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6.7
Partial Differential Algebraic Equations
Very similar to the definition of data fitting problems based on partial differential equations
outlined in the previous section, we have to define fitting criteria, differential equations,
initial and boundary conditions, coupling and transition equations and constraints in a
suitable format. For defining variables, we need the following rules:
1. The first names are the identifiers for n independent parameters to be estimated, p1 ,
. . ., pn .
2. The subsequent names identify the np state variables of the system, u1 , . . ., unp , where
first the differential, then the algebraic variables must be listed.
3. In a similar way, the corresponding variables denoting the first and second spatial
derivatives of differential and algebraic variables are to be declared in this order, u1x ,
. . ., unp x and u1xx , . . ., unp xx .
4. Next, names of nc variables belonging to coupled differential algebraic equations are
defined, v1 , . . ., vnc , where first the differential, then the algebraic variables must be
given.
5. If flux functions are to be inserted into the right-hand side formulation of the PDAE,
then np identifiers for the fluxes and their spatial derivatives are given, f1 , . . ., fnp and
f1 x , . . ., fnp x .
6. Then a name is to be defined for the space or spatial variable x.
7. The last name identifies the independent time variable t for which measurements are
available.
8. Any other variables are not allowed to be declared.
Model functions are defined in the following format:
1. If flux functions are to be used, then na np functions f1i (p, u, ux, x, t), . . ., fni p (p, u, ux, x, t)
defining the flux are inserted, one set for each integration area, i = 1, . . ., ma . They
may depend on x, t, u, ux , and p.
2. Functions for the right-hand side of partial differential equations
F1i (p, f i, fxi , u, ux, v, x, t), . . . , Fni p (p, f i, fxi , u, ux, v, x, t)
are defined next, one set for each integration area, i = 1, . . ., ma . Each function
may depend on x, t, v, u, ux , uxx , p, and, optionally, also on the flux functions and
their derivatives. First, the differential equations, then the algebraic equations must
be defined.
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