HOME

TheInfoList



OR:

In mathematics a partial differential algebraic equation (PDAE) set is an incomplete system of
partial differential equation In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a multivariable function. The function is often thought of as an "unknown" to be solved for, similarly to ...
s that is closed with a set of
algebraic equation In mathematics, an algebraic equation or polynomial equation is an equation of the form :P = 0 where ''P'' is a polynomial with coefficients in some field, often the field of the rational numbers. For many authors, the term ''algebraic equation'' ...
s.


Definition

A general PDAE is defined as: : 0 = \mathbf F \left( \mathbf x, \mathbf y, \frac, \frac, \ldots, \mathbf z \right), where: * F is a set of arbitrary functions; * x is a set of independent variables; * y is a set of dependent variables for which partial derivatives are defined; and * z is a set of dependent variables for which no partial derivatives are defined. The relationship between a PDAE and a
partial differential equation In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a multivariable function. The function is often thought of as an "unknown" to be solved for, similarly to ...
(PDE) is analogous to the relationship between an
ordinary differential equation In mathematics, an ordinary differential equation (ODE) is a differential equation whose unknown(s) consists of one (or more) function(s) of one variable and involves the derivatives of those functions. The term ''ordinary'' is used in contrast ...
(ODE) and a differential algebraic equation (DAE). PDAEs of this general form are challenging to solve. Simplified forms are studied in more detail in the literature. Even as recently as 2000, the term "PDAE" has been handled as unfamiliar by those in related fields.


Solution methods

Semi-discretization is a common method for solving PDAEs whose independent variables are those of time and space, and has been used for decades.de Dieuvleveult, C.; Erhel, J.; Kern, M.. 2009. "A global strategy for solving reactive transport equations," Journal of Computational Physics, v. 228, pp. 6395–6410. This method involves removing the spatial variables using a
discretization In applied mathematics, discretization is the process of transferring continuous functions, models, variables, and equations into discrete counterparts. This process is usually carried out as a first step toward making them suitable for numerical ...
method, such as the finite volume method, and incorporating the resulting linear equations as part of the algebraic relations. This reduces the system to a DAE, for which conventional solution methods can be employed.


References

Partial differential equations Differential equations Multivariable calculus Numerical analysis {{applied-math-stub