week2 - Week 2 Math 716 1 Linear 1st order PDEs in two...

Info iconThis preview shows pages 1–3. 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: Week 2, Math 716 1 Linear 1st order PDEs in two independent variables Consider first special class of linear 1st order PDEs in two independent variables ( x 1 ,x 2 ). a 1 ( x 1 ,x 2 ) u x 1 + a 2 ( x 1 ,x 2 ) u x 2 = c ( x 1 ,x 2 ) (1) where a 1 , a 2 and c are continuous function in some domain Ω ⊂ R 2 . Denote x = ( x 1 ,x 2 ). We will assume 1. On some differentiable curve Γ = { x : x = x ( s ) } characterized by a real parameter s in some interval, the tangent d x ds is never parallel to a ( x ( s )), where a ≡ ( a 1 ,a 2 ). The significance of this non-characteristic condition will be clearer later. 2. On the non-characteristic curve Γ, we specify initial condition: u ( x ( s )) = u ( s ) (2) We now seek a solution in a domain Ω adjoining Γ. We notice (1) geometrically implies specified directional derivative of u along a , since (1) is simply a · ∇ u = c (3) We introduce a set of characteristic curve X ( t ; s ) parametrized by real scalar t ∈ I for some interval I containing 0, for each s . ∂ X ∂t = a ( X ( t ; s )) with initial condition X ( s ; 0) = x ( s ) (4) A unique solution C 1 solution X ( t ; s ) for each s is guaranteed locally from theory of ODEs for sufficiently small size of interval I . Notice, that this is not the case if the non-characteristic condition 1 is not met, since u = u on Γ implies the tangential derivative of u on Γ is also known; i.e. d x ( s ) ds · ∇ u = u ′ ( s ). This is generally incompatible with (3) at any point s where d x ( s ) ds is parallel to a . On a curve X ( t ; s ) for fixed s , equation (3) implies ∂ ∂t u ( X ( t ; s )) = c ( X ( t ; s )) (5) The theory of ODEs guarantees a locally unique solution to (5) for t ∈ I . Denote this solution by u = U ( t ; s ) (6) Since the non-charactertistic condition 1 above implies that the inverse function theorem applies, i.e. the Jacobian ∂ ( X 1 ,X 2 ) ∂ ( t,s ) = ( a 1 ,a 2 ) · ( X 2 s , − X 1 s ) negationslash = 0 at t = 0, it is possible to invert the relation x = X ( t ; s ), locally near the initial curve Γ to obtain ( t,s ) = ( T ( x ) ,S ( x )). Thus, using (6), we have solution to the initial value problem (consisting of PDE and given initial condition): u ( x ) = U ( T ( x ) ,S ( x )) (7) 1 The method of characteristics introduced here is not limited to two independent variable. Indeed, in general, in n independent variables, the initial data is given on a non-characteristic n − 1 dimensional surface, characterized by real parameters ( s 1 ,s 2 ,..s n − 1 ) ≡ s so that vector a is no where tangent to this surface. Then the procedure above generalizes, if we replace scalar s by vector s . 1.1 Example of an explicit calculation Consider for instance u x 1 + x 2 u x 2 = 0 with initial condition u (0 ,x 2 ) = f ( x 2 ) (8) for some differentiable function f ( x 2 ), and the domain of u is all of R 2 . It is clear that the curve Γ = { ( x 1 ,x 2 ) : x 1 = 0 } , characterized by parameter s = x 2 is everywhere non-characteristic...
View Full Document

This note was uploaded on 07/26/2011 for the course MATH 716 taught by Professor Tanveer,s during the Spring '08 term at Ohio State.

Page1 / 11

week2 - Week 2 Math 716 1 Linear 1st order PDEs in two...

This preview shows document pages 1 - 3. Sign up to view the full document.

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