Mathematics

Introduction to PDE Problems

pdepe solves systems of PDEs in one spatial variable and time , of the form

(14-4)

The PDEs hold for and . The interval must be finite. can be 0, 1, or 2, corresponding to slab, cylindrical, or spherical symmetry, respectively. If , then must also hold.

In Equation 14-4, is a flux term and is a source term. The coupling of the partial derivatives with respect to time is restricted to multiplication by a diagonal matrix . The diagonal elements of this matrix are either identically zero or positive. An element that is identically zero corresponds to an elliptic equation and otherwise to a parabolic equation. There must be at least one parabolic equation. An element of that corresponds to a parabolic equation can vanish at isolated values of if they are mesh points. Discontinuities in and/or due to material interfaces are permitted provided that a mesh point is placed at each interface.

At the initial time , for all the solution components satisfy initial conditions of the form

(14-5)

At the boundary or , for all the solution components satisfy a boundary condition of the form

(14-6)

is a diagonal matrix with elements that are either identically zero or never zero. Note that the boundary conditions are expressed in terms of the flux rather than . Also, of the two coefficients, only can depend on .

PDE Function Summary

MATLAB Partial Differential Equation Solver