Introduction to PDE Problems :: Differential Equations (Mathematics)

Mathematics

Introduction to PDE Problems

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

equation showing the form of systems of partial differential equations in one spatial variable (5-3)

The PDEs hold for t sub 0 less than or equal t less than or equal t sub f and a less than or equal x less than or equal b . The interval [a,b] must be finite. can be 0, 1, or 2, corresponding to slab, cylindrical, or spherical symmetry, respectively. If m greater than 0 , then a greater than or equal 0 must also hold.

In Equation 5-3, f(x,t, u, partial derivative of u with repect to x) is a flux term and s(x,t, u, partial derivative of u with repect to x) is a source term. The flux term must depend on . The coupling of the partial derivatives with respect to time is restricted to multiplication by a diagonal matrix c(x,t, u, partial derivative of u with repect to x) . 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 t = t sub 0 , for all the solution components satisfy initial conditions of the form

u(x, t sub 0) = u sub 0 (x) (5-4)

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

p(x,t,u) + q(x,t) f (x,t,u, partial derivative of u with respect to x) = 0 (5-5)

q(x,t) 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 partial derivative of u with respect to x . Also, of the two coefficients, only can depend on .

PDE Function Summary MATLAB Partial Differential Equation Solver