Assignment 4- Solutions to MAPLE problems.

Question 1(b)

The first 10 zeros of J2

 > for m from 1 to 10 do a[m]:=evalf(BesselJZeros(2,m)); od;

Evaluating J1 at the zeros of J3

 > for m from 1 to 10 do b[m]:=evalf(BesselJ(3,a[m]));od;

Calculating the coefficient B2[m].  Note that we are using the values of J3 calculated above.

 > for m from 1 to 10 do B2[m]:=2*evalf(int(s*(1-s^2)*BesselJ(2,a[m]*s),s=0..1))/a[m]/b[m]^2;od;

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

 > u3:=sum(B2[j]*sin(2*phi)*sin(a[j]*t)*BesselJ(2,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);

Question 2(b)

 > u2:=1+rho^2*sin(2*phi);

 > plot3d(u2,rho=0..1,phi=0..2*Pi,coords=z_cylindrical,axes=BOXED);

Question 4(b)

First 10 zeros of J1

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

 > for m from 1 to 10 do b[m]:=evalf(BesselJ(2,a[m]));od;

 > for m from 1 to 10 do A1[m]:=-2*evalf(int(s^2*BesselJ(1,a[m]*s),s=0..1))/a[m]^2/b[m]^2;od;

 > u4:=sum(A1[k]*BesselJ(1,a[k]*rho)*cos(phi),k=1..10);

 > plot3d(u4,rho=0..1,phi=0..2*Pi,coords=z_cylindrical,axes=BOXED);

 >

 >