Partial differential algebraic equation

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In mathematics a partial differential algebraic equation (PDAE) set is an incomplete system of partial differential equations that is closed with a set of algebraic equations.

Mathematics Field of study concerning quantity, patterns and change

Mathematics includes the study of such topics as quantity, structure, space, and change.

Partial differential equation differential equation that contains unknown multivariable functions and their partial derivatives

In mathematics, a partial differential equation (PDE) is a differential equation that contains unknown multivariable functions and their partial derivatives. PDEs are used to formulate problems involving functions of several variables, and are either solved by hand, or used to create a computer model. A special case is ordinary differential equations (ODEs), which deal with functions of a single variable and their derivatives.

In mathematics, an algebraic equation or polynomial equation is an equation of the form

Contents

Definition

A general PDAE is defined as:

where:

The relationship between a PDAE and a partial differential equation (PDE) is analogous to the relationship between an ordinary differential equation (ODE) and a differential algebraic equation (DAE).

In mathematics, an ordinary differential equation (ODE) is a differential equation containing one or more functions of one independent variable and the derivatives of those functions. The term ordinary is used in contrast with the term partial differential equation which may be with respect to more than one independent variable.

PDAEs of this general form are challenging to solve. Simplified forms are studied in more detail in the literature. [1] [2] [3] Even as recently as 2000, the term "PDAE" has been handled as unfamiliar by those in related fields. [4]

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. [5] [6] This method involves removing the spatial variables using a discretization 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.

Time dimension in which events can be ordered from the past through the present into the future

Time is the indefinite continued progress of existence and events that occur in apparently irreversible succession from the past, through the present, to the future. Time is a component quantity of various measurements used to sequence events, to compare the duration of events or the intervals between them, and to quantify rates of change of quantities in material reality or in the conscious experience. Time is often referred to as a fourth dimension, along with three spatial dimensions.

Space general framework of distances and directions according to a physical observer in its proper time

Space is the boundless three-dimensional extent in which objects and events have relative position and direction. Physical space is often conceived in three linear dimensions, although modern physicists usually consider it, with time, to be part of a boundless four-dimensional continuum known as spacetime. The concept of space is considered to be of fundamental importance to an understanding of the physical universe. However, disagreement continues between philosophers over whether it is itself an entity, a relationship between entities, or part of a conceptual framework.

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 evaluation and implementation on digital computers. Dichotomization is the special case of discretization in which the number of discrete classes is 2, which can approximate a continuous variable as a binary variable.

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References

  1. ↑ Wagner, Y. 2000. "A further index concept for linear PDAEs of hyperbolic type," Mathematics and Computers in Simulation, v. 53, pp. 287–291.
  2. ↑ W. S. Martinson, P. I. Barton. (2002) "Index and characteristic analysis of linear PDAE systems," SIAM Journal on Scientific Computing, v. 24, n. 3, pp. 905–923.
  3. ↑ Lucht, W.; Strehmel, K.. 1998. "Discretization based indices for semilinear partial differential algebraic equations," Applied Numerical Mathematics, v. 28, pp. 371–386.
  4. ↑ Simeon, B.; Arnold, M.. 2000. "Coupling DAEs and PDEs for simulating the interaction of pantograph and catenary," Mathematical and Computer Modelling of Dynamical Systems, v. 6, pp. 129–144.
  5. ↑ Jacob, J.; Le Lann, J; Pinguad, H.; Capdeville, B.. 1996. "A generalized approach for dynamic modelling and simulation of biofilters: application to waste-water denitrification," Chemical Engineering Journal, v. 65, pp. 133–143.
  6. ↑ 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.


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