Question

Use MATLAB to plot the function of Problem 2: (Advanced engineering mathematics 7th edition textbook - chapter 13 section 13.2 Problem 5) f(x) = cosh(ix) -15x51 for the given range of x. Plot, on the same figure, the Fourier expansion of the function by using: a. 2 terms (n=2) b. 5 terms (n = 5) c. 20 terms (n = 20)

Answer 1

MATLAB Code:

close all
clear
clc

syms t k L n
evalin(symengine, 'assume(k, Type::Integer)');
a = @(f,t,k,L) int(f*cos(k*pi*t/L)/L, t, -L, L);
b = @(f,t,k,L) int(f*sin(k*pi*t/L)/L, t, -L, L);
fs = @(f,t,n,L) a(f,t,0,L)/2 + symsum(a(f,t,k,L)*cos(k*pi*t/L) + b(f,t,k,L)*sin(k*pi*t/L), k, 1, n); % Fourier series summation

f = cosh(pi*t)*(heaviside(t + 1) - heaviside(t - 1)); % Interval [-1, 1]

figure, hold on
xlabel('t'), ylabel('f(t) & Fourier Series'), title('Fourier Series Representation')

L = 1;
t_vals = linspace(-L, L, 500); % 500 samples from -L to L
plot(t_vals, subs(f, t_vals)) % Plot f(t)

n = 2; % Number of terms
g = fs(f,t,n,L);
plot(t_vals, subs(g, t_vals))

n = 5;
g = fs(f,t,n,L);
plot(t_vals, subs(g, t_vals))

n = 20;
g = fs(f,t,n,L);
plot(t_vals, subs(g, t_vals))

legend('f(t)', 'Fourier Series (n = 2)', 'Fourier Series (n = 5)', 'Fourier Series (n = 20)', 'Location', 'north')
hold off

Plot:

Fourier Series Representation f(t) Fourier Series (n = 2) Fourier Series (n = 5) Fourier Series (n = 20) f(t) & Fourier Series 1 -0.8 -0.6 -0.4 -0.2 0.2 0.4 0.6 0.8 1