name: differential-equation-solver description: Solve linear and nonlinear differential equations using chebops. Use when solving boundary value problems, ODEs, integral equations, or PDEs with spectral methods.

Differential Equation Solver

Linear Differential/Integral Equations

Basic Structure

Solve equations of the form L(u) = f where:

• L is a linear differential or integral operator
• u is the unknown function
• f is the forcing function

Step-by-Step Process

1. Define the operator:

   L = chebop(@(x,u) diff(u,2) + x.^3*u, [-3,3]);
   

2. Specify boundary conditions:

   L.bc = 0;                                    % Dirichlet (zero at both ends)
   L.bc = 100;                                  % Dirichlet (value at both ends)
   L.bc = @(u) diff(u);                         % Neumann (derivative conditions)
   L.bc = 'periodic';                           % Periodic BCs
   

   Alternative single-line syntax:

   L = chebop(op, domain, bc_left, bc_right);
   

3. Solve:

   u = L \ f;
   

4. Verify solution:

   norm(L(u) - f)    % Should be near machine precision
   

Example: Second-Order ODE

% Define operator: u'' + x^3*u = f
L = chebop(@(x,u) diff(u,2) + x.^3*u, [-1,1]);
L.bc = 0;                    % u(-1) = u(1) = 0
f = chebfun(@(x) exp(x), [-1,1]);
u = L \ f;                   % Solve

Time-Dependent PDEs (Operator Exponential)

For PDEs of the form du/dt = Lu:

1. Define spatial operator:

   L = chebop(@(x,u) diff(u,2), domain);  % Second derivative
   

2. Compute operator exponential for time t:

   E = expm(t * L);
   

3. Apply to initial condition:

   u_t = E * u_0;
   

Example: Heat equation (u_t = u_xx):

L = chebop(@(x,u) diff(u,2), [-1,1]);
L.bc = 0;
u_0 = chebfun(@(x) exp(-10*x.^2), [-1,1]);
u_at_t = expm(0.1 * L) * u_0;  % Solution at t = 0.1

Nonlinear Equations

Automated Nonlinear Backslash

u = L \ f;    % Chebfun uses automatic differentiation

Chebfun automatically constructs the Frechet derivative (Jacobian).

Manual Newton Iteration

% Construct Jacobian explicitly
J = chebop(@(x,u) linearized_operator, ...);

% Iterate
u_new = u - J \ (L(u) - f);

Unknown Parameters

Include parameters as unknowns with zero derivatives:

% Solve with unknown temperature T
L = chebop(@(x,u,T) diff(u,2) + u.^2 - T, [0,1]);
L.bc = @(u,T) [u(0); diff(u,1) - T];
f = 0;

uT = L \ f;      % Returns chebmatrix

% Extract results
u = uT{1};       % Solution function
T = uT{2};       % Parameter value

Important: Different initial guesses may converge to different solutions.

Use When

• Solving boundary value problems (BVPs)
• Working with linear or nonlinear ODEs
• Solving integral equations
• Computing time evolution of PDEs
• Finding eigenvalues/eigenfunctions
• Problems with unknown parameters