Page 25 - Computational Modeling in Biomedical Engineering and Medical Physics

Physical, mathematical, and numerical modeling 11

order ODE, of the order equal to the number of existing reactive elements plus one.
Summing up a consistent Cauchy problem may thus be formulated. Cauchy problem
solvers are available for both linear and nonlinear ODEs (Vetterling et al., 1997).

Boundary and initial values problems
The physicals model for a system that executes a process under internal and external
constraints lead to the mathematical models represented through PDEs, built out of
the mathematical equations that present the physical laws, the boundary and initial
conditions. Two general classes of problems may be distinguished: well-posed and
ill-posed problems.
In Hadamard’s definition (Hadamard, 1902, 1923), a problem is well-posed if (1) it
has a solution (here, a physical solution); (2) the solution is unique; and (3) the solu-
tion depends continuously on the problem data. If any of these conditions is not ful-
filled, then the problem is ill posed.
In practice, even though the existence of a solution is an important requirement
for exact data, the condition (1) is sometimes satisfied provided the concept of solution
is relaxed.
Condition (2) is more important. Should a problem admit multiple solutions then
the question that arises is which of them is relevant in a particular situation. One may
decide to use additional information in order to restrict the set of admissible solutions.
It is very probable that in a practical application the existing measurement data even if,
in the limit, available in an infinite number of points does not completely determine
the solution. Although condition (2) is fulfilled in the continuous formulation sense of
the problem when data are known everywhere, the nonunicity of the solution may
occur due to the discretization of the computational domain, when numerical meth-
ods are used to solve the problem.
Condition (3) is motivated by the fact that in applications the problem data are
obtained thorough measurements prone to errors. It is wishfully expectable that small
errors do not amplify the errors in the solution. If not observed, condition (3) may
produce significant numerical concerns because the numerical methods may become
unstable. This difficulty may be partially alleviated by the usage of the regularization
techniques (Vetterling et al., 1997). However, no mathematical artifact may fully “cure”
the intrinsically unstable nature of ill-posed problems that are not complying with
condition (3). In general regularization methods may recover partial information on
the solutions and their application is actually an accepted compromise between the
accuracy and the stability of the solution.
From the perspective of the input data, the mathematical models distinguish
between direct and inverse problems. In direct problems the structure of the system, the
materials and their properties, the internal sources and the boundary interactions are