)},s(t));

`>`	invfourier(rhs(%),w2,y);

-2*Pi*(-54*sin(a*sqrt(w1^2)*t)*w1^4*Dirac(w1)+5*sin(a*sqrt(w1^2)*t)*w1^4*Dirac(w1)*y+99*I*sin(a*sqrt(w1^2)*t)*w1^4*Dirac(3,w1)-61*sin(a*sqrt(w1^2)*t)*w1^4*Dirac(2,w1)*y-50*I*sin(a*sqrt(w1^2)*t)*w1^4*Di...

`>`	sol3:=invfourier(%,w1,x);

`>`	is(sol1=sol3);

Checking the Solution

`>`	u:=unapply(sol1,(x,y,t)):

`>`	simplify(rhs(eq)-lhs(eq));

`>`	simplify(rhs(ic)-lhs(ic));

`>`	simplify(rhs(ict)-lhs(ict));

>

Cauchy problem for 3d wave equation

Problem

`>`	restart;with(linalg):with(student):

Warning, the protected names norm and trace have been redefined and unprotected

`>`	*eq:=diff(u(x,y,z,t),t,t)=a^2laplacian(u(x,y,z,t),[x,y,z]);**

`>`	ic:=u(x,y,z,0)=u0(x,y,z);ict:=D[4](u)(x,y,z,0)=u1(x,y,z);

eq := diff(u(x,y,z,t),`$`(t,2)) = a^2*(diff(u(x,y,z,t),`$`(x,2))+diff(u(x,y,z,t),`$`(y,2))+diff(u(x,y,z,t),`$`(z,2)))

`>`	u0:=unapply(randpoly([x,y,z]),x,y,z);

`>`	u1:=unapply(randpoly([x,y,z]),x,y,z);

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

Solving(1 method)

We use Kirchhoff formula for solution:

u(x,y,z,t) = Int(u1,S = S .. ``)/(4*Pi*a^2*t)+Diff(Int(u0,S = S .. ``)/t,t)/(4*Pi*a^2)

where S={( ): (xi[1]-x)^2+(xi[2]-y)^2+(xi[3]-z)^2 = (a*t)^2 }.

`>`	*sol:=Int(u1,S)/(4Pia^2t)+Diff(Int(u0,S)/t,t)/(4Pia^2);**

sol := 1/4*Int(u1,S)/Pi/a^2/t+1/4*Diff(Int(u0,S)/t,t)/Pi/a^2

For computing integrals we use spherical coordinates:

`>`	T:={xi[1]=x+rcos(phi)sin(theta),xi[2]=y+rsin(phi)sin(theta), xi[3]=z+r*cos(theta)};

`>`	linalg[jacobian]([x+rcos(phi)sin(theta),y+rsin(phi)sin(theta), z+r*cos(theta)],[phi,r,theta]);

matrix([[-r*sin(phi)*sin(theta), cos(phi)*sin(theta), r*cos(phi)*cos(theta)], [r*cos(phi)*sin(theta), sin(phi)*sin(theta), r*sin(phi)*cos(theta)], [0, cos(theta), -r*sin(theta)]])

`>`	J:=simplify(det(%));

`>`	subs(Int(u1,S)=Doubleint(subs(T,u1(xi[1],xi[2],xi[3]))J, theta=0..Pi,phi=0..2Pi), Int(u0,S)=Doubleint(subs(T,u0(xi[1],xi[2],xi[3]))J, theta=0..Pi,phi=0..2Pi), r=a*t,sol);

1/4*Int(Int((77*(y+a*t*sin(phi)*sin(theta))*(z+a*t*cos(theta))^2+66*(x+a*t*cos(phi)*sin(theta))^3*(y+a*t*sin(phi)*sin(theta))+54*(x+a*t*cos(phi)*sin(theta))*(y+a*t*sin(phi)*sin(theta))*(z+a*t*cos(theta...

`>`	sol1:=simplify(value(%));

Solving(2 method)

`>`	L:=f->laplacian(f,[x,y,z]);

`>`	d:=max(degree(u0(x,y,z)),degree(u1(x,y,z)));

`>`	Sum(a^(2k)t^(2k)/(2k)!(L@@k)(u0(x,y,z))+ a^(2k)t^(2k+1)/(2k+1)!(L@@k)(u1(x,y,z)),k=0..d);

Sum(a^(2*k)*t^(2*k)/(2*k)!*`@@`(L,k)(79*y+56*x^2*z^2+49*x*y^2*z+63*y^2*z^2+57*x^3*y^2-59*x^2*y^3)+a^(2*k)*t^(2*k+1)/(2*k+1)!*`@@`(L,k)(77*y*z^2+66*x^3*y+54*x*y*z^2-5*x*z^3+99*y^2*z^2-61*x*z^4),k = 0 .....

`>`	sol2:=simplify(value(%));

`>`	is(sol1=sol2);

>

Solving(3 method)

`>`	with(inttrans):

Warning, the name hilbert has been redefined

`>`	fourier(eq,x,w1);

diff(fourier(u(x,y,z,t),x,w1),`$`(t,2)) = a^2*(-w1^2*fourier(u(x,y,z,t),x,w1)+diff(fourier(u(x,y,z,t),x,w1),`$`(y,2))+diff(fourier(u(x,y,z,t),x,w1),`$`(z,2)))

`>`	fourier(%,y,w2);

diff(fourier(fourier(u(x,y,z,t),x,w1),y,w2),`$`(t,2)) = a^2*(-w1^2*fourier(fourier(u(x,y,z,t),x,w1),y,w2)-w2^2*fourier(fourier(u(x,y,z,t),x,w1),y,w2)+diff(fourier(fourier(u(x,y,z,t),x,w1),y,w2),`$`(z,2...

`>`	fourier(%,z,w3);

diff(fourier(fourier(fourier(u(x,y,z,t),x,w1),y,w2),z,w3),`$`(t,2)) = a^2*(-w1^2*fourier(fourier(fourier(u(x,y,z,t),x,w1),y,w2),z,w3)-w2^2*fourier(fourier(fourier(u(x,y,z,t),x,w1),y,w2),z,w3)-w3^2*four...

`>`	ode:=subs(fourier(fourier(fourier(u(x,y,z,t),x,w1),y,w2),z,w3)=s(t),%);

ode := diff(s(t),`$`(t,2)) = a^2*(-w1^2*s(t)-w2^2*s(t)-w3^2*s(t))

`>`	dsolve({ode,s(0)=fourier(fourier(fourier(u0(x,y,z),x,w1),y,w2),z,w3), D(s)(0)=fourier(fourier(fourier(u1(x,y,z),x,w1),y,w2),z,w3)},s(t));

`>`	invfourier(rhs(%),w3,z):

`>`	invfourier(%,w2,y):

`>`	sol3:=invfourier(%,w1,x);

`>`	is(sol1=sol3);

>

Checking the Solution

`>`	u:=unapply(sol1,(x,y,z,t)):

`>`	simplify(rhs(eq)-lhs(eq));

`>`	simplify(rhs(ic)-lhs(ic));

`>`	simplify(rhs(ict)-lhs(ict));

>

Non-homogenous wave equation

Problem

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

`>`	*eq:=diff(u(x,y,z,t),t,t)-a^2laplacian(u(x,y,z,t),[x,y,z])=F(x,y,z,t);#-infty<x,y,z<infty,t>0**

`>`	ic:=u(x,y,z,0)=0;#-infty<x,y,z<infty

`>`	ict:=D[4](u)(x,y,z,0)=0;#-infty<x,y,z<infty

The function F(x,y,z,t) may be polynomial respect to x,y,z .

eq := diff(u(x,y,z,t),`$`(t,2))-a^2*(diff(u(x,y,z,t),`$`(x,2))+diff(u(x,y,z,t),`$`(y,2))+diff(u(x,y,z,t),`$`(z,2))) = F(x,y,z,t)

`>`	*randpoly([x,y,z,t,tsin(t),cos(t),exp(t)]):**

`>`	F:=unapply(%,[x,y,z,t]);

procedure

> wsol:=proc(u)
option `Copyright Aleksas Domarkas, 2002`;
local dsol;
dsol:=proc(F)
local g,f,d,L;
if has(F, t) and not has(F,[x,y,z]) then g:=unapply(F,t); f:=1;
elif not has(F,t) then g:=1;f:=F;
else f:=select(has,F,[x,y,z]); g:=unapply(F/f,t); end if;
d:=max(degree(f),1);
L:=f->linalg[laplacian](f,[x,y,z]);
sum(a^(2*k)/(2*k+1)!*(L@@k)(f)*Int((t-tau)^(2*k+1)*g(tau),tau=0..t),k=0..d);value(%);
end proc:
if type(u,`+`) then map(dsol,u) else dsol(u) end if:
end proc:

solution

`>`	sol:=wsol(F(x,y,z,t));

>

test

`>`	u:=unapply(sol,(x,y,z,t)):

`>`	simplify(rhs(eq)-lhs(eq)):if %<>0 then combine(%,trig) else % fi;

`>`	simplify(rhs(ic)-lhs(ic));

`>`	simplify(rhs(ict)-lhs(ict));

>

Decomposition method

`>`	restart;with(linalg,laplacian):with(inttrans,fourier,invfourier):

Proble m

From V.S.Vladimirov(ed.), Exercises book on Equations of Mathematical Physics, Nauka, Moscow, 1982(in Russian) 12.38.7.

`>`	*eq:=diff(u(x,y,z,t),t,t)-a^2laplacian(u(x,y,z,t),[x,y,z])=exp(z)sin(y)cos(x);#-infty<x,y,z<infty,t>0**

`>`	*ic:=u(x,y,z,0)=x^2exp(y+z);#-infty<x,y,z<infty**

`>`	*ict:=D[4](u)(x,y,z,0)=sin(x)exp(y+z);#-infty<x,y,z<infty**

eq := diff(u(x,y,z,t),`$`(t,2))-a^2*(diff(u(x,y,z,t),`$`(x,2))+diff(u(x,y,z,t),`$`(y,2))+diff(u(x,y,z,t),`$`(z,2))) = exp(z)*sin(y)*cos(x)

`>`	eq1:=eq;

`>`	ic1:=u(x,y,z,0)=0;

`>`	ict1:=D[4](u)(x,y,z,0)=0;

eq1 := diff(u(x,y,z,t),`$`(t,2))-a^2*(diff(u(x,y,z,t),`$`(x,2))+diff(u(x,y,z,t),`$`(y,2))+diff(u(x,y,z,t),`$`(z,2))) = exp(z)*sin(y)*cos(x)

Solving of I

`>`	*subs(u(x,y,z,t)=v(t)rhs(eq1),eq1);**

diff(v(t)*exp(z)*sin(y)*cos(x),`$`(t,2))-a^2*(diff(v(t)*exp(z)*sin(y)*cos(x),`$`(x,2))+diff(v(t)*exp(z)*sin(y)*cos(x),`$`(y,2))+diff(v(t)*exp(z)*sin(y)*cos(x),`$`(z,2))) = exp(z)*sin(y)*cos(x)

`>`	simplify(%/rhs(eq1));

diff(v(t),`$`(t,2))+a^2*v(t) = 1

`>`	dsolve({%,v(0)=0,D(v)(0)=0},v(t));

v(t) = -cos(a*t)/a^2+1/(a^2)

`>`	*sol1:=rhs(%)rhs(eq1);**

>

sol1 := (-cos(a*t)/a^2+1/(a^2))*exp(z)*sin(y)*cos(x)

`>`	*eq2:=diff(u(x,y,z,t),t,t)-a^2laplacian(u(x,y,z,t),[x,y,z])=0;**

`>`	ic2:=u(x,y,z,0)=0;

`>`	*ict2:=D[4](u)(x,y,z,0)=sin(x)exp(y+z);**

eq2 := diff(u(x,y,z,t),`$`(t,2))-a^2*(diff(u(x,y,z,t),`$`(x,2))+diff(u(x,y,z,t),`$`(y,2))+diff(u(x,y,z,t),`$`(z,2))) = 0

Solving of II

`>`	*subs(u(x,y,z,t)=v(t)rhs(ict2),eq2);**

diff(v(t)*sin(x)*exp(y+z),`$`(t,2))-a^2*(diff(v(t)*sin(x)*exp(y+z),`$`(x,2))+diff(v(t)*sin(x)*exp(y+z),`$`(y,2))+diff(v(t)*sin(x)*exp(y+z),`$`(z,2))) = 0

`>`	simplify(%/rhs(ict2));

diff(v(t),`$`(t,2))-a^2*v(t) = 0

`>`	dsolve({%,v(0)=0,D(v)(0)=1},v(t));convert(%,trig): combine(%,trig);

v(t) = -1/2/a*exp(-a*t)+1/2/a*exp(a*t)

`>`	*sol2:=rhs(%)rhs(ict2);**

>

sol2 := sinh(a*t)/a*sin(x)*exp(y+z)

III

`>`	eq3:=eq2;

`>`	*ic3:=u(x,y,z,0)=x^2exp(y+z);**

`>`	ict3:=D[4](u)(x,y,z,0)=0;

eq3 := diff(u(x,y,z,t),`$`(t,2))-a^2*(diff(u(x,y,z,t),`$`(x,2))+diff(u(x,y,z,t),`$`(y,2))+diff(u(x,y,z,t),`$`(z,2))) = 0

Solving of III

`>`	*subs(u(x,y,z,t)=v(x,t)exp(y+z),eq3);**

diff(v(x,t)*exp(y+z),`$`(t,2))-a^2*(diff(v(x,t)*exp(y+z),`$`(x,2))+diff(v(x,t)*exp(y+z),`$`(y,2))+diff(v(x,t)*exp(y+z),`$`(z,2))) = 0

`>`	eqn:=simplify(%/exp(y+z));

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

`>`	fourier(eqn,x,w);

diff(fourier(v(x,t),x,w),`$`(t,2))+a^2*w^2*fourier(v(x,t),x,w)-2*a^2*fourier(v(x,t),x,w) = 0

`>`	subs(fourier(v(x,t),x,w)=s(t),%);

diff(s(t),`$`(t,2))+a^2*w^2*s(t)-2*a^2*s(t) = 0

`>`	dsolve({%,s(0)=fourier(x^2,x,w),D(s)(0)=0},s(t));

`>`	subs(s(t)=fourier(v(x,t),x,w),%);

`>`	invfourier(%,w,x);

`>`	convert(%,trig): combine(%,trig);

`>`	*sol3:=rhs(%)exp(y+z);**

>

Solution

`>`	sol:=sol1+sol2+sol3;

sol := (-cos(a*t)/a^2+1/(a^2))*exp(z)*sin(y)*cos(x)+sinh(a*t)/a*sin(x)*exp(y+z)+(1/2*sinh(sqrt(2)*a*t)*a*sqrt(2)*t+cosh(sqrt(2)*a*t)*x^2)*exp(y+z)

Checking the Solution

`>`	u:=unapply(sol,(x,y,z,t)):

`>`	simplify(rhs(eq)-lhs(eq));

`>`	simplify(rhs(ic)-lhs(ic));

`>`	simplify(rhs(ict)-lhs(ict));

>

Radial-symmetric solution

For radial-symmetric solutions of linear elliptic equations see radsymell.mws

Problem

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

[vlad] 12.38.9.

`>`	*eq:=diff(u(x,y,z,t),t,t)-a^2laplacian(u(x,y,z,t),[x,y,z])=0;**

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

`>`	ic:=u(x,y,z,0)=cos((x^2+y^2+z^2)^(1/2));

`>`	ict:=D[4](u)(x,y,z,0)=cos((x^2+y^2+z^2)^(1/2));

Solving problem

`>`	*eqn:=diff(u(r,t),t,t)-a^2expand(laplacian(u(r,t),[r,theta,phi],coords=spherical));**

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

Solution we search in form u(r,t) = v(r,t)/r :

`>`	subs(u(r,t)=v(r,t)/r,eqn);

diff(v(r,t)/r,`$`(t,2))-a^2*(2/r*diff(v(r,t)/r,r)+diff(v(r,t)/r,`$`(r,2)))

`>`	*eq1:=simplify(%r);**

eq1 := diff(v(r,t),`$`(t,2))-a^2*diff(v(r,t),`$`(r,2))

`>`	*ic1:=v(r,0)=rcos(r);**

`>`	*ict1:=D[2](v)(r,0)=rcos(r);**

`>`	phi:=unapply(rhs(ic1),r);

`>`	psi:=unapply(rhs(ict1),r);

By d'Alembert's formula :

`>`	*v(r,t)=(phi(r+at)+phi(r-at))/2+Int(psi(xi),xi=r-at .. r+at)/2/a;*

v(r,t) = 1/2*(r+a*t)*cos(r+a*t)+1/2*(r-a*t)*cos(-r+a*t)+1/2*Int(xi*cos(xi),xi = r-a*t .. r+a*t)/a

`>`	value(rhs(%))/r;

(1/2*(r+a*t)*cos(r+a*t)+1/2*(r-a*t)*cos(-r+a*t)+1/2*(cos(r+a*t)+sin(r+a*t)*r+sin(r+a*t)*a*t-cos(-r+a*t)+sin(-r+a*t)*r-sin(-r+a*t)*a*t)/a)/r

`>`	expand(%);

`>`	collect(%,{sin(r),cos(r)});

`>`	applyop(collect,[1,1],%,r):applyop(collect,[2,1],%,r);

(1/a*sin(a*t)+cos(a*t))*cos(r)+(t*cos(a*t)-a*t*sin(a*t)-1/a*sin(a*t))/r*sin(r)

Solution

`>`	sol:=subs(r=sqrt(x^2+y^2+z^2),%);

sol := (1/a*sin(a*t)+cos(a*t))*cos(sqrt(x^2+y^2+z^2))+(t*cos(a*t)-a*t*sin(a*t)-1/a*sin(a*t))/(x^2+y^2+z^2)^(1/2)*sin(sqrt(x^2+y^2+z^2))

>

Checking the Solution

`>`	u:=unapply(sol,(x,y,z,t)):

`>`	simplify(rhs(eq)-lhs(eq));

`>`	simplify(rhs(ic)-lhs(ic));

`>`	simplify(rhs(ict)-lhs(ict));

>

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.

Back to contents

Maple

Maple Add-Ons

Student Success Platform

Improving Retention Rates

MapleSim

MapleSim Add-Ons

Systems Engineering

Consulting Services

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:

Cauchy problems for wave equations (updated to Maple 7)

Maple

Powerful math software that is easy to use

Maple Add-Ons

Student Success Platform

Improving Retention Rates

MapleSim

Advanced System Level Modeling

MapleSim Add-Ons

Systems Engineering

Consulting Services

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:

Cauchy problems for wave equations (updated to Maple 7)