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3. Consider one dimensional the heat equation $\frac{\partial u}{\partial t} = k \frac{\partial^2 u}{\partial x^2}$ a. Determine the infinite space Green's function for this equation b. Solve this equation using infinite space Green's function and the concept of superposition subject to the following boundary and intial conditions $u(0,t) = 0$ and $\lim_{x \to \infty} u(x,t) \to 0$ $u(x, 0) = f(x)$ c. possible to determine the time for which this semi-infinite medium solution fails to represent for a given rod lenght L?

          3. Consider one dimensional the heat equation
$\frac{\partial u}{\partial t} = k \frac{\partial^2 u}{\partial x^2}$
a. Determine the infinite space Green's function for this equation
b. Solve this equation using infinite space Green's function and the concept of
superposition subject to the following boundary and intial conditions
$u(0,t) = 0$ and $\lim_{x \to \infty} u(x,t) \to 0$  $u(x, 0) = f(x)$
c. possible to determine the time for which this semi-infinite medium solution fails to
represent for a given rod lenght L?

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3. Consider one dimensional the heat equation
(∂ u)/(∂ t) = k (∂^2 u)/(∂ x^2)
a. Determine the infinite space Green's function for this equation
b. Solve this equation using infinite space Green's function and the concept of
superposition subject to the following boundary and intial conditions
u(0,t) = 0 and limx →∞ u(x,t) → 0 u(x, 0) = f(x)
c. possible to determine the time for which this semi-infinite medium solution fails to
represent for a given rod lenght L?

Added by Juan P.

University Physics with Modern Physics

Hugh D. Young 14th Edition

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Solved by Expert Rachel Gore

Step 1

To determine the Green's function for the one-dimensional heat equation in infinite space, we need to solve the equation: ∂G/∂t = α(∂²G/∂x²) subject to the boundary condition G(x, t=0) = 0. To solve this equation, we can use separation of variables. Let's Show more…

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Consider the one-dimensional heat equation: âˆ‚u/âˆ‚t = Î±(âˆ‚Â²u/âˆ‚xÂ²) a. Determine the Green's function for this equation in infinite space. b. Solve this equation using the Green's function for infinite space and the concept of superposition, subject to the following boundary and initial conditions: u(0,t) = 0 and u(L,t) = 0 u(x,0) = f(x) c. Is it possible to determine the time for which this semi-infinite medium solution fails to represent for a given rod length L?

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