Lec1 - Math 124A – Viktor Grigoryan 1 Introduction Recall that an ordinary differential equation(ODE contains an independent variable x and a

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

This note was uploaded on 01/10/2011 for the course MATH 124A taught by Professor Ponce during the Fall '08 term at UCSB.

Page1 / 3

Lec1 - Math 124A – Viktor Grigoryan 1 Introduction Recall that an ordinary differential equation(ODE contains an independent variable x and a

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online