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Unformatted text preview: 1 Definition of PDE A partial differential equation (PDE) is an equation in which there appear partial derivatives of some unknown function with respect to two or more independent variables. Its two or more since if there were only one independent variable we would have an ordinary differential equation or ODE. As an example consider the PDE u x + u y- u = 0 . Here, u is a function of two variables x and y and, for the purpose of this example, we take as a constant. As a short hand notation we often write the partial derivatives using subscripts. So in the example above, we could define u x u x u y u y , and the PDE would then be written u x + u y- u = 0 . The solution of this PDE is some function u ( x,y ) that exists in some region R of the x- y plane, for which the partial derivatives of u ( x,y ) , namely u x and u y , are defined (meaning they exist) for each point ( x,y ) of the region R , and for which the equation becomes an identity, meaning that at each point ( x,y ) , if we add u x and u y , and then subtract u we get the right hand side 0. If this happens, we say u ( x,y ) satisfies the PDE in the region R . We can extend the subscript notation to include multiple partial derivatives. For exam- ple u x u x u y u y u xx 2 u x 2 u yy 2 u y 2 u xy 2 u xy etc. Note that in using this notation were assuming you can take the partial derivatives in any order which is not always true at every point. As you recall from calculus, its not always true that 2 u xy = 2 u yx , so where order matters we will be careful to indicate that u xy is not u yx . Otherwise you can assume when we use this notation that it doesnt matter in which order you take the partial derivatives. 1 2 PDE terminology There are several descriptive terms we apply to PDEs so as to try to characterize them: Order The order of a PDE is the order of the highest partial derivative appearing in the expression for the PDE. Linearity A PDE is called linear if the equation is a linear relation in the dependent vari- able and its derivatives. A linear PDE with 2 independent variables x and y can be generically written in the form N X n =0 M X m =0 a nm ( x,y ) n + m u n x m y = g ( x,y ) , where the term with n = m = 0 represents u ( x,y ) . Nonlinearity A PDE is nonlinear if it isnt linear. For example if you have a PDE of the form F ( u,u x ,u y ,x,y ) = g ( x,y ) , and F is not linear in u or all the derivatives of u , then the PDE is nonlinear. In order for a PDE to be linear F must be linear in u and all the derivatives of u . Homogeneity You can generally write any PDE in the form F ( u,u x ,u y ,x,y ) = g ( x,y ) If g then the PDE is said to be homogeneous. Otherwise it is inhomogeneous....
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This note was uploaded on 12/12/2009 for the course ACM 95c taught by Professor Nilesa.pierce during the Spring '09 term at Caltech.
- Spring '09