Question

Suppose the ends of a bar are insulated, and the left half is initially (t=0) at constant temperature T while the right half is initially at temperature zero: $f(x) = \begin{cases} T & \text{for } 0 \le x \le L/2\\ 0 & \text{for } L/2 < x \le L \end{cases}$ The heat equation for this scenario has a general solution $u(x, t) = a_0 + \sum_{n=1}^{\infty} a_n \cos(\frac{n\pi x}{L})e^{-n^2 \pi^2 kt/L^2}$ Compute the coefficients $a_0$ and $a_n$ to find the particular solution of the heat equation.

          Suppose the ends of a bar are insulated, and the left half is initially (t=0) at constant temperature T while the right half is initially at temperature zero:
$f(x) = \begin{cases} T & \text{for } 0 \le x \le L/2\\ 0 & \text{for } L/2 < x \le L \end{cases}$
The heat equation for this scenario has a general solution
$u(x, t) = a_0 + \sum_{n=1}^{\infty} a_n \cos(\frac{n\pi x}{L})e^{-n^2 \pi^2 kt/L^2}$
Compute the coefficients $a_0$ and $a_n$ to find the particular solution of the heat equation.

Suppose the ends of a bar are insulated, and the left half is initially (t=0) at constant temperature T while the right half is initially at temperature zero:
f(x) = T for 0 ≤ x ≤ L/2
0 for L/2 < x ≤ L
The heat equation for this scenario has a general solution
u(x, t) = a0 + ∑n=1^∞ an cos((nπ x)/(L))e^-n^2 π^2 kt/L^2
Compute the coefficients a0 and an to find the particular solution of the heat equation.

Added by Brian P.

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