, QUESTION 1 Consider the heat flow in an horizontal rod of
length L units and heat conductivity 1. We assume that
initially the rod was submerged in a meduim where the
temperature at each point x of the rod is described by the
function f (x) . We also suppose that the left and the right
ends of the rod are in contact with media which
temperatures change with time and are described by the
functions g1 (t) and g2 (t) respectively. (a) Write down the
initial-boundary problem satisfied by the temperature
distribution u (x, t) in the rod at any point x and time t
(Explain all the meaning of the variables and parameters
used). (5 Marks) (b) Suppose that f, g1, g2 are bounded,
there exist constants m and M such that for all x in the
domain of g1 and g2, and all t 0, we have m f (x)
M;m g1 (x) M;m g2 (x) M; and the temperature u
(x, t) solution of the IBVP described above satisfies the
inequalities m u (x, t) M; for all x and t 0. Show that
the solution u (x, t) of the heat problem described above is
unique. (Explain clearly all the steps (10 Marks) (c) Suppose
that u1 (x, t) and u2 (x, t) are solutions of the heat problem
above (with different initial and boundary conditions) are
such that u1 (0, t) u2 (0, t) , u1 (L, t) u2 (L, t) , and u1 (x,
0) u2 (x, 0) . Show that u1 (x, t) u2 (x, t) for all 0 x L
and all t 0. (10 Marks) [25 Marks] 7 Downloaded by
Corona Virus () lOMoARcPSD|
length L units and heat conductivity 1. We assume that
initially the rod was submerged in a meduim where the
temperature at each point x of the rod is described by the
function f (x) . We also suppose that the left and the right
ends of the rod are in contact with media which
temperatures change with time and are described by the
functions g1 (t) and g2 (t) respectively. (a) Write down the
initial-boundary problem satisfied by the temperature
distribution u (x, t) in the rod at any point x and time t
(Explain all the meaning of the variables and parameters
used). (5 Marks) (b) Suppose that f, g1, g2 are bounded,
there exist constants m and M such that for all x in the
domain of g1 and g2, and all t 0, we have m f (x)
M;m g1 (x) M;m g2 (x) M; and the temperature u
(x, t) solution of the IBVP described above satisfies the
inequalities m u (x, t) M; for all x and t 0. Show that
the solution u (x, t) of the heat problem described above is
unique. (Explain clearly all the steps (10 Marks) (c) Suppose
that u1 (x, t) and u2 (x, t) are solutions of the heat problem
above (with different initial and boundary conditions) are
such that u1 (0, t) u2 (0, t) , u1 (L, t) u2 (L, t) , and u1 (x,
0) u2 (x, 0) . Show that u1 (x, t) u2 (x, t) for all 0 x L
and all t 0. (10 Marks) [25 Marks] 7 Downloaded by
Corona Virus () lOMoARcPSD|
