By Michael Renardy

Partial differential equations are primary to the modeling of traditional phenomena. the need to appreciate the ideas of those equations has regularly had a fashionable position within the efforts of mathematicians and has encouraged such different fields as complicated functionality thought, practical research, and algebraic topology. This ebook, intended for a starting graduate viewers, offers an intensive advent to partial differential equations.

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Be an orthonorma1 basis such that ql is in the direction of Vd. To simplify notation, we shall write q for ql. We have * where The matrix B can be represented as Let D 2 u denote the matrix of the second derivatives a2u/axiaxj. 26)). If u and its normal derivative are prescribed, these terms can therefore be considered known. , that the surface 4 = 0 is noncharacteristic. 3 Higher-Order Equations and Systems The generalization of the definitions above to equations of higher order than second is straightforward.

U N ) , and the functions a:j and bi are assumed analytic. 46); below we shall discuss the reduction algorithm in detail. We shall start the section by reviewing some basic facts about real analytic functions. 21 Real Analytic Functions Analytic functions are functions which can be represented locally by power series. 2. T h e Cauchy-Kovalevskaya Theorem 47 section and write the power series of a function of n variables in the form where a = (01,. . 6). We note the following facts about power series: 1.

How can we say that a solution satisfies a differential equation at a point at which it is not differentiable? In later chapters we will examine this question more fully, and especially in the context of generalized wave equations we will get some fairly detailed answers. 2. Elementary Partial Differential Equations 33 Energy conservation In this section we derive a result for solutions of the wave equation known as conservation of energy. We prove a version here that holds for the one-dimensional wave equation with iixed ends defined above and leave generalizations for later chapters.

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