B5b4 - B5b Applied Partial Dierential Equations 41 4...

Info iconThis preview shows pages 1–4. Sign up to view the full content.

View Full Document Right Arrow Icon
B5b Applied Partial Differential Equations 4–1 4 First-order nonlinear equations 4.1 Introduction Now we consider general first-order nonlinear scalar PDEs, that is ones that are not necessarily quasi-linear. The general form of such an equation is F ( p,q,u,x,y ) = 0 , (4.1) where we use ∂u ∂x = p, ∂y = q (4.2) as shorthand, so that ∂p = ∂q . (4.3) The case of quasilinear equations corresponds to F being a linear function of p and q , i.e. F ( p,q,u,x,y, ) = a ( x,y,u ) p + b ( ) q - c ( ) . (4.4) 4.2 Charpit’s equations If we differentiate (4.1) with respect to x and y , we obtain ∂F + = - - p , (4.5a) + = - - q , (4.5b) or, using (4.3), + = - - p , (4.6a) + = - - q . (4.6b) So, if we define characteristics or rays as curves ( x ( τ ) ,y ( τ ) ) satisfying d x d τ = , d y d τ = (4.7)
Background image of page 1

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

View Full DocumentRight Arrow Icon
4–2 OCIAM Mathematical Institute University of Oxford then, along these curves, d p d τ = - ∂F ∂x - p ∂u , d q d τ = - ∂y - q . (4.8) We therefore have a system of four ODEs for x , y , p and q satisfied along the rays. Recall, though, that in general F depends on u also, so to close the system we also need an ODE for u along the rays, namely d u d τ = d x d τ + d y d τ = p ∂p + q ∂q . (4.9) In summary, we have the following system of ODEs for x , y , p , q and u , known as Charpit’s equations : d x d τ = , (4.10a) d y d τ = , (4.10b) d p d τ = - - p , (4.10c) d q d τ = - - q , (4.10d) d u d τ = p + q . (4.10e) It is easily verified that these reduce to the usual characteristic equations d x d τ = a, d y d τ = b, d u d τ = c, (4.11) for quasi-linear equations where F takes the form (4.4). 4.3 Boundary data As for quasilinear scalar equations, Cauchy data specifies u along some curve Γ in the ( x,y )-plane: x = x 0 ( s ) , y = y 0 ( s ) , u = u 0 ( s ) , (4.12) for s in some (possibly infinite) interval. We also require initial conditions for p and q , say p = p 0 ( s ), q = q 0 ( s ), which are obtained by differentiating u 0 with respect to s and using the PDE (4.1): d u 0 d s = p 0 d x 0 d s + q 0 d y 0 d s , F ( p 0 ,q 0 ,u 0 ,x 0 ,y 0 ) = 0 . (4.13) By the implicit function theorem, the two equations (4.13) may be solved uniquely (in prin- ciple, if not explicitly) for p 0 and q 0 provided the condition d x 0 d s 0 - d y 0 d s 0 6 = 0 (4.14)
Background image of page 2
B5b Applied Partial Differential Equations 4–3 is satisfied. This is the same as insisting that Γ not be parallel to a ray. Charpit’s method consists of solving the ODEs (4.10) for ( p,q,u,x,y ), with (4.12) and (4.13) as initial data at τ = 0. This gives ( ) all as functions of s and τ and, in principle, allows us to reconstruct the solution surface u = u ( x,y ). Example 4.1 Sugar on a spoon Consider sugar piled up on a spoon such that its height is given by u ( ) . At criticality, just before the sugar would start to slide off the spoon, the sugar makes a constant angle of repose γ with the horizontal, that is | u | 2 = ± ∂u ∂x ² 2 + ± ∂y ² 2 = tan 2 γ. (4.15) After normalisation, this can be written as the Eikonal equation ± ² 2 + ± ² 2 = 1 , (4.16) which is of the form (4.1) with F ( p,q ) = 1 2 ( p 2 + q 2 - 1 ) . (4.17) Charpit’s equations for this particular F are d x d τ = p, d y d τ = q, d p d τ = 0 , d q d τ = 0 , d u d τ = p 2 + q 2 = 1 . (4.18) Notice that p and q are constant along rays and, hence, given by their boundary values: p = p 0 ( s ) , q = q 0 ( s ) . (4.19) The remaining ODEs are then readily integrated to give x = x 0 ( s ) + p 0 ( s ) τ, y = y 0 ( s ) + q 0 ( s ) u = u 0 ( s ) + τ.
Background image of page 3

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

View Full DocumentRight Arrow Icon
Image of page 4
This is the end of the preview. Sign up to access the rest of the document.

Page1 / 10

B5b4 - B5b Applied Partial Dierential Equations 41 4...

This preview shows document pages 1 - 4. Sign up to view the full document.

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