partial differential equation in matlab

To solve a partial differential equation (PDE) in MATLAB, one can use the pdepe function which solves a system of time-dependent PDEs using the method of lines. The method of lines is a technique for converting a PDE into a system of ordinary differential equations (ODEs), which can be solved using MATLAB's ODE solvers.

To use pdepe, one will need to define the PDE, the boundary conditions, and initial conditions. This can be done using MATLAB functions. Then, pdepe can be called to solve the system of ODEs created from the PDE.

Here's an example of solving the heat equation using the finite difference method:

main.m
% Define the PDE
function [c,f,s] = heat_eqn(x,t,u,DuDx)
c = 1;
f = DuDx;
s = 0;

% Define the boundary conditions
function [pl,ql,pr,qr] = heat_bc(xl,ul,xr,ur,t)
pl = ul - 1; % u(x=0,t)=1
ql = 0;
pr = ur; % u(x=L,t)=0
qr = 1;

% Define the initial condition
function u0 = heat_ic(x)
u0 = sin(pi*x);

% Define the spatial mesh
x = linspace(0,1,100);

% Define the time vector
t = linspace(0,1,100);

% Call pdepe to solve the PDE
sol = pdepe(0,@heat_eqn,@heat_ic,@heat_bc,x,t);

% Extract the solution
u = sol(:,:,1); % sol(:,i,1) is the solution at time t(i)
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In this example, we define the heat equation as the PDE, with boundary conditions of fixed temperature at the left end of the domain and insulated at the right end, and an initial condition of a sinusoidal temperature distribution. We then define the spatial mesh and time vector, and call pdepe to solve the PDE. The solution u is extracted from the output of pdepe.

This is just a simple example, and there are many other ways to solve PDEs in MATLAB using different numerical methods.