Solving Partial Differential Equations in MATLAB

MATLAB offers two main approaches for solving partial differential equations (PDEs):

1. pdepe – Built-in solver for 1-D parabolic and elliptic PDEs in one spatial variable (x) and time (t). Suitable for heat conduction, diffusion, wave equation, and reaction-diffusion systems.
2. Partial Differential Equation Toolbox – Uses finite element method (FEM) for 2-D and 3-D problems (structural mechanics, heat transfer, electromagnetics, general scalar/vector PDEs).

Using pdepe (1-D PDEs)

PDE must be written in the standard form: c(u,x,t) * ∂u/∂t = ∂/∂x [ f(u,x,t,u_x) ] + s(u,x,t,u,u_x)

Steps:

1. Define PDE coefficients in a function [c,f,s] = mypde(x,t,u,dudx).
2. Define initial condition u0(x) = myic(x).
3. Define boundary conditions [pl,ql,pr,qr] = mybc(xa,ua,xb,ub,t).
4. Create spatial and time meshes: x = linspace(0,1,50); t = linspace(0,1,100);.
5. Solve: sol = pdepe(m, @mypde, @myic, @mybc, x, t); (m = 0 for slab/Cartesian, 1 cylindrical, 2 spherical).
6. Extract solution: u = sol(:,:,k) for k-th equation.
7. Plot: surf(x,t,u) or pcolor(x,t,u).

Using Partial Differential Equation Toolbox (2-D/3-D)

Steps:

1. Create model: model = createpde(N); (N = number of equations).
2. Define geometry (built-in shapes, CSG, or import STL).
3. Specify PDE coefficients (c, a, f, d terms).
4. Apply boundary conditions (Dirichlet, Neumann, generalized).
5. Generate mesh: generateMesh(model);.
6. Solve:
   • Stationary: result = solvepde(model);
   • Transient: solvepdetransient
   • Eigenvalue: solvepdeeig
7. Visualize: pdeplot(model,'XYData',u,'Contour','on'); or pdeplot3D.

Ideal for engineering and physics simulations.