Solução de Equações Diferenciais por Séries de Fourier de T(x) = dado inicial

por

Milton Procópio de Borba

> restart;

> Ate:=20:

> T:=piecewise(x>=0 and x<=1,60*x, x>=1 and x<=2,20*x+40, x>=2 and x<=3,-80*x+240);

T := PIECEWISE([60*x, -x <= 0 and x-1 <= 0],[20*x+4...

> plot(T,x=0..3,color=blue);

[Maple Plot]

> s[n]:=sin(n*Pi*x/3);

> bn:=2*int(T*s[n],x=0..3)/3;

s[n] := sin(1/3*n*Pi*x)

bn := 120*(-4*sin(n*Pi)+5*sin(2/3*n*Pi)+2*sin(1/3*n...

> for i to Ate do

> B:=subs(n=i,bn):

> b[i]:=evalf(B)

> od;

B := 120*(-4*sin(Pi)+5*sin(2/3*Pi)+2*sin(1/3*Pi))/P...

b[1] := 73.70724392

B := 30*(-4*sin(2*Pi)+5*sin(4/3*Pi)+2*sin(2/3*Pi))/...

b[2] := -7.897204706

B := 40/3*(-4*sin(3*Pi)+5*sin(2*Pi)+2*sin(Pi))/Pi^2...

b[3] := 0.

B := 15/2*(-4*sin(4*Pi)+5*sin(8/3*Pi)+2*sin(4/3*Pi)...

b[4] := 1.974301176

B := 24/5*(-4*sin(5*Pi)+5*sin(10/3*Pi)+2*sin(5/3*Pi...

b[5] := -2.948289757

B := 10/3*(-4*sin(6*Pi)+5*sin(4*Pi)+2*sin(2*Pi))/Pi...

b[6] := 0.

B := 120/49*(-4*sin(7*Pi)+5*sin(14/3*Pi)+2*sin(7/3*...

b[7] := 1.504229468

B := 15/8*(-4*sin(8*Pi)+5*sin(16/3*Pi)+2*sin(8/3*Pi...

b[8] := -.4935752941

B := 40/27*(-4*sin(9*Pi)+5*sin(6*Pi)+2*sin(3*Pi))/P...

b[9] := 0.

B := 6/5*(-4*sin(10*Pi)+5*sin(20/3*Pi)+2*sin(10/3*P...

b[10] := .3158881882

B := 120/121*(-4*sin(11*Pi)+5*sin(22/3*Pi)+2*sin(11...

b[11] := -.6091507762

B := 5/6*(-4*sin(12*Pi)+5*sin(8*Pi)+2*sin(4*Pi))/Pi...

b[12] := 0.

B := 120/169*(-4*sin(13*Pi)+5*sin(26/3*Pi)+2*sin(13...

b[13] := .4361375380

B := 30/49*(-4*sin(14*Pi)+5*sin(28/3*Pi)+2*sin(14/3...

b[14] := -.1611674430

B := 8/15*(-4*sin(15*Pi)+5*sin(10*Pi)+2*sin(5*Pi))/...

b[15] := 0.

B := 15/32*(-4*sin(16*Pi)+5*sin(32/3*Pi)+2*sin(16/3...

b[16] := .1233938235

B := 120/289*(-4*sin(17*Pi)+5*sin(34/3*Pi)+2*sin(17...

b[17] := -.2550423665

B := 10/27*(-4*sin(18*Pi)+5*sin(12*Pi)+2*sin(6*Pi))...

b[18] := 0.

B := 120/361*(-4*sin(19*Pi)+5*sin(38/3*Pi)+2*sin(19...

b[19] := .2041751909

B := 3/10*(-4*sin(20*Pi)+5*sin(40/3*Pi)+2*sin(20/3*...

b[20] := -.7897204706e-1

> ser_Cal:=0:ser_Ond:=0:

> for n to Ate do

> ser_Cal:=ser_Cal + exp(-t*(n*Pi/3)^2)*b[n]*sin(n*Pi*x/3):

> ser_Ond:=ser_Ond + cos(t*n*Pi/3)*b[n]*sin(n*Pi*x/3):

> od:

> with(plots):

> G1 := plot(T,x=0..3,color=blue,style=point):

> ser:=subs( t=0, ser_Cal);

> G2:=plot(ser,x=0..3):

> display({G1,G2}); 

Warning, the name changecoords has been redefined

ser := 73.70724392*exp(0)*sin(1/3*Pi*x)-7.897204706...
ser := 73.70724392*exp(0)*sin(1/3*Pi*x)-7.897204706...
ser := 73.70724392*exp(0)*sin(1/3*Pi*x)-7.897204706...
ser := 73.70724392*exp(0)*sin(1/3*Pi*x)-7.897204706...

[Maple Plot]

Gráfico da Equação do Calor na Barra

> plot3d(ser_Cal,x=0..3,t=0..5,axes=boxed);

[Maple Plot]

> animate(ser_Cal,x=0..3,t=0..1,frames=500);

Gráfico da Equação da Corda Vivrante

> plot3d(ser_Ond,x=0..3,t=0..5,grid=[10,100], axes=boxed);

[Maple Plot]

> animate(ser_Ond,x=0..3,t=0..6,frames=30);

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