Ordinary & Partial Differential Equations - Reynolds (2000) - Chapter 8 - Introduction to Partia

Ordinary & Partial Differential Equations - Reynolds (2000) - Chapter 8 - Introduction to Partia

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CHAPTER 8 Introduction to Partial Diferential Equations M any of the natural phenomena that we wish to mathematically model involve several independent variables. For example, the outdoor tem- perature T depends not only upon time, but also upon spatial location. If x and y denote latitude and longitude and t denotes time, then the function T ( x , y , t ) describes how temperature varies in space and time. Weather reports usually render this function by using animations in which the variable t increases. In- stead of plotting T ( x , y , t ) as a surface in three dimensions for each ±xed t , the maps are usually color-coded, with red corresponding to high temperature and blue corresponding to low temperature. Mathematical models of phenomena incorporating several independent vari- ables frequently lead to equations involving partial derivatives. Usually, the independent variables correspond to time and position. Before de±ning what we mean by a partial differential equation 1 , let us establish notation. If u is a quantity that depends upon a single spatial variable (e.g., latitude) as well as time, we will usually write u = u ( x , t ) . Here x denotes the spatial variable and t denotes time. When three spatial dimensions are involved, we will write u = u ( x , y , z , t ) . Instead of using the Leibniz notation for partial derivatives, we use subscripts as follows: u x = u x u t = u t u xx = 2 u x 2 u tt = 2 u t 2 u xt =( u x ) t = t ± u x ² = 2 u t x . 1 All of our subsequent presentation is based heavily on the text of Strauss [ 10 ]. 218
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introduction to partial differential equations 219 Unless otherwise stated, we will always assume that our functions u are suf- Fciently smooth to ensure that mixed partial derivatives are equal. That is, u xt = u tx and u xxt = u xtx = u txx , and so on. Roughly speaking, a partial differential equation is any equation involving partial derivatives of some function u . With the above notational conventions in mind, we state a more precise deFnition. Defnition 8 . 0 . 2 . Suppose u = u ( x , t ) is a function of two variables. An equation of the form F ( x , t , u , u x , u t , u xx , u , u tt , . . . )= 0 is called a partial differential equation ( pde ) . In this deFnition, it is understood that the function F has only Fnitely many arguments. The deFnition is easily extended to allow for more than two in- dependent variables. The order of a pde is the order of the highest derivative present in the equation. ±or example, the equation u t + u x = 0 is a Frst-order pde , and the equation u - ( u x ) 8 = 0 is a third-order pde . The most general form of a Frst-order pde with three independent variables t , x , and y would be F ( t , x , y , u , u t , u x , u y 0. Here are some well-known examples of pde s. The transport or advection equation: Let u = u ( x , t ) . Then the equation u t + cu x = 0 where c is a constant is called the simple advection equation . It can be used to model the transport of a pollutant being carried (but not diffusing) in a long, straight river with velocity c .
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This note was uploaded on 02/27/2012 for the course MATH 532 taught by Professor Reynolds during the Fall '11 term at VCU.

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Ordinary & Partial Differential Equations - Reynolds (2000) - Chapter 8 - Introduction to Partia

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