Transcription of First Order Partial Differential Equations
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First Order Partial Differential Equations
1 First Order Partial Differential Equations The profound study of nature is the most fertile source of mathematical discover-ies. - Joseph Fourier (1768-1830) begin our study of Partial Differential equationswithfirstorder Partial Differential doing so, we need to define a (see the appendix on Differential Equations ) that ann-th orderordinary Differential equation is an equation for an unknown functiony(x)n-th Order ordinary Differential equationthat expresses a relationship between the unknown function and its firstnderivatives. One could write this generally asF(y(n)(x),y(n 1)(x), .. ,y (x),y(x),x) =0.( )Herey(n)(x)represents thenth derivative ofy(x). Furthermore, and initialvalue problem consists of the Differential equation plus the values of theInitial value 1 derivatives at a particular value of the independent variable, sayx0:y(n 1)(x0) =yn 1,y(n 2)(x0) =yn 2.
Semilinear first order partial differential differential equation in the form equation. a(x,y)ux +b(x,y)uy = f(x,y,u).(1.7) Here the left side of the equation is linear in u, ux and uy. However, the right hand side can be nonlinear in u. For the most part, we will introduce the Method of Characteristics for solving quasilinear equations.
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