Find the locations of its maximum and its minimum

Reference no: EM133029884

Questions -

Q1. Consider the solution 1 - x² - 2kt of the diffusion equation. Find the locations of its maximum and its minimum in the closed rectangle {0 ≤ x ≤ 1, 0 ≤ t ≤ T}.

Q2. Consider a solution of the diffusion equation u_t = u_xx in {0 ≤ x ≤ l, 0 ≤ t ≤ ∞}.

(a) Let M(T) = the maximum of u(x, t) in the closed rectangle {0 ≤ x ≤ l, 0 ≤ t ≤ T}. Does M(T) increase or decrease as a function of T?

(b) Let m(T) = the minimum of u(x, t) in the closed rectangle {0 ≤ x ≤ l, 0 ≤ t ≤ T}. Does m(T) increase or decrease as a function of T?

Q3. Consider the diffusion equation u_t = u_xx in the interval (0, 1) with u(0, t)= u(1, t) = 0 and u(x, 0) = 1 - x². Note that this initial function does not satisfy the boundary condition at the left end, but that the solution will satisfy it for all t > 0.

Reference no: EM133029884

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