Partial differential equation
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(Redirected from Partial differential equations)
Curator: Dr. Andrei D. Polyanin, Institute for Problems in Mechanics, Moscow, Russia
Curator: Dr. William E. Schiesser, Lehigh University, 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
1 First-Order Partial Differential Equations
1.1 General Form of First-Order Partial Differential Equation
1.2 Quasilinear Equations. Characteristic System. General Solution
1.2.1 General form of first-order quasilinear PDE
1.2.2 Characteristic system. General solution
1.3 Cauchy Problem: Two Formulations. Solving the Cauchy Problem
1.3.1 Generalized Cauchy problem
1.3.2 Classical Cauchy problem
1.3.3 Existence and uniqueness theorem
1.3.4 Procedure of solving the Cauchy problem
2 Second-Order Partial Differential Equations
2.1 Linear, Semilinear, and Nonlinear Second-Order PDEs
2.1.1 Linear second-order PDEs and their properties. Principle of linear superposition
2.1.2 Semilinear and nonlinear second-order PDEs
2.2 Some Linear Equations Encountered in Applications
2.2.1 Heat equation (a parabolic equation)
2.2.2 Wave equation (a hyperbolic equation)
2.2.3 Laplace equation (an elliptic equation)
2.3 Classification of Second-Order Partial Differential Equations
2.3.1 Types of equations
2.3.2 Characteristic equations
2.3.3 Canonical form of parabolic equations
2.3.4 Two canonical forms of hyperbolic equations
2.3.5 Canonical form of elliptic equations
2.4 Basic Problems for PDEs of Mathematical Physics
2.4.1 Cauchy problem and boundary value problems for parabolic equations
2.4.2 Cauchy problem and boundary value problems for hyperbolic equations
2.4.3 Boundary value problems for elliptic equations
2.5 Some Nonlinear Equations Encountered in Applications
2.5.1 Nonlinear heat equation
2.5.2 Kolmogorov–Petrovskii–Piskunov equation
2.5.3 Burgers equation
2.5.4 Nonlinear wave equation
2.5.5 Nonlinear Klein–Gordon equation
2.5.6 Nonlinear Laplace equation
2.5.7 Monge–Ampère equation
2.6 Simplest Types of Exact Solutions of Nonlinear PDEs
2.6.1 Preliminary remarks
2.6.2 Traveling-wave solutions
2.6.3 Self-similar solutions
2.7 Cauchy Problem and Boundary Value Problems for Nonlinear Equations
3 Higher-Order Partial Differential Equations
3.1 Higher-Order Linear Partial Differential Equations
3.1.1 Equation of transverse vibration of elastic rod
3.1.2 Biharmonic equation
3.2 Higher-Order Nonlinear Partial Differential Equations
3.2.1 Korteweg–de Vries equation
3.2.2 Equation of a steady laminar boundary layer on a flat plate
3.2.3 Boussinesq equation
3.2.4 Equation of motion of a viscous fluid
4 Approximate and Numerical Methods
4.1 Parabolic PDE
4.2 Hyperbolic PDE
4.3 Elliptic PDE
4.4 Remarks
4.5 Appendix 1. MATLAB program for a parabolic PDE
4.6 Appendix 2. MATLAB program for a hyperbolic PDE
4.7 Appendix 3. MATLAB program for an elliptic PDE
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 |
