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Unformatted text preview: Math 124A – September 28, 2009 Viktor Grigoryan 1 Introduction Recall that an ordinary differential equation (ODE) contains an independent variable x and a dependent variable u , which is the unknown in the equation. The defining property of an ODE is that derivatives of the unknown function u = du dx enter the equation. Thus, an equation that relates the independent variable x , the dependent variable u and derivatives of u is called an ordinary differential equation . Some examples of ODEs are: u ( x ) = u u 00 + 2 xu = e x u 00 + x ( u ) 2 + sin u = ln x In general, and ODE can be written as F ( x,u,u ,u 00 ,... ) = 0. In contrast to ODEs, a partial differential equation (PDE) contains partial derivatives of the depen- dent variable, which is an unknown function in more than one variable x,y,... . Denoting the partial derivative of ∂u ∂x = u x , and ∂u ∂y = u y , we can write the general first order PDE for u ( x,y ) as F ( x,y,u ( x,y ) ,u x ( x,y ) ,u y ( x,y )) = F ( x,y,u,u x ,u y ) = 0 . (1) Although one can study PDEs with as many independent variables as one wishes, we will be primarily concerned with PDEs in two independent variables. A solution to the PDE (1) is a function u ( x,y ) which satisfies (1) for all values of the variables x and y . Some examples of PDEs (of physical significance) are: u x + u y = 0 transport equation (2) u t + uu x = 0 inviscid Burger’s equation (3) u xx + u yy = 0 Laplace’s equation (4) u tt- u xx = 0 wave equation (5) u t- u xx = 0 heat equation (6) u t + uu x + u xxx = 0 KdV equation (7) iu t- u xx = 0 Shr¨ odinger’s equation (8) It is generally nontrivial to find the solution of a PDE, but once the solution is found, it is easy to...
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