Question

2. Using the MATLAB "integral" command, numerically determine the Fourier Cosine series of the following function. Assume each case has an even extension (b,-0) Last Name N-Z: f= 2xcos (Vx+4), 0<x<3 (Hint: after extension L-3) Have your code plot both the analytical function (as a red line) and the numerical Fourier series (in blue circles -spaced appropriately). Use the Legend command to identify the two items. It is suggested to use a series with 15 terms.

Answer 1

clc; clear; close; Defining f(x) f = @(x) 2*x.^(1.1) .* cos(3qrt(x+4)); L = 3; L = 3 Coefficient a a0 = (2/L) * integral (1,0,1); $ Using 15 terms, calculating al, a2, a3...an and storing them in vector a for n = 1 : 15 integrand = @ (x) F(x). *C03 (n*pi*x/L); integrand for an for current n a(n) = (2/L) * integral (integrand, 0, 1); adding coefficient an in a vector end Defining fourier series as a function of x vector of n values 1 to 15 fourier = @(x) a0./2 + sum(a.*003 (n. *x)); fourier series Ploting x = linspace(0,3); $x vector from 0 to 3 (100 elements) Evaluating f(x) using fourier expansion for every x in vector x fourier_val3 = arrayfun (Fourier,x); $vector of fourier series evaluated at every x Ploting f(x) as red line and fourier evaluations as blue circles plot (x, f(x),'-r',X, Fourier_vals,'ob') Labeling the plot and setting legend xlabel('x'); legend('f(x)','fourier cosine of f(x)', 'location', 'best')

text :-

clc; clear; close;

% Defining f(x)
f = @(x) 2*x.^(1.1) .* cos(sqrt(x+4));
L = 3;          % L = 3

% Coefficient a0
a0 = (2/L) * integral(f,0,L);

% Using 15 terms, calculating a1,a2,a3...an and storing them in vector a
for n = 1 : 15
    integrand = @(x) f(x).*cos(n*pi*x/L);       % integrand for an for current n
    a(n) = (2/L) * integral(integrand,0,L);     % adding coefficient an in a vector
end

% Defining fourier series as a function of x
n = 1:15;           % vector of n values 1 to 15
fourier = @(x) a0./2 + sum(a.*cos(n.*x));       % fourier series

% Ploting
x = linspace(0,3);          % x vector from 0 to 3 (100 elements)
% Evaluating f(x) using fourier expansion for every x in vector x
fourier_vals = arrayfun(fourier,x);     % vector of fourier series evaluated at every x
% Ploting f(x) as red line and fourier evaluations as blue circles
plot(x,f(x),'-r',x,fourier_vals,'ob')
% Labeling the plot and setting legend
xlabel('x');
legend('f(x)','fourier cosine of f(x)','location','best')

Result :-

DO ORDINADO o -f(x) fourier cosine of f(x) .000000 Spoooo O 0.5 1 1.5 NE 2 2.5