ellipweb_1.mw

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> restart:with(plots):a:=1:b:=1:

Warning, the name changecoords has been redefined

> lambda[n]:=(n*Pi/a)^2;

lambda[n] := n^2*Pi^2

> X[n](x):=sin(n*Pi/a*x);X[n](s):=subs(x=s,X[n](x)):X[m](x):=subs(n=m,X[n](x)):

X[n](x) := sin(n*Pi*x)

> Int(X[n](x)*X[m](x),x=0..a)= (1/2)*delta(n,m);

Int(sin(n*Pi*x)*sin(m*Pi*x), x = (0 .. 1)) = 1/2*delta(n, m)

> y1(y):=sinh(sqrt(lambda)*(b-y));y2(y):=cosh(sqrt(lambda)*(b-y));

y1(y) := sinh(lambda^(1/2)*(1-y))

y2(y) := cosh(lambda^(1/2)*(1-y))

> Y[n](y):=A(n)*cosh(n*Pi/a*(b-y))+B(n)*sinh(n*Pi/a*(b-y));

Y[n](y) := A(n)*cosh(n*Pi*(1-y))+B(n)*sinh(n*Pi*(1-y))

> eval(subs(y=b,Y[n](y)))=0;

A(n) = 0

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> Y[n](y):=B(n)*sinh(n*Pi/a*(b-y));

Y[n](y) := B(n)*sinh(n*Pi*(1-y))

> u[n](x,y):=X[n](x)*Y[n](y):

> u(x,y):=Sum(u[n](x,y),n=1..infinity);

u(x, y) := Sum(sin(n*Pi*x)*B(n)*sinh(n*Pi*(1-y)), n = (1 .. infinity))

> f(x):=x*(1-x);

f(x) := x*(1-x)

> f(x)=subs(y=0,u(x,y));

x*(1-x) = Sum(sin(n*Pi*x)*B(n)*sinh(n*Pi), n = (1 .. infinity))

> B(n):=(1/sinh(n*Pi*b/a))*Int(f(x)*X[n](x),x=0..a);B(n):=expand(value(%)):

B(n) := Int(x*(1-x)*sin(n*Pi*x), x = (0 .. 1))/sinh(n*Pi)

> B(n):=simplify(subs({sin(n*Pi)=0,cos(n*Pi)=(-1)^n},B(n)));

B(n) := -2*(-1+(-1)^n)/(sinh(n*Pi)*n^3*Pi^3)

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> u[n](x,y):=eval(X[n](x)*Y[n](y)):

> u(x,y):=Sum(u[n](x,y),n=1..infinity);

u(x, y) := Sum(-2*sin(n*Pi*x)*(-1+(-1)^n)*sinh(n*Pi*(1-y))/(sinh(n*Pi)*n^3*Pi^3), n = (1 .. infinity))

> u(x,y):=sum(u[n](x,y),n=1..5):

> plot3d(u(x,y),x=0..a,y=0..b,axes=framed,thickness=3);

[Plot]

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