1 - First Order Scalar Semilinear Equations

1 - First Order Scalar Semilinear Equations - MATH 220:...

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Unformatted text preview: MATH 220: FIRST ORDER SCALAR SEMILINEAR EQUATIONS First order scalar semilinear equations have the form (1) a ( x, y ) u x + b ( x, y ) u y = c ( x, y, u ); here we assume that a, b, c are at least C 1 , given real valued functions. Let V be the vector field on R 2 given by V ( x, y ) = ( a ( x, y ) , b ( x, y )) , so a ( x, y ) u x + b ( x, y ) u y is the directional derivative of u along V . Let γ = γ ( s ) be an integral curve of V , i.e. γ ( s ) = ( x ( s ) , y ( s )) has tangent vector V = V ( x ( s ) , y ( s )) for each s . Explicitly, this says that (2) x ′ ( s ) = a ( x ( s ) , y ( s )) , y ′ ( s ) = b ( x ( s ) , y ( s )) . Now let v ( s ) = u ( γ ( s )) = u ( x ( s ) , y ( s )). Thus, by the chain rule, v ′ ( s ) = x ′ ( s ) u x ( x ( s ) , y ( s )) + y ′ ( s ) u y ( x ( s ) , y ( s )) = a ( x ( s ) , y ( s )) u x ( x ( s ) , y ( s )) + b ( x ( s ) , y ( s )) u y ( x ( s ) , y ( s )) = c ( x ( s ) , y ( s ) , u ( x ( s ) , y ( s ))) , where in the last step we used the PDE. Thus, v ′ ( s ) = c ( x ( s ) , y ( s ) , v ( s )) , i.e. v satisfies an ODE along each integral curve of V . To solve the PDE, we parameterize the integral curves by an additional param- eter r , i.e. the integral curves are γ r = γ r ( s ) = ( x r ( s ) , y r ( s )), where r is in an interval (or the whole real line), and each γ r is an integral curve for V , i.e. (3) x ′ r ( s ) = a ( x r ( s ) , y r ( s )) , y ′ r ( s ) = b ( x r ( s ) , y r ( s )) , so v r ( s ) = u ( γ r ( s )) solves (4) v ′ r ( s ) = c ( x r ( s ) , y r ( s ) , v r ( s )) . Note that here the subscript r denotes a parameter, not a derivative! We may equally well write x r ( s ) = x ( r, s ), y r ( s ) = y ( r, s ), and we will do so; we adopted the subscript notation to emphasize that along each integral curve r is fixed, i.e. is a constant. Which parameterization should we use? We are normally also given initial con- ditions along a curve Γ = Γ( r ) with Γ( r ) = (Γ 1 ( r ) , Γ 2 ( r )), namely u (Γ( r )) = φ ( r ) , where φ is a given function. For example, we are given an initial condition on the x axis: u ( x, 0) = φ ( x ), in which case we may choose r = x , Γ( r ) = ( r, 0). Then we want the integral curve with parameter r to go through Γ( r ) at ‘time’ 0, i.e. we want γ r (0) = Γ( r ) , and we want v r (0) = u ( γ r (0)) = φ ( r ) . 1 2 Combining these, we have two groups of ODEs: a system for the integral curves of V , also called (projected) characteristic curves , with initial conditions given by Γ, and a scalar ODE along the integral curves with initial condition given by φ : x ′ r ( s ) = a ( x r ( s ) , y r ( s )) , x r (0) = Γ 1 ( r ) , y ′ r ( s ) = b ( x r ( s ) , y r ( s )) , y r (0) = Γ 2 ( r ) , (5) and (6) v ′ r ( s ) = c ( x r ( s ) , y r ( s ) , v r ( s )) , v r (0) = φ ( r ) ....
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This note was uploaded on 03/16/2010 for the course CME 303 taught by Professor Vasy during the Fall '10 term at Stanford.

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1 - First Order Scalar Semilinear Equations - MATH 220:...

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