Approximate Methods
From Scholarpedia
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Curator: Dr. Andrei D. Polyanin, Institute for Problems in Mechanics, Moscow, Russia
Curator: Prof. William E. Schiesser, Lehigh University, USA
Curator: Dr. Alexei I. Zhurov, Cardiff University, UK, and Institute for Problems in Mechanics, Moscow, Russia.
Contents |
First-Order Partial Differential Equations
Second-Order Partial Differential Equations
Higher-Order Partial Differential Equations
Approximate and Numerical Methods
The preceding discussion pertains to the exact or analytical solution of PDEs. For example, in the case of Eqs. () and (), an exact solution would be a function 
which, when substituted into Eq. () or (), would satisfy it identically along with all of the associated initial and boundary conditions.
Although analytical solutions are exact, they also may not be available, simply because we do not know how to derive such solutions. This could be because the PDE system has too many PDEs, or they are too complicated, e.g., nonlinear, or both, to be amenable to analytical solution. In this case, we may have to resort to an approximate solution. That is, we seek an analytical or numerical approximation to the exact solution.
Perturbation methods are a very important subset of approximate analytical methods. They may be applied if the problem involves small (or large) parameters, which are used for constructing solutions in the form of asymptotic expansions. For books on perturbation methods, see Google Book Search. These and other methods for PDEs are also outlined in Zwillinger (1997).
Unlike exact and approximate analytical methods, methods to compute numerical PDE solutions are in principle not limited by the number or complexity of the PDEs. This generality combined with the availability of high performance computers makes the calculation of numerical solutions feasible for a broad spectrum of PDEs (such as the Navier–Stokes equations) that are beyond analysis by analytical methods. The development and implementation (as computer codes) of numerical methods or algorithms for PDE systems is a very active area of research. Here we indicate in the external links just two readily available links to Scholarpedia.
