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;

a[1] := 5.135622302

a[2] := 8.417244140

a[3] := 11.61984117

a[4] := 14.79595178

a[5] := 17.95981949

a[6] := 21.11699705

a[7] := 24.27011231

a[8] := 27.42057355

a[9] := 30.56920450

a[10] := 33.71651951

Evaluating J1 at the zeros of J3

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

b[1] := .3396687427

b[2] := -.2713825895

b[3] := .2324431290

b[4] := -.2065407495

b[5] := .1877264152

b[6] := -.1732647388

b[7] := .1617009319

b[8] := -.1521808560

b[9] := .1441657581

b[10] := -.1372968020

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;

B2[1] := .2898165852

B2[2] := .8519971820e-1

B2[3] := .4907526544e-1

B2[4] := .2813989216e-1

B2[5] := .2000270414e-1

B2[6] := .1391737250e-1

B2[7] := .1084347271e-1

B2[8] := .8284433682e-2

B2[9] := .6800806946e-2

B2[10] := .5491101622e-2

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);

u3 := .2898165852*sin(2*phi)*sin(5.135622302*t)*BesselJ(2,5.135622302*rho)+.8519971820e-1*sin(2*phi)*sin(8.417244140*t)*BesselJ(2,8.417244140*rho)+.4907526544e-1*sin(2*phi)*sin(11.61984117*t)*BesselJ(2...
u3 := .2898165852*sin(2*phi)*sin(5.135622302*t)*BesselJ(2,5.135622302*rho)+.8519971820e-1*sin(2*phi)*sin(8.417244140*t)*BesselJ(2,8.417244140*rho)+.4907526544e-1*sin(2*phi)*sin(11.61984117*t)*BesselJ(2...
u3 := .2898165852*sin(2*phi)*sin(5.135622302*t)*BesselJ(2,5.135622302*rho)+.8519971820e-1*sin(2*phi)*sin(8.417244140*t)*BesselJ(2,8.417244140*rho)+.4907526544e-1*sin(2*phi)*sin(11.61984117*t)*BesselJ(2...
u3 := .2898165852*sin(2*phi)*sin(5.135622302*t)*BesselJ(2,5.135622302*rho)+.8519971820e-1*sin(2*phi)*sin(8.417244140*t)*BesselJ(2,8.417244140*rho)+.4907526544e-1*sin(2*phi)*sin(11.61984117*t)*BesselJ(2...
u3 := .2898165852*sin(2*phi)*sin(5.135622302*t)*BesselJ(2,5.135622302*rho)+.8519971820e-1*sin(2*phi)*sin(8.417244140*t)*BesselJ(2,8.417244140*rho)+.4907526544e-1*sin(2*phi)*sin(11.61984117*t)*BesselJ(2...
u3 := .2898165852*sin(2*phi)*sin(5.135622302*t)*BesselJ(2,5.135622302*rho)+.8519971820e-1*sin(2*phi)*sin(8.417244140*t)*BesselJ(2,8.417244140*rho)+.4907526544e-1*sin(2*phi)*sin(11.61984117*t)*BesselJ(2...
u3 := .2898165852*sin(2*phi)*sin(5.135622302*t)*BesselJ(2,5.135622302*rho)+.8519971820e-1*sin(2*phi)*sin(8.417244140*t)*BesselJ(2,8.417244140*rho)+.4907526544e-1*sin(2*phi)*sin(11.61984117*t)*BesselJ(2...
u3 := .2898165852*sin(2*phi)*sin(5.135622302*t)*BesselJ(2,5.135622302*rho)+.8519971820e-1*sin(2*phi)*sin(8.417244140*t)*BesselJ(2,8.417244140*rho)+.4907526544e-1*sin(2*phi)*sin(11.61984117*t)*BesselJ(2...
u3 := .2898165852*sin(2*phi)*sin(5.135622302*t)*BesselJ(2,5.135622302*rho)+.8519971820e-1*sin(2*phi)*sin(8.417244140*t)*BesselJ(2,8.417244140*rho)+.4907526544e-1*sin(2*phi)*sin(11.61984117*t)*BesselJ(2...
u3 := .2898165852*sin(2*phi)*sin(5.135622302*t)*BesselJ(2,5.135622302*rho)+.8519971820e-1*sin(2*phi)*sin(8.417244140*t)*BesselJ(2,8.417244140*rho)+.4907526544e-1*sin(2*phi)*sin(11.61984117*t)*BesselJ(2...

>    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]

Question 2(b)

>    u2:=1+rho^2*sin(2*phi);

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

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

[Maple Plot]

Question 4(b)

First 10 zeros of J1

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

a[1] := 3.831705970

a[2] := 7.015586670

a[3] := 10.17346814

a[4] := 13.32369194

a[5] := 16.47063005

a[6] := 19.61585851

a[7] := 22.76008438

a[8] := 25.90367209

a[9] := 29.04682854

a[10] := 32.18967991

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

b[1] := .4027593957

b[2] := -.3001157525

b[3] := .2497048768

b[4] := -.2183594071

b[5] := .1964653715

b[6] := -.1800633754

b[7] := .1671846005

b[8] := -.1567249862

b[9] := .1480111099

b[10] := -.1406057982

>    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;

A1[1] := -.8826891498e-1

A1[2] := .1929964061e-1

A1[3] := -.7606692902e-2

A1[4] := .3872436822e-2

A1[5] := -.2278312770e-2

A1[6] := .1471575835e-2

A1[7] := -.1014640329e-2

A1[8] := .7341887986e-3

A1[9] := -.5513654102e-3

A1[10] := .4264585156e-3

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

u4 := -.8826891498e-1*BesselJ(1,3.831705970*rho)*cos(phi)+.1929964061e-1*BesselJ(1,7.015586670*rho)*cos(phi)-.7606692902e-2*BesselJ(1,10.17346814*rho)*cos(phi)+.3872436822e-2*BesselJ(1,13.32369194*rho)...
u4 := -.8826891498e-1*BesselJ(1,3.831705970*rho)*cos(phi)+.1929964061e-1*BesselJ(1,7.015586670*rho)*cos(phi)-.7606692902e-2*BesselJ(1,10.17346814*rho)*cos(phi)+.3872436822e-2*BesselJ(1,13.32369194*rho)...
u4 := -.8826891498e-1*BesselJ(1,3.831705970*rho)*cos(phi)+.1929964061e-1*BesselJ(1,7.015586670*rho)*cos(phi)-.7606692902e-2*BesselJ(1,10.17346814*rho)*cos(phi)+.3872436822e-2*BesselJ(1,13.32369194*rho)...
u4 := -.8826891498e-1*BesselJ(1,3.831705970*rho)*cos(phi)+.1929964061e-1*BesselJ(1,7.015586670*rho)*cos(phi)-.7606692902e-2*BesselJ(1,10.17346814*rho)*cos(phi)+.3872436822e-2*BesselJ(1,13.32369194*rho)...
u4 := -.8826891498e-1*BesselJ(1,3.831705970*rho)*cos(phi)+.1929964061e-1*BesselJ(1,7.015586670*rho)*cos(phi)-.7606692902e-2*BesselJ(1,10.17346814*rho)*cos(phi)+.3872436822e-2*BesselJ(1,13.32369194*rho)...
u4 := -.8826891498e-1*BesselJ(1,3.831705970*rho)*cos(phi)+.1929964061e-1*BesselJ(1,7.015586670*rho)*cos(phi)-.7606692902e-2*BesselJ(1,10.17346814*rho)*cos(phi)+.3872436822e-2*BesselJ(1,13.32369194*rho)...
u4 := -.8826891498e-1*BesselJ(1,3.831705970*rho)*cos(phi)+.1929964061e-1*BesselJ(1,7.015586670*rho)*cos(phi)-.7606692902e-2*BesselJ(1,10.17346814*rho)*cos(phi)+.3872436822e-2*BesselJ(1,13.32369194*rho)...
u4 := -.8826891498e-1*BesselJ(1,3.831705970*rho)*cos(phi)+.1929964061e-1*BesselJ(1,7.015586670*rho)*cos(phi)-.7606692902e-2*BesselJ(1,10.17346814*rho)*cos(phi)+.3872436822e-2*BesselJ(1,13.32369194*rho)...
u4 := -.8826891498e-1*BesselJ(1,3.831705970*rho)*cos(phi)+.1929964061e-1*BesselJ(1,7.015586670*rho)*cos(phi)-.7606692902e-2*BesselJ(1,10.17346814*rho)*cos(phi)+.3872436822e-2*BesselJ(1,13.32369194*rho)...
u4 := -.8826891498e-1*BesselJ(1,3.831705970*rho)*cos(phi)+.1929964061e-1*BesselJ(1,7.015586670*rho)*cos(phi)-.7606692902e-2*BesselJ(1,10.17346814*rho)*cos(phi)+.3872436822e-2*BesselJ(1,13.32369194*rho)...

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

[Maple Plot]

>   

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