Consider the function

for the concentration profile as a function of time. In practice, we need to truncate the sum with a finite number k terms,

(a) Some functions are very difficult to represent as a Fourier series. The initial condition for this problem is one such function. Write a MATLAB program that automatically plots the value of c(x, 0) for different values of k. It is convenient to use the linspace function in MATLAB to generate values of x for this problem. You should choose values of k that illustrate the difficulty in making the plot. To help get you started with modular programming, your program should include a subfunction out=getc(x,t,k) that returns the value of c(x, t) for some position x and time t using k terms in the sum. Explain what happens to the behavior of the plot as a function of k. Also explain why the difference between the sum and the initial condition is different at x = 0 and x = 1.

(b) Now that we know how to plot the hardest solution, modify your program from part (a) to plot the concentration profile as a function of time. Pick a number of times for the plot that illustrate both the short-time and long-time behavior of the solution – in other words, it should be clear how the concentration profile is evolving as a function of time but the plot needs to be readable! Explain why your plot satisfies the boundary conditions and what happens at the steady state for the problem.

Transcribed Image Text:

c(x, t) = Σ n=0 4(-1)" (2n + 1)π COS Dit]exp - (2 - + ] (2n + 1)π 2 (2n + 1)π.χ 2