mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

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 involves a multivariable function and one or more of its partial derivatives. The function is often thought of as an "unknown" that solves the equation, similar to ho ...

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 example, x^5-3x+1=0 is an algebraic equati ...

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

(PDE) is analogous to the relationship between an

ordinary differential equation In mathematics, an ordinary differential equation (ODE) is a differential equation (DE) dependent on only a single independent variable (mathematics), variable. As with any other DE, its unknown(s) consists of one (or more) Function (mathematic ...

(ODE) and a

differential algebraic equation In mathematics, a differential-algebraic system of equations (DAE) is a system of equations that either contains differential equations and algebraic equations, or is equivalent to such a system. The set of the solutions of such a system is a ...

(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 Time is the continuous progression of existence that occurs in an apparently irreversible process, irreversible succession from the past, through the present, and into the future. It is a component quantity of various measurements used to sequ ...

and

space Space is a three-dimensional continuum containing positions and directions. In classical physics, physical space is often conceived in three linear dimensions. Modern physicists usually consider it, with time, to be part of a boundless ...

, 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 numeri ...

method, such as the

finite volume method The finite volume method (FVM) is a method for representing and evaluating partial differential equations in the form of algebraic equations. In the finite volume method, volume integrals in a partial differential equation that contain a divergen ...

, 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

{{Reflist Partial differential equations Multivariable calculus Numerical analysis