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| In [[mathematics]] a '''partial differential algebraic equation (PDAE)''' set is an incomplete system of [[partial differential equation]]s that is closed with a set of [[algebraic equation]]s.
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| == Definition ==
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| A general PDAE is defined as:
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| : <math>0 = \mathbf F \left( \mathbf x, \mathbf y, \frac{\partial y_i}{\partial x_j}, \frac{\partial^2 y_i}{\partial x_j \partial x_k}, \ldots, \mathbf z \right),</math>
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| where:
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| * '''F''' is a set of arbitrary functions;
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| * '''x''' is a set of independent variables;
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| * '''y''' is a set of dependent variables for which partial derivatives are defined; and
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| * '''z''' is a set of dependent variables for which no partial derivatives are defined.
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| 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).
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| PDAEs of this general form are challenging to solve. Simplified forms are studied in more detail in the literature.<ref>Wagner, Y. 2000. "A further index concept for linear PDAEs of hyperbolic type," Mathematics and Computers in Simulation, v. 53, pp. 287–291.</ref><ref>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.</ref><ref>Lucht, W.; Strehmel, K.. 1998. "Discretization based indices for semilinear partial differential algebraic equations," Applied Numerical Mathematics, v. 28, pp. 371–386.</ref> Even as recently as 2000, the term "PDAE" has been handled as unfamiliar by those in related fields.<ref>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.</ref>
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| == 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.<ref>Jacob, J.; Le Lann, J; Pinguad, H.; Capdeville, B.. 1996. "A generalized approach for dynamic modelling
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| and simulation of biofilters: application to waste-water denitrification," Chemical Engineering Journal, v.
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| 65, pp. 133–143.</ref><ref>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.</ref> 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 [[differential algebraic equation|DAE]], for which conventional solution methods can be employed.
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| == References == | |
| {{Reflist}}
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| [[Category:Partial differential equations]]
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| [[Category:Differential equations]]
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| [[Category:Multivariable calculus]]
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| [[Category:Numerical analysis]]
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| {{applied-math-stub}}
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