User:Leo Trottier/PDE/Partial Differential Equation

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.

First-Order Partial Differential Equations

Main article: First-order partial differential equations

A first-order partial differential equation with $$n$$independent variables has the general form $F\biggl(x_1,x_2,\dots, x_n,w,\frac{\partial w}{\partial x_1}, \frac{\partial w}{\partial x_2},\dots,\frac{\partial w}{\partial x_n}\biggl)=0,$ where $$w=w(x_1,x_2,\dots, x_n)$$ is the unknown function and $$F(\dots)$$ is a given function.

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Second-Order Partial Differential Equations

Main article: Second-order partial differential equations

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Higher-Order Partial Differential Equations

Main article: Higher-order partial differential equations

Apart from second-order PDEs, higher-order equations also quite often arise in applications. Below are only a few important examples of such equations with some of their solutions.

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Approximate and Numerical Methods

Main article: Approximate and numerical methods for partial differential equations

The preceding discussion pertains to the exact or analytical solution of PDEs. For example, in the case of a heat equation or a wave equation, an exact solution would be a function $$w=f(x,t)$$ which, when substituted into the respective equation 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 an 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).

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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.