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);

u1 := sin(2*Pi*x)*sinh(2*Pi*y)/sinh(2*Pi)+8/Pi^3*(sinh(Pi*(2-x))*sin(Pi*y)/sinh(2*Pi)+1/27*sinh(3*Pi*(2-x))*sin(3*Pi*y)/sinh(6*Pi)+1/125*sinh(5*Pi*(2-x))*sin(5*Pi*y)/sinh(10*Pi)+1/343*sinh(7*Pi*(2-x))*...
u1 := sin(2*Pi*x)*sinh(2*Pi*y)/sinh(2*Pi)+8/Pi^3*(sinh(Pi*(2-x))*sin(Pi*y)/sinh(2*Pi)+1/27*sinh(3*Pi*(2-x))*sin(3*Pi*y)/sinh(6*Pi)+1/125*sinh(5*Pi*(2-x))*sin(5*Pi*y)/sinh(10*Pi)+1/343*sinh(7*Pi*(2-x))*...
u1 := sin(2*Pi*x)*sinh(2*Pi*y)/sinh(2*Pi)+8/Pi^3*(sinh(Pi*(2-x))*sin(Pi*y)/sinh(2*Pi)+1/27*sinh(3*Pi*(2-x))*sin(3*Pi*y)/sinh(6*Pi)+1/125*sinh(5*Pi*(2-x))*sin(5*Pi*y)/sinh(10*Pi)+1/343*sinh(7*Pi*(2-x))*...
u1 := sin(2*Pi*x)*sinh(2*Pi*y)/sinh(2*Pi)+8/Pi^3*(sinh(Pi*(2-x))*sin(Pi*y)/sinh(2*Pi)+1/27*sinh(3*Pi*(2-x))*sin(3*Pi*y)/sinh(6*Pi)+1/125*sinh(5*Pi*(2-x))*sin(5*Pi*y)/sinh(10*Pi)+1/343*sinh(7*Pi*(2-x))*...
u1 := sin(2*Pi*x)*sinh(2*Pi*y)/sinh(2*Pi)+8/Pi^3*(sinh(Pi*(2-x))*sin(Pi*y)/sinh(2*Pi)+1/27*sinh(3*Pi*(2-x))*sin(3*Pi*y)/sinh(6*Pi)+1/125*sinh(5*Pi*(2-x))*sin(5*Pi*y)/sinh(10*Pi)+1/343*sinh(7*Pi*(2-x))*...

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

[Maple Plot]

>   

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);

u3 := -32/Pi^5*(4/5*sin(1/2*Pi*x)*sin(Pi*y)+4/351*sin(3/2*Pi*x)*sin(Pi*y)+4/3625*sin(5/2*Pi*x)*sin(Pi*y)+4/18179*sin(7/2*Pi*x)*sin(Pi*y)+4/61965*sin(9/2*Pi*x)*sin(Pi*y)+4/166375*sin(11/2*Pi*x)*sin(Pi*y...
u3 := -32/Pi^5*(4/5*sin(1/2*Pi*x)*sin(Pi*y)+4/351*sin(3/2*Pi*x)*sin(Pi*y)+4/3625*sin(5/2*Pi*x)*sin(Pi*y)+4/18179*sin(7/2*Pi*x)*sin(Pi*y)+4/61965*sin(9/2*Pi*x)*sin(Pi*y)+4/166375*sin(11/2*Pi*x)*sin(Pi*y...
u3 := -32/Pi^5*(4/5*sin(1/2*Pi*x)*sin(Pi*y)+4/351*sin(3/2*Pi*x)*sin(Pi*y)+4/3625*sin(5/2*Pi*x)*sin(Pi*y)+4/18179*sin(7/2*Pi*x)*sin(Pi*y)+4/61965*sin(9/2*Pi*x)*sin(Pi*y)+4/166375*sin(11/2*Pi*x)*sin(Pi*y...
u3 := -32/Pi^5*(4/5*sin(1/2*Pi*x)*sin(Pi*y)+4/351*sin(3/2*Pi*x)*sin(Pi*y)+4/3625*sin(5/2*Pi*x)*sin(Pi*y)+4/18179*sin(7/2*Pi*x)*sin(Pi*y)+4/61965*sin(9/2*Pi*x)*sin(Pi*y)+4/166375*sin(11/2*Pi*x)*sin(Pi*y...

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

[Maple Plot]

Question 4(a)

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

[Maple Plot]

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;

a[1] := 2.404825558

a[2] := 5.520078110

a[3] := 8.653727913

a[4] := 11.79153444

a[5] := 14.93091771

a[6] := 18.07106397

a[7] := 21.21163663

a[8] := 24.35247153

a[9] := 27.49347913

a[10] := 30.63460647

Evaluating J1 at the zeros of J0

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

b[1] := .5191474972

b[2] := -.3402648066

b[3] := .2714522999

b[4] := -.2324598313

b[5] := .2065464331

b[6] := -.1877288030

b[7] := .1732658942

b[8] := -.1617015507

b[9] := .1521812138

b[10] := -.1441659777

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;

A[1] := -.4849645420e-1

A[2] := 1.873089122

A[3] := -1.107436953

A[4] := -.1709462688

A[5] := -.1453154235

A[6] := -.5869626100e-1

A[7] := -.5944045266e-1

A[8] := -.3073755420e-1

A[9] := -.3271341972e-1

A[10] := -.1913088671e-1

>    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;

B[1] := -.2789337882

B[2] := .3173576708

B[3] := -.1211711423e-1

B[4] := .2934988466e-2

B[5] := -.1125484358e-2

B[6] := .5398827062e-3

B[7] := -.2965381410e-3

B[8] := .1785251499e-3

B[9] := -.1148825410e-3

B[10] := .7777785414e-4

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);

u3 := (-.4849645420e-1*cos(2.404825558*t)-.2789337882*sin(2.404825558*t))*BesselJ(0,2.404825558*rho)+(1.873089122*cos(5.520078110*t)+.3173576708*sin(5.520078110*t))*BesselJ(0,5.520078110*rho)+(-1.10743...
u3 := (-.4849645420e-1*cos(2.404825558*t)-.2789337882*sin(2.404825558*t))*BesselJ(0,2.404825558*rho)+(1.873089122*cos(5.520078110*t)+.3173576708*sin(5.520078110*t))*BesselJ(0,5.520078110*rho)+(-1.10743...
u3 := (-.4849645420e-1*cos(2.404825558*t)-.2789337882*sin(2.404825558*t))*BesselJ(0,2.404825558*rho)+(1.873089122*cos(5.520078110*t)+.3173576708*sin(5.520078110*t))*BesselJ(0,5.520078110*rho)+(-1.10743...
u3 := (-.4849645420e-1*cos(2.404825558*t)-.2789337882*sin(2.404825558*t))*BesselJ(0,2.404825558*rho)+(1.873089122*cos(5.520078110*t)+.3173576708*sin(5.520078110*t))*BesselJ(0,5.520078110*rho)+(-1.10743...
u3 := (-.4849645420e-1*cos(2.404825558*t)-.2789337882*sin(2.404825558*t))*BesselJ(0,2.404825558*rho)+(1.873089122*cos(5.520078110*t)+.3173576708*sin(5.520078110*t))*BesselJ(0,5.520078110*rho)+(-1.10743...
u3 := (-.4849645420e-1*cos(2.404825558*t)-.2789337882*sin(2.404825558*t))*BesselJ(0,2.404825558*rho)+(1.873089122*cos(5.520078110*t)+.3173576708*sin(5.520078110*t))*BesselJ(0,5.520078110*rho)+(-1.10743...
u3 := (-.4849645420e-1*cos(2.404825558*t)-.2789337882*sin(2.404825558*t))*BesselJ(0,2.404825558*rho)+(1.873089122*cos(5.520078110*t)+.3173576708*sin(5.520078110*t))*BesselJ(0,5.520078110*rho)+(-1.10743...
u3 := (-.4849645420e-1*cos(2.404825558*t)-.2789337882*sin(2.404825558*t))*BesselJ(0,2.404825558*rho)+(1.873089122*cos(5.520078110*t)+.3173576708*sin(5.520078110*t))*BesselJ(0,5.520078110*rho)+(-1.10743...
u3 := (-.4849645420e-1*cos(2.404825558*t)-.2789337882*sin(2.404825558*t))*BesselJ(0,2.404825558*rho)+(1.873089122*cos(5.520078110*t)+.3173576708*sin(5.520078110*t))*BesselJ(0,5.520078110*rho)+(-1.10743...
u3 := (-.4849645420e-1*cos(2.404825558*t)-.2789337882*sin(2.404825558*t))*BesselJ(0,2.404825558*rho)+(1.873089122*cos(5.520078110*t)+.3173576708*sin(5.520078110*t))*BesselJ(0,5.520078110*rho)+(-1.10743...
u3 := (-.4849645420e-1*cos(2.404825558*t)-.2789337882*sin(2.404825558*t))*BesselJ(0,2.404825558*rho)+(1.873089122*cos(5.520078110*t)+.3173576708*sin(5.520078110*t))*BesselJ(0,5.520078110*rho)+(-1.10743...
u3 := (-.4849645420e-1*cos(2.404825558*t)-.2789337882*sin(2.404825558*t))*BesselJ(0,2.404825558*rho)+(1.873089122*cos(5.520078110*t)+.3173576708*sin(5.520078110*t))*BesselJ(0,5.520078110*rho)+(-1.10743...
u3 := (-.4849645420e-1*cos(2.404825558*t)-.2789337882*sin(2.404825558*t))*BesselJ(0,2.404825558*rho)+(1.873089122*cos(5.520078110*t)+.3173576708*sin(5.520078110*t))*BesselJ(0,5.520078110*rho)+(-1.10743...
u3 := (-.4849645420e-1*cos(2.404825558*t)-.2789337882*sin(2.404825558*t))*BesselJ(0,2.404825558*rho)+(1.873089122*cos(5.520078110*t)+.3173576708*sin(5.520078110*t))*BesselJ(0,5.520078110*rho)+(-1.10743...
u3 := (-.4849645420e-1*cos(2.404825558*t)-.2789337882*sin(2.404825558*t))*BesselJ(0,2.404825558*rho)+(1.873089122*cos(5.520078110*t)+.3173576708*sin(5.520078110*t))*BesselJ(0,5.520078110*rho)+(-1.10743...
u3 := (-.4849645420e-1*cos(2.404825558*t)-.2789337882*sin(2.404825558*t))*BesselJ(0,2.404825558*rho)+(1.873089122*cos(5.520078110*t)+.3173576708*sin(5.520078110*t))*BesselJ(0,5.520078110*rho)+(-1.10743...
u3 := (-.4849645420e-1*cos(2.404825558*t)-.2789337882*sin(2.404825558*t))*BesselJ(0,2.404825558*rho)+(1.873089122*cos(5.520078110*t)+.3173576708*sin(5.520078110*t))*BesselJ(0,5.520078110*rho)+(-1.10743...
u3 := (-.4849645420e-1*cos(2.404825558*t)-.2789337882*sin(2.404825558*t))*BesselJ(0,2.404825558*rho)+(1.873089122*cos(5.520078110*t)+.3173576708*sin(5.520078110*t))*BesselJ(0,5.520078110*rho)+(-1.10743...

>    addcoords(z_cylindrical,[z,rho,phi],[rho*cos(phi),rho*sin(phi),z]);

>    t:=0;

t := 0

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

[Maple Plot]

>    t:=1/2;

t := 1/2

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

[Maple Plot]

>    t:=3/4;

t := 3/4

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

[Maple Plot]

>    t:=1;

t := 1

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

[Maple Plot]

>    t:=1/4;

t := 1/4

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

[Maple Plot]

>    t:=1/8;

t := 1/8

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

[Maple Plot]

>    t:=1/16;

t := 1/16

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

[Maple Plot]

>