% The derivative du/dx is approximated with finite differences, using % single-sided differences on the edges and centered differences in between.

% M1 and M2 are the portions of the mass matrix for Burgers' equation. % Boundary values of solution u and mesh x % Extract the components of y for the solution u and mesh x This formulation shows that M 1 and M 2 form the left side of Burgers' equations (the first N equations in the system), while M 3 and M 4 form the left side of the mesh equations (the last N equations in the system). The mass matrix is composed of four blocks, each of which is a square matrix of order N: Set the left side of the system of equations equal to M y ′ to extract the form of the mass matrix. The left side of the system of equations involves linear combinations of first derivatives, so a mass matrix is required to represent all of the terms. You can either include the required functions as local functions at the end of a file (as done here), or save them as separate, named files in a directory on the MATLAB path. To solve this equation in MATLAB®, write a derivative function, a mass matrix function, a function for the sparsity pattern of the Jacobian dF / dy, and a function for the sparsity pattern of d ( Mv ) / dy.

Supplying these sparsity patterns helps the solver operate more efficiently.įinite differences are used to approximate several partial derivatives. The sparsity patterns of the Jacobian dF / dy and the derivative of the mass matrix multiplied with a vector d ( Mv ) / dy are supplied to the solver as functions. The derivative function not only includes the equations for Burgers' equation, but also a set of equations governing the moving mesh selection. The mass matrix is provided to the ode15s solver as a function. The system of equations is expressed using a mass matrix formulation, M y ′ = f ( t, y ).

Solve matrix vector equation matlab