Partial differential equation
From Scholarpedia
| Andrei D. Polyanin et al. (2008), Scholarpedia, 3(10):4605. | revision #55842 [link to/cite this article] | |||||||||||||||||||
Curator: Dr. Andrei D. Polyanin, Institute for Problems in Mechanics, Moscow, Russia
Curator: Dr. William E. Schiesser, Lehigh University and University of Pennsylvania, USA
Curator: Dr. Alexei I. Zhurov, Cardiff University, UK, and Institute for Problems in Mechanics, Moscow, Russia.
A partial differential equation (or briefly a PDE) is a mathematical equation that involves two or more independent variables, an unknown function (dependent on those variables), and partial derivatives of the unknown function with respect to the independent variables. The order of a partial differential equation is the order of the highest derivative involved. A solution (or a particular solution) to a partial differential equation is a function that solves the equation or, in other words, turns it into an identity when substituted into the equation. A solution is called general if it contains all particular solutions of the equation concerned.
The term exact solution is often used for second- and higher-order nonlinear PDEs to denote a particular solution (see also Preliminary remarks at Second-Order Partial Differential Equations).
Partial differential equations are used to mathematically formulate, and thus aid the solution of, physical and other problems involving functions of several variables, such as the propagation of heat or sound, fluid flow, elasticity, electrostatics, electrodynamics, etc.
Contents |
First-Order Partial Differential Equations
Second-Order Partial Differential Equations
Higher-Order Partial Differential Equations
Approximate and Numerical Methods
References
- R. Courant and D. Hilbert, Methods of Mathematical Physics. Volume 2. Partial Differential Equations, Wiley-VCH, 1989.
- L. C. Evans, Partial Differential Equations, American Mathematical Society, Providence, 1998.
- S. J. Farlow, Partial Differential Equations for Scientists and Engineers, Dover Publications Inc., 1993.
- F. John, Partial Differential Equations. Fourth Edition, Springer, 1991.
- J. Jost, Partial Differential Equations, Springer-Verlag, New York, 2002.
- I. G. Petrovskii, Partial Differential Equations, W. B. Saunders Co., Philadelphia, 1967.
- Y. Pinchover and J. Rubinstein, An Introduction to Partial Differential Equations, Cambridge University Press, Cambridge, 2005.
- A. D. Polyanin, Handbook of Linear Partial Differential Equations for Engineers and Scientists, Chapman & Hall/CRC Press, Boca Raton, 2002.
- A. D. Polyanin and V. F. Zaitsev, Handbook of Nonlinear Partial Differential Equations, Chapman & Hall/CRC Press, Boca Raton, 2004.
- A. D. Polyanin, V. F. Zaitsev, and A. Moussiaux, Handbook of First Order Partial Differential Equations, Taylor & Francis, London, 2002.
- D. L. Powers, Boundary Value Problems, Fifth Edition: and Partial Differential Equations, Elsevier Academic Press, 2005.
- W. E. Schiesser, Computational Mathematics in Engineering and Applied Science: ODEs, DAEs, and PDEs, CRC Press, Boca Raton, 1993.
- I. Stakgold, Boundary Value Problems of Mathematical Physics, Vols. I, II, SIAM, Philadelphia, 2000.
- A. N. Tikhonov and A. A. Samarskii, Equations of Mathematical Physics, Dover Publ., New York, 1990.
- D. Zwillinger, Handbook of Differential Equations (3rd edition), Academic Press, Boston, 1997.
Internal references
- Ian Gladwell (2008) Boundary value problem. Scholarpedia, 3(1):2853.
External links
- Partial Differential Equations: Exact Solutions at EqWorld: The World of Mathematical Equations [1]
- Partial Differential Equations: Index of PDEs at EqWorld: The World of Mathematical Equations [2]
- Partial Differential Equations: Methods at EqWorld: The World of Mathematical Equations [3]
- Partial Differential Equation at Wolfram MathWorld by Eric Weisstein [4]
- Example problems with solutions at ExampleProblems.com [5]
- General reference for numerical methods at Scholarpedia [6]
- Introduction to numerical methods for partial differential equations at Scholarpedia [7]
| Andrei D. Polyanin, William E. Schiesser, Alexei I. Zhurov (2008) Partial differential equation. Scholarpedia, 3(10):4605, (go to the first approved version) Created: 2 August 2007, reviewed: 29 September 2008, accepted: 10 October 2008 |
