Editing Equations, Formulas and Algorithms (section)

== Spectral Methods ==
Spectral methods approximate the solution of differential equations using expansions in terms of basis functions, such as Fourier series or Chebyshev polynomials. These methods offer high accuracy and convergence rates and are used in various scientific and engineering applications.

'''Example: Spectral Method'''
Consider the one-dimensional wave equation:
<math mode="display"> \frac{\partial^2 u}{\partial t^2} = c^2 \frac{\partial^2 u}{\partial x^2} </math>
where <math> u(x, t) </math> is the displacement of a vibrating string, <math> c </math> is the wave speed, <math> x </math> is the spatial coordinate, and <math> t </math> is time.

'''Source Code Example (MATLAB with Chebfun):'''
<syntaxhighlight lang="matlab">
% Define domain and initial conditions
domain = [-1, 1]; % Define the spatial domain
initial_condition = chebfun(@(x) exp(-20*x.^2), domain); % Define initial condition

% Define Chebfun objects for functions and derivatives
c = 1; % Wave speed
wave_equation = @(t, u) -c^2 * diff(u, 2); % Define the wave equation
t_span = [0, 1]; % Time span for simulation

% Solve the wave equation using Chebfun's time-stepping capabilities
u = pdepe(0, wave_equation, initial_condition, @(x, t) 0, domain, t_span);

% Plot solution
figure;
plot(u(end, :));
title('Solution of the One-dimensional Wave Equation');
xlabel('Spatial Domain');
ylabel('Solution');
</syntaxhighlight>