Question

3. The last question is about physical interpretation of PDES. Assume throughout that u(r, t) describes the temperature in a thin rod, with r [0, L] andt2 0. Note: when I ask for physical interpretation, I mean things like "u(0,t) = 0 means that fixing one of the ends at constant temperature" (a) Consider the IBVP we are O (3 points) at Or2 u(0, t) Ou (L, t) u(x,0) =0 Give a physical interpretation of the boundary conditions and of the initial condi tion (b) Consider the PDE (2 points) D at Give a physical interpretation of source term. (c) Consider the PDE (5 points) u u D at Explain why we should consider u for the source term as a source term. Give a physical interpretation Total for Question 3: 10 points

Answer 1

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The last question is about physical interpretation of PDES. Assume throughout that u(x, t) describes the temperature in a thin rod, with e [0, L andt > 0. Note: when I ask for physical interpretation, I mean things like "u(0, t) = 0 means that we are fixing one of the ends at constant temperature"

EXPLANTION ::-

(A)

Consider the IBVP д'и D =0 да? u(0, t) t (L, t) 0 дх u (х, 0) — 0. Give a physical interpretation of the boundary conditions and of the initial condi tion

SOL ::-

u(0, t) = t means "Temperature at the initial end of the rod at any time 't' is equal to t".

(L, t) 0
means " Heat flow at the end of the rod at any time is fixed to a constant."

u(x, 0) = 0 means "At the beginning i.e, at t=0 there is no temperature in the rod".

(B)

Consider the PDE ди D = 5. Give a physical interpretation of source term

SOL :-

The source term is 5, meaning there is a fixed source of thermal energy of 5 units throughout.

(C)

Consider the PDE ди = u2 дх? at Explain why we should consider u2 as a source term. Give a physical interpretation for the source term

SOL ::-

We are to consider
u
as the source term since it is the non-homogeneous term in the equation.

Physical interpretation of it is that there is a source of thermal energy that is dependent on the temperature in the rod.  

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