Example 8.6.mw

Example 8.6 

Example (8.1.6) is solved in Maple and the results obtained are given below: 

> restart; -1; with(inttrans); -1; with(plots); -1
 

> eq:=diff(u(x,t),t)=diff(u(x,t),x$2)+2/x*diff(u(x,t),x);
 

diff(u(x, t), t) = `+`(diff(diff(u(x, t), x), x), `/`(`*`(2, `*`(diff(u(x, t), x))), `*`(x))) (1)
 

> u(x,0):=0;
 

0 (2)
 

> bc1:=diff(u(x,t),x)=0;
 

diff(u(x, t), x) = 0 (3)
 

> bc2:=u(x,t)=1;
 

u(x, t) = 1 (4)
 

> eqs:=laplace(eq,t,s):
 

> eqs:=subs(laplace(u(x,t),t,s)=U(x),eqs);
 

`*`(s, `*`(U(x))) = `+`(diff(diff(U(x), x), x), `/`(`*`(2, `*`(diff(U(x), x))), `*`(x))) (5)
 

> bc1:=laplace(bc1,t,s):
 

> bc1:=subs(diff(laplace(u(x,t),t,s),x)=D(U)(0),laplace(u(x,t),t,s)=U(0),bc1);
 

(D(U))(0) = 0 (6)
 

> bc2:=laplace(bc2,t,s):
 

> bc2:=subs(diff(laplace(u(x,t),t,s),x)=D(U)(1),laplace(u(x,t),t,s)=U(1),bc2);
 

U(1) = `/`(1, `*`(s)) (7)
 

> U(x):=rhs(dsolve({eqs,bc2},U(x)));
 

`+`(`/`(`*`(_C1, `*`(sinh(`*`(`^`(s, `/`(1, 2)), `*`(x))))), `*`(x)), `-`(`/`(`*`(`+`(`*`(s, `*`(_C1, `*`(exp(`+`(`*`(8, `*`(`^`(s, `/`(1, 2))))))))), `-`(`*`(s, `*`(_C1, `*`(exp(`+`(`*`(6, `*`(`^`(s,... (8)
 

> U(x):=subs(_C2=0,U(x));
 

`+`(`/`(`*`(_C1, `*`(sinh(`*`(`^`(s, `/`(1, 2)), `*`(x))))), `*`(x)), `-`(`/`(`*`(`+`(`*`(s, `*`(_C1, `*`(exp(`+`(`*`(8, `*`(`^`(s, `/`(1, 2))))))))), `-`(`*`(s, `*`(_C1, `*`(exp(`+`(`*`(6, `*`(`^`(s,... (9)
 

> convert(U(x),exp);
 

`+`(`/`(`*`(_C1, `*`(`+`(`*`(`/`(1, 2), `*`(exp(`*`(`^`(s, `/`(1, 2)), `*`(x))))), `-`(`*`(`/`(1, 2), `*`(exp(`+`(`-`(`*`(`^`(s, `/`(1, 2)), `*`(x))))))))))), `*`(x)), `-`(`/`(`*`(`+`(`*`(s, `*`(_C1, ... (10)
 

> U1s:=exp(s^(1/2))/s/(exp(s^(1/2))^2-1)*exp(s^(1/2)*x)/x;
 

`/`(`*`(exp(`*`(`^`(s, `/`(1, 2)))), `*`(exp(`*`(`^`(s, `/`(1, 2)), `*`(x))))), `*`(s, `*`(`+`(`*`(`^`(exp(`*`(`^`(s, `/`(1, 2)))), 2)), `-`(1)), `*`(x)))) (11)
 

> U2s:=-exp(s^(1/2))/s/(exp(s^(1/2))^2-1)*exp(-s^(1/2)*x)/x;
 

`+`(`-`(`/`(`*`(exp(`*`(`^`(s, `/`(1, 2)))), `*`(exp(`+`(`-`(`*`(`^`(s, `/`(1, 2)), `*`(x))))))), `*`(s, `*`(`+`(`*`(`^`(exp(`*`(`^`(s, `/`(1, 2)))), 2)), `-`(1)), `*`(x)))))) (12)
 

> U1S:=series(subs(exp(s^(1/2))=1/S,U1s),S):
 

> U1S:=subs(S=exp(-s^(1/2)),U1S):
 

> simplify(U1S);
 

`/`(`*`(`+`(exp(`*`(`^`(s, `/`(1, 2)), `*`(`+`(x, `-`(1))))), exp(`*`(`^`(s, `/`(1, 2)), `*`(`+`(x, `-`(3))))), exp(`*`(`^`(s, `/`(1, 2)), `*`(`+`(x, `-`(5))))), `*`(O(exp(`+`(`-`(`*`(7, `*`(`^`(s, `/... (13)
 

> U1S:=1/x*Sum(exp(s^(1/2)*(x-2*n+1))/s,n=1..infinity);
 

`/`(`*`(Sum(`/`(`*`(exp(`*`(`^`(s, `/`(1, 2)), `*`(`+`(x, `-`(`*`(2, `*`(n))), 1))))), `*`(s)), n = 1 .. infinity)), `*`(x)) (14)
 

> u1s:=exp(s^(1/2)*(x-2*n+1))/s;
 

`/`(`*`(exp(`*`(`^`(s, `/`(1, 2)), `*`(`+`(x, `-`(`*`(2, `*`(n))), 1))))), `*`(s)) (15)
 

> u1t:=invlaplace(u1s,s,t);
 

invlaplace(`/`(`*`(exp(`*`(`^`(s, `/`(1, 2)), `*`(`+`(x, `-`(`*`(2, `*`(n))), 1))))), `*`(s)), s, t) (16)
 

> U1t:=1/x*Sum(u1t,n=1..infinity);
 

`/`(`*`(Sum(invlaplace(`/`(`*`(exp(`*`(`^`(s, `/`(1, 2)), `*`(`+`(x, `-`(`*`(2, `*`(n))), 1))))), `*`(s)), s, t), n = 1 .. infinity)), `*`(x)) (17)
 

> U2S:=series(subs(exp(s^(1/2))=1/S,U2s),S):
 

> U2S:=subs(S=exp(-s^(1/2)),U2S):
 

> simplify(U2S);
 

`+`(`-`(`/`(`*`(`+`(exp(`+`(`-`(`*`(`^`(s, `/`(1, 2)), `*`(`+`(x, 1)))))), exp(`+`(`-`(`*`(`^`(s, `/`(1, 2)), `*`(`+`(x, 3)))))), exp(`+`(`-`(`*`(`^`(s, `/`(1, 2)), `*`(`+`(x, 5)))))), `-`(`*`(O(exp(`... (18)
 

> U2S:=-1/x*Sum(exp(-s^(1/2)*(x+2*n-1))/s,n=1..infinity);
 

`+`(`-`(`/`(`*`(Sum(`/`(`*`(exp(`+`(`-`(`*`(`^`(s, `/`(1, 2)), `*`(`+`(x, `*`(2, `*`(n)), `-`(1)))))))), `*`(s)), n = 1 .. infinity)), `*`(x)))) (19)
 

> u2s:=exp(-s^(1/2)*(x+2*n-1))/s:
 

> u2t:=invlaplace(u2s,s,t);
 

invlaplace(`/`(`*`(exp(`*`(`+`(`-`(x), `-`(`*`(2, `*`(n))), 1), `*`(`^`(s, `/`(1, 2)))))), `*`(s)), s, t) (20)
 

> U2t:=-1/x*Sum(u2t,n=1..infinity);
 

`+`(`-`(`/`(`*`(Sum(invlaplace(`/`(`*`(exp(`*`(`+`(`-`(x), `-`(`*`(2, `*`(n))), 1), `*`(`^`(s, `/`(1, 2)))))), `*`(s)), s, t), n = 1 .. infinity)), `*`(x)))) (21)
 

> Ut:=U1t+U2t;
 

`+`(`/`(`*`(Sum(invlaplace(`/`(`*`(exp(`*`(`^`(s, `/`(1, 2)), `*`(`+`(x, `-`(`*`(2, `*`(n))), 1))))), `*`(s)), s, t), n = 1 .. infinity)), `*`(x)), `-`(`/`(`*`(Sum(invlaplace(`/`(`*`(exp(`*`(`+`(`-`(x... (22)
 

> u:=subs(infinity=N,Ut):
 

> u:=subs(N=20,u);
 

`+`(`/`(`*`(Sum(invlaplace(`/`(`*`(exp(`*`(`^`(s, `/`(1, 2)), `*`(`+`(x, `-`(`*`(2, `*`(n))), 1))))), `*`(s)), s, t), n = 1 .. 20)), `*`(x)), `-`(`/`(`*`(Sum(invlaplace(`/`(`*`(exp(`*`(`+`(`-`(x), `-`... (23)
 

> plot3d(u,x=1e-6..1,t=1e-6..0.1,axes=boxed,title="Figure Exp. 8.11.",labels=[x,t,"u"],orientation=[-150,60]);
 

Plot
 

> plot([subs(t=1e-6,u),subs(t=1e-2,u),subs(t=0.05,u),subs(t=0.1,u)],x=0..1,axes=boxed,title="Figure Exp. 8.12.",thickness=5,labels=[x,"u"]);
 

Plot_2d
 

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