A tube of length L contains a solution with sugar...

A tube of length $L$ contains a solution with sugar concentration at time $t=0$ given by $$ n(x, 0)=n_0+n_1\left\{\cos \frac{\pi x}{L}+\frac{1}{9} \cos \frac{3 \pi x}{L}+\frac{1}{25} \cos \frac{5 \pi x}{L}\right\} . $$ Assume that $n(x, t)$ obeys a one-dimensional diffusion equation with diffusion constant $D$. (a) Write down the diffusion equation for $n(x, t)$. (b) Calculate $n(x, t)$ for $t>0$. FIGURE CAN'T COPY.

A tube of length $L$ contains a solution with sugar concentration at time $t=0$ given by
$$
n(x, 0)=n_0+n_1\left\{\cos \frac{\pi x}{L}+\frac{1}{9} \cos \frac{3 \pi x}{L}+\frac{1}{25} \cos \frac{5 \pi x}{L}\right\} .
$$

Assume that $n(x, t)$ obeys a one-dimensional diffusion equation with diffusion constant $D$.
(a) Write down the diffusion equation for $n(x, t)$.
(b) Calculate $n(x, t)$ for $t>0$.
FIGURE CAN'T COPY.

Key Concepts

Exponential Decay of Fourier Modes

Each Fourier mode in the series solution to the diffusion equation decays exponentially with time. The decay rate depends on the square of the spatial frequency (or eigenvalue) associated with that mode. Mathematically, a mode with spatial frequency k decays as exp(?Dk²t). This property is crucial in determining the time evolution of the concentration profile from its initial state.

Initial Condition and Fourier Series

When solving the diffusion equation with a nonuniform initial condition, it is common to expand the initial distribution in terms of a Fourier series. This expansion expresses the initial state as a sum of cosine (or sine) functions that are eigenfunctions of the Laplacian operator under appropriate boundary conditions. Each term in the Fourier series represents a specific spatial frequency component of the initial concentration profile.

One-Dimensional Diffusion Equation

This is a partial differential equation that models the spread of particles (or, in this context, sugar molecules) along a single spatial dimension. It has the form ?n/?t = D ?²n/?x², where n(x, t) is the concentration, t is time, x is the spatial coordinate, and D is the diffusion constant. The equation encapsulates how concentration gradients drive the diffusion process over time.

Separation of Variables

This is a method used to solve partial differential equations by assuming that the solution can be written as a product of functions, each depending on a single variable. In the context of the diffusion equation, this approach allows us to separate the spatial part from the time part, where the spatial functions align with the Fourier series eigenfunctions and the time evolution is governed by exponential decay factors.

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