A differential equation is an equation involving one or more dependent variables (called yi or ui), their derivatives, and one or more independent variables (called t, x, and y). Users will typically need to relabel their own model variables so that they correspond to the variables used in the solvers described here. A differential equation with one independent variable is called an ordinary differential equation (ODE). A system of equations involving derivatives in one independent variable and other dependent variables is called a differential-algebraic system. A differential equation with more than one independent variable is called a partial differential equation (PDE).

The order of a differential equation is the highest order of any of the derivatives in the equation. Some of the routines in this chapter require the user to reduce higher-order problems to systems of first-order differential equations.

It is convenient to use the vector notation below. We denote the number of equations as the value N. The problem statement is abbreviated by writing it as a system of first-order ODEs

The problem becomes

with initial values y(t0). Values of y(t) for t > t0 or t < t0 are required. The routines IVPRK, IVMRK, and IVPAG, solve the IVP for systems of ODEs of the form yʹ = f(t, y) with y(t = t0) specified. Here, f is a user supplied function that must be evaluated at any set of values (t, y1, …, yN); i = 1, …, N. The routines IVPAG, and DAESL, will also solve implicit systems of the form Ayʹ = f(t, y) where A is a user supplied matrix. For IVPAG, the matrix A must be nonsingular.

The system yʹ = f(t, y) is said to be stiff if some of the eigenvalues of the Jacobian matrix {∂fi/∂yj} have large, negative real parts. This is often the case for differential equations representing the behavior of physical systems such as chemical reactions proceeding to equilibrium where subspecies effectively complete their reaction in different epochs. An alternate model concerns discharging capacitors such that different parts of the system have widely varying decay rates (or time constants). This definition of stiffness, based on the eigenvalues of the Jacobian matrix, is not satisfactory. Users typically identify stiff systems by the fact that numerical differential equation solvers such as IVPRK, are inefficient, or else they fail. The most common inefficiency is that a large number of evaluations of the functions fi are required. In such cases, use routine IVPAG, or DAESL. For more about stiff systems, see Gear (1971, Chapter 11) or Shampine and Gear (1979).

In the boundary value problem (BVP) for ODEs, constraints on the dependent variables are given at the endpoints of the interval of interest, [a, b]. The routines BVPFD and BVPMS solve the BVP for systems of the form yʹ(t) = f(t, y), subject to the conditions

Here, f and h = [h1, …, hN]T are user-supplied functions.

IVOAM solves systems of ordinary differential equations of order one, order two, or mixed order one and two.

Frequently, it is not possible or not convenient to express the model of a dynamical system as a set of ODEs. Rather, an implicit equation is available in the form

The gi are user-supplied functions. The system is abbreviated as

With initial value y(t0). Any system of ODEs can be trivially written as a differential-algebraic system by defining

The routine DAESL solves differential-algebraic systems of index 1 or index 0. For a definition of index of a differential-algebraic system, see (Brenan et al. 1989). Also, see Gear and Petzold (1984) for an outline of the computing methods used.

The routine MMOLCH solves the IVP problem for systems of the form

subject to the boundary conditions

and subject to the initial conditions

The routines FPS2H and FPS3H solve Laplace’s, Poisson’s, or Helmholtz’s equation in two or three dimensions. FPS2H uses a fast Poisson method to solve a PDE of the form

over a rectangle, subject to boundary conditions on each of the four sides. The scalar constant c and the function f are user specified. FPS3H solves the three-dimensional analogue of this problem.

The following table summarizes the types of problems handled by the routines in this chapter. With the exception of FPS2H and FPS3H, the routines can handle more than one differential equation.