Partial differential equations are central to mathematics, whether pure or applied. They arise in mathematical models whose dependent variables vary continuously as functions of several independent variables, usually space and time. Their most striking attribute is their universality, a property which has enabled us to motivate every mathematical idea in this book by real-world examples from fluid or solid mechanics, electromagnetism, probability, finance and a host of other areas of application. Moreover, this applicability is growing day by day because of the flexibility and power of modern software tailored to suitable discretised approximations of the equations. Equally dramatic is the way in which the equations that arise in all these areas of application can so easily motivate the study of fundamental mathematical questions of great depth and significance and, conversely, benefit from the results of such investigations.
Whether or not it is in the context of a model of a physical situation, the analysis of a partial differential equation has many objectives. One of our principal goals will be to investigate the question of well-posedness. We will give a more precise definition of this in Chapter 2 but, roughly speaking, a partial differential equation problem is said to be well posed if it has a solution, that solution is unique, and it only changes by a small amount in response to small changes in the input data. The first two criteria are reasonable requirements of a sensible model of a physical situation, and the third is often expected on the basis of experimental observations. When thinking of well-posedness, we must also remember that it is often impossible to find explicit solutions to problems of practical interest, so that approximation schemes, and in particular numerical solutions, are of vital importance in practice. Hence, the question of well-posedness is intimately connected with the central question of scientific computation in partial differential equations: given the data for a problem with a certain accuracy, to what accuracy does the computed output of a numerical solution solve the problem? It is because the answer to this question is so important for modern quantitative science that well-posedness is a principal mathematical theme of this book.
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