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

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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)

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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)
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.

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• Summer '11
• Arubi
• Boundary value problem, wave equation, rays, Laplace operator, Hyperbolic partial differential equation, Helmholtz equation

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

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