Environmental Engineering Reference
In-Depth Information
Chapter 9
Ordinary Differential Equations:
Dynamical Systems
Ordinary differential equations (ode) are differential equations for functions which
depend on one independent variable only. These 'odes' are simpler than partial
differential equations which contain more than one independent variable. In almost
all models or simulations independent variables are either time and/or space.
In environmental modeling, two situations can be distinguished in which odes
appear. The first situation deals with systems in which spatial differences can be
neglected and the temporal development is questioned. In chemistry, the continu-
ously stirred reactor is an often used concept for which an approach is allowed with
time
t
as the only independent variable.
One such situation a was already described In Chap. 5.1, with degradation or
decay as the only relevant process for the transient change of some species concen-
tration. Below (Chap. 10.1) an example is presented dealing with the determination
of a reaction kinetic, using data from batch experiments. In the field situation such
ideal systems are seldom appropriate, but sometimes the assumption of no space-
dependence may be approximately fulfilled. The long-term accumulation of a sub-
stance in lakes can be modeled with the idea of an ideally mixed reservoir, for
example.
In a second different constellation time is neglected and a steady state is sought
for a system which can be described by a single space variable. Such models are
common for aquatic sediments, where parameters and variables show characteristic
changes normal to the water-sediment interface, usually in vertical direction. Also
in streams the one-dimensional approach can be applied under certain circumstances:
the space coordinate is taken along the steady streamline following the river down-
stream. Surface water infiltrating an aquifer has a pronounced direction along the
flow path. Here the 1D formulation is justified, because the transverse gradients
almost vanish.
Aside of analytical solutions,
are
introduced in this chapter, one (
ode15s
) designed for the solution of initial value
problems, the other (
bvp4c
) for the solution of boundary value problems (bvp). In
initial value problems boundary conditions are formulated for one value of the
independent variable only (typically:
t ¼
two numerical solvers of MATLAB
®
0), whereas in boundary value problems
