MATH3271 - FW03

Assignment 2

Question 1(b)(i)

>    restart;

>    u1:=0.1*sin(2*Pi*x)*cos(2*Pi*t);

u1 := .1*sin(2*Pi*x)*cos(2*Pi*t)

>    for i from 0 to 5 do

>    t:=i/10;

>    s||i:=u1;

>    od;

>   

t := 0

s0 := .1*sin(2*Pi*x)

t := 1/10

s1 := .1*sin(2*Pi*x)*cos(1/5*Pi)

t := 1/5

s2 := .1*sin(2*Pi*x)*cos(2/5*Pi)

t := 3/10

s3 := -.1*sin(2*Pi*x)*cos(2/5*Pi)

t := 2/5

s4 := -.1*sin(2*Pi*x)*cos(1/5*Pi)

t := 1/2

s5 := -.1*sin(2*Pi*x)

>    plot([s0,s1,s2,s3,s4,s5],x=0..1,colour=black);

>    t:='t';


[Maple Plot]

t := 't'

Question 1(b)(ii)

>    u2:=sin(Pi*x)*sin(Pi*t)/(2*Pi)-8/Pi^3*sum(sin(2*k*Pi*x)*sin(2*k*Pi*t)/(2*k-1)^2/(2*k+1)^2,k=1..10):

>    for i from 0 to 5 do

>    t:=i/10:

>    s||i:=u2:

>    od:

>   

>    plot([s0,s1,s2,s3,s4,s5],x=0..1,colour=black);

[Maple Plot]

>    t:='t';

t := 't'

Question 1 (b)(iii)

>    u3:=-16/Pi*sum(sin((2*k+1)*Pi*x)*cos((2*k+1)*Pi*t)/(2*k-1)/(2*k+1)/(2*k+3), k = 0..10) + 5/6/Pi^3*sum(sin(0.4*n*Pi)*sin(n*Pi*x)*sin(n*Pi*t)/n^3,n=1..10):

>    for i from 0 to 10 do

>    t:=i/20:

>    s[i]:=u3:

>    od:

>   

>    plot([s[j] $ j = 0..10],x=0..1,colour=black);

[Maple Plot]

>    t:='t';

t := 't'

Question 3(b)(i)

>    u31:=20-10*x+240/Pi^3*sum(sin((2*k+1)*Pi*x)*exp(-(2*k+1)^2*Pi^2*t)/(2*k+1)^3,k=0..10);

u31 := 20-10*x+240/Pi^3*(sin(Pi*x)*exp(-Pi^2*t)+1/27*sin(3*Pi*x)*exp(-9*Pi^2*t)+1/125*sin(5*Pi*x)*exp(-25*Pi^2*t)+1/343*sin(7*Pi*x)*exp(-49*Pi^2*t)+1/729*sin(9*Pi*x)*exp(-81*Pi^2*t)+1/1331*sin(11*Pi*x)...
u31 := 20-10*x+240/Pi^3*(sin(Pi*x)*exp(-Pi^2*t)+1/27*sin(3*Pi*x)*exp(-9*Pi^2*t)+1/125*sin(5*Pi*x)*exp(-25*Pi^2*t)+1/343*sin(7*Pi*x)*exp(-49*Pi^2*t)+1/729*sin(9*Pi*x)*exp(-81*Pi^2*t)+1/1331*sin(11*Pi*x)...
u31 := 20-10*x+240/Pi^3*(sin(Pi*x)*exp(-Pi^2*t)+1/27*sin(3*Pi*x)*exp(-9*Pi^2*t)+1/125*sin(5*Pi*x)*exp(-25*Pi^2*t)+1/343*sin(7*Pi*x)*exp(-49*Pi^2*t)+1/729*sin(9*Pi*x)*exp(-81*Pi^2*t)+1/1331*sin(11*Pi*x)...
u31 := 20-10*x+240/Pi^3*(sin(Pi*x)*exp(-Pi^2*t)+1/27*sin(3*Pi*x)*exp(-9*Pi^2*t)+1/125*sin(5*Pi*x)*exp(-25*Pi^2*t)+1/343*sin(7*Pi*x)*exp(-49*Pi^2*t)+1/729*sin(9*Pi*x)*exp(-81*Pi^2*t)+1/1331*sin(11*Pi*x)...

>    for i from 0 to 5 do t:=i/20: s[i]:=u31: od:

>    plot([s[j]$j=0..5],x=0..1,colour=black);

[Maple Plot]

>    t:='t';

t := 't'

Question 3 (b)(ii)

>    u32:=20-30/Pi^2*sum(cos(2*k*Pi*x)*exp(-4*k^2*Pi^2*t)/k^2,k=1..5)+40/Pi^2*sum(cos((2*k+1)*Pi*x)*exp(-(2*k+1)^2*Pi^2*t)/(2*k+1)^2,k=0..5):

>    for i from 0 to 5 do t:=i/20: s[i]:=u32: od:

>    plot([s[j]$j=0..5],x=0..1,colour=black);

[Maple Plot]

Question 4

>    for i from 1 to 5 do fsolve(tan(x)+x,x,(2*i-1)*Pi/2..(2*i+1)*Pi/2);od;

2.028757838

4.913180439

7.978665712

11.08553841

14.20743673

Question 5 (c)

>    t:='t';

t := 't'

>    u5:=sin(Pi*x)*sin(2*Pi*y)*cos(sqrt(5)*Pi*t);

u5 := sin(Pi*x)*sin(2*Pi*y)*cos(5^(1/2)*Pi*t)

>    t:=0;

t := 0

>    plot3d(u5,x=0..1,y=0..1);

[Maple Plot]

>    t:=1/sqrt(20);

t := 1/10*5^(1/2)

>    plot3d(u5,x=0..1,y=0..1);

[Maple Plot]

>    t:=1/2;

t := 1/2

>    plot3d(u5,x=0..1,y=0..1);

[Maple Plot]

>