2. Consider the 'Robin problem', defined as: solve the inhomogeneous heat equation du + s(r, 1), for 0 0, with initial/boundary conditions u(r,0) = uo(z), u(0,t)- au, (0,t) s(). u(L, t) + Bu, (0, t) = 9(4) r>0 t>0 t>0 where a, 320 (a) Show that if u,(x,t) and u(r, t) are two solution of the Robin problem then their difference w(x,t) = u(1,t)- ua(r,t) satisfies the homogeneous heat equation with all initial/boundary conditions replaced by their homogeneous counterparts. (b) Define the energy of a solution w(z, t) of the homogeneous heat equation by E(t) = | w*(r,t) dz. Prove that E'() = -aw,"(0, t) - Bu,"(L, t)- (c) Deduce that w(r, t) as defined in part (a) must vanish identically so the Robin problem has a unique solution.
2. Consider the 'Robin problem', defined as: solve the inhomogeneous heat equation du + s(r, 1), for 0 0, with initial/boundary conditions u(r,0) = uo(z), u(0,t)- au, (0,t) s(). u(L, t) + Bu, (0, t) = 9(4) r>0 t>0 t>0 where a, 320 (a) Show that if u,(x,t) and u(r, t) are two solution of the Robin problem then their difference w(x,t) = u(1,t)- ua(r,t) satisfies the homogeneous heat equation with all initial/boundary conditions replaced by their homogeneous counterparts. (b) Define the energy of a solution w(z, t) of the homogeneous heat equation by E(t) = | w*(r,t) dz. Prove that E'() = -aw,"(0, t) - Bu,"(L, t)- (c) Deduce that w(r, t) as defined in part (a) must vanish identically so the Robin problem has a unique solution.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
3
Transcribed Image Text:2. Consider the 'Robin problem', defined as: solve the inhomogeneous heat equation
u
+ s(r, t),
du
for 0 <z < L
where o > 0, with initial/boundary conditions
u(r,0) = uo(r),
I>0
u(0, t) – au, (0, t) = f(t),
u(L, t) + Bu, (0, t) = g(t)
t>0
t>0
where a, 320
(a) Show that if u (z, t) and u2(r, t) are two solution of the Robin problem then their difference
w(r, t) = u(x, t) – uz(r, t) satisfies the homogeneous heat equation with all initial/boundary
conditions replaced by their homogeneous counterparts.
(b) Define the energy of a solution w(r, t) of the homogeneous heat equation by
E(t) = ! | w*(x, t) dr.
Prove that
E'() = -aw,"(0,t) - Bu,"(L, t) - ow(a,t) dr.
(c) Deduce that w(r, t) as defined in part (a) must vanish identically so the Robin problem has a
unique solution.
Transcribed Image Text:2. Consider the 'Robin problem', defined as: solve the inhomogeneous heat equation
u
+ s(r, t),
du
for 0 <z < L
where o > 0, with initial/boundary conditions
u(r,0) = uo(r),
I>0
u(0, t) – au, (0, t) = f(t),
u(L, t) + Bu, (0, t) = g(t)
t>0
t>0
where a, 320
(a) Show that if u (z, t) and u2(r, t) are two solution of the Robin problem then their difference
w(r, t) = u(x, t) – uz(r, t) satisfies the homogeneous heat equation with all initial/boundary
conditions replaced by their homogeneous counterparts.
(b) Define the energy of a solution w(r, t) of the homogeneous heat equation by
E(t) = ! | w*(x, t) dr.
Prove that
E'() = -aw,"(0,t) - Bu,"(L, t) - ow(a,t) dr.
(c) Deduce that w(r, t) as defined in part (a) must vanish identically so the Robin problem has a
unique solution.
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