Assinment 3 - Solutions to MAPLE problems.

Question i (b)

 > u1:=sin(2*Pi*x)*sinh(2*Pi*y)/sinh(2*Pi)+8/Pi^3*sum(sinh((2*k+1)*Pi*(2-x))*sin((2*k+1)*Pi*y)/(2*k+1)^3/sinh(2*(2*k+1)*Pi),k=0..10);

 > plot3d(u1,x=0..2,y=0..1,axes=BOXED);

 >

Question 2 (c)

 > u3:=-32/Pi^5*sum(sin((2*k+1)*Pi*x/2)*sin(Pi*y)/((k+1/2)^2+1)/(2*k+1)^3,k=0..10);

 > plot3d(u3,x=0..2,y=0..3,axes=BOXED);

Question 4(a)

 > plot([BesselJ(0,x),BesselJ(1,x)],x=0..20,colour=black);

The zeros of J1 lie between consecutive zeros of J0.

Question 4(b)

The first 10 zeros of J0

 > for m from 1 to 10 do a[m]:=evalf(BesselJZeros(0,m)); od;

Evaluating J1 at the zeros of J0

 > for m from 1 to 10 do b[m]:=evalf(BesselJ(1,a[m]));od;

Question 4(c)

Calculating the coefficients A[m] and B[m].  Note that we are using the values of J1 calculated above.

 > for m from 1 to 10 do A[m]:=2*evalf(int(s*sin(2*Pi*s)*BesselJ(0,a[m]*s),s=0..1))/b[m]^2;od;

 > for m from 1 to 10 do B[m]:=2*evalf(int(s*cos(3*Pi*s/2)*BesselJ(0,a[m]*s),s=0..1))/a[m]/b[m]^2;od;

Question 4(d)

Calculate the tenth partial sum and plot for several values of t.

 > u3:=sum((A[j]*cos(a[j]*t)+B[j]*sin(a[j]*t))*BesselJ(0,a[j]*rho),j=1..10);

 > t:=0;

 > plot3d(u3,rho=0..1,phi=0..2*Pi,coords=z_cylindrical,axes=BOXED);

 > t:=1/2;

 > plot3d(u3,rho=0..1,phi=0..2*Pi,coords=z_cylindrical,axes=BOXED);

 > t:=3/4;

 > plot3d(u3,rho=0..1,phi=0..2*Pi,coords=z_cylindrical,axes=BOXED);

 > t:=1;

 > plot3d(u3,rho=0..1,phi=0..2*Pi,coords=z_cylindrical,axes=BOXED);

 > t:=1/4;

 > plot3d(u3,rho=0..1,phi=0..2*Pi,coords=z_cylindrical,axes=BOXED);

 > t:=1/8;

 > plot3d(u3,rho=0..1,phi=0..2*Pi,coords=z_cylindrical,axes=BOXED);

 > t:=1/16;

 > plot3d(u3,rho=0..1,phi=0..2*Pi,coords=z_cylindrical,axes=BOXED);

 >