solve the heat equation in matlab

To solve the heat equation in Matlab, we can use the Partial Differential Equation (PDE) Toolbox. The toolbox provides a wide variety of numerical methods to solve the heat equation, such as finite difference, finite element and boundary element methods. In this example, we will use the finite difference method.

First, we need to define the parameters of the problem. Let's consider a 1D heat equation:

φt = αφxx

where φ is the temperature distribution, t is time, x is the spatial variable, and α is the thermal diffusivity.

We will solve this equation on a finite domain [0, L] with initial condition φ(x,0) = f(x) and boundary conditions φ(0,t) = 0 and φ(L,t) = 0.

To solve this problem in Matlab, we need to define the domain, the grid points, the initial condition, and the boundary conditions. Then, we can use the pdepe function to solve the problem.

main.m
% Define the parameters
L = 1;          % length of the domain
Tfinal = 1;     % final time
N = 100;        % number of grid points
alpha = 1;      % thermal diffusivity

% Define the domain and the grid
x = linspace(0,L,N);
t = linspace(0,Tfinal,N);

% Define the initial condition
f = sin(pi*x);

% Define the boundary conditions
bc = @(xl,ul,xr,ur,t) [ul; ur];

% Define the PDE
pde = @(x,t,u,DuDx) alpha*DuDx;

% Solve the PDE
sol = pdepe(0,pde,f,bc,x,t);

% Extract the solution
u = sol(:,:,1);

% Plot the solution
[X,T] = meshgrid(x,t);
surf(X,T,u)
xlabel('x')
ylabel('t')
zlabel('u(x,t)')
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In this example, we defined the domain as [0, L], the number of grid points as N=100, and the thermal diffusivity as α=1. We also defined the initial condition as f(x) = sin(πx) and the boundary conditions as φ(0,t) = 0 and φ(L,t) = 0. Then, we used the pdepe function to solve the PDE and plotted the solution using the surf function.

The finite difference method is just one of the many numerical methods available in the PDE Toolbox. The toolbox also provides tools to solve 2D and 3D PDEs, nonlinear PDEs, and systems of PDEs, among other features. By choosing the appropriate numerical method and parameters, we can solve a wide variety of PDEs in Matlab.