Preface

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Solutions of PDEs

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Experience from solving ordinary differential equations (ODEs) suggests that solution techniques for PDEs are varied, and depend on such equation characteristics as linearity and order. It is reasonable, and correct, to assume methods for solving PDEs are more numerous and equation specific due to the increased complexity of multivariable functions.

In keeping with the similarities for solving ODEs, solution techniques for PDEs may be divided into two general categories: analytic and numerical. Analytical techniques are those that strive to find "exact" formulas for the dependent variable u as a function of all independent variables. Values for u can be determined for all times and all locations within the temporal and spacial domain of the problem. Analytical methods include separation of variables, where a PDE of n independent variables is transformed to n ODEs, integral transformations, where a PDE in n −1 variables (a PDE in two variables would be transformed to an ODE), calculus of variations, where the solution to PDE is the same function that minimizes a companion energy function, and eigenfunction expansion, where the solution to the PDE is presented as an infinite sum of eigenfunctions that solve a corresponding eigenvalue problem.

Numerical methods result in approximate values of the dependent variable u at prescribed, discrete locations within a finite domain of the independent variables. There are various numerical methods, and many are specific to PDE type. A unifying feature is the transformation of the PDE into an alternate expression that commonly includes a system of linear algebraic equations. Such techniques include the finite difference method, the finite element method, the finite volume method and the finite analytic method. A somewhat more detailed description of these numerical methods approaches in Part VII of thsi tutorial.

This tutorial focuses on analytical methods of separation of variables and integral transformations that are suitable for several classes of linear partial differential equations. The numerical part of this tutorial is devoted to finite difference schemes.

The set of functions used to formulate a PDE, which might include coefficients or terms in the equation itself as well as boundary and initial conditions, is collectively referred to as the input data. The most basic question for any PDE is whether a solution exists for a given set of data. However, for most purposes we want to require something more. A PDE problem is said to be well posed if, for a given set of data:

1. A solution exists.
2. The solution is uniquely determined by the data.
3. The solution depends continuously on the data

These criteria were formulated by Jacques Hadamard in 1902. The first two properties hold for ODE under rather general assumptions, but not necessarily for PDE. It is easy to find nonlinear equations that admit no solutions, and even in the linear case there is no guarantee.

The third condition, continuous dependence on the input data, is sometimes called stability. One practical justification for this requirement is it is not possible to specify input data with absolute accuracy. Stability implies that the effects of small variations in the data can be controlled.

For certain PDE, especially the classical linear cases, we have a good under- standing of the requirements for well-posedness. For other important problems, for example, in fluid mechanics, well-posedness remains a difficult unsolved conjecture. Furthermore, many interesting problems are known not to be well posed. For exam- ple, problems in image processing are frequently ill posed, because information is lost due to noise or technological limitations. References