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By H. T. Clifford

ISBN-10: 0121767507

ISBN-13: 9780121767501

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Example text

In addition, boundary layer correctors are required (Ref. [38]). CHAPTER 4 The Telegraph Equations For a deeper understanding of the solution behavior, we continue the discussion of the advection system. We demonstrate that the advection system can be transformed into two independent telegraph equations. A solution formula for the telegraph equation on unbounded domains is presented. By an extension of the data, this formula turns out to be applicable even in boundary cases. The telegraph equations form a singularly perturbed problem for the heat equation.

1) are provided at the inflow boundaries, that is, u (·, xL ) = uL (·) in (0, T ), v (·, xR ) = vR (·) in (0, T ). In the sequel, we are interested in approximations of solutions of the heat equation. For this reason we use the data of the heat equation for the advection system. Let Di Di Neu Neu Rob Rob , rR rL , rR , rL and rR be the boundary data for the heat equation rL belonging to the Dirichlet, Neumann or Robin problem. The following choices for the boundary conditions are possible for the advection system.

37). 5). Choose r0 = r0Rob and fLR = fLR in this case. In the Dirichlet case with r = rDi , we get in addition rDi (T, ·) 2 0 + T νπ 2 2|Ω|2 rDi (s, ·) 0 2 0 ds + ν 2 T |rDi (s, ·)|21 0 ≤ r0Di 2 0 + |Ω|2 νπ 2 T 0 Di 2 fLR 0. 38) Proof. Multiplying the heat equation by r yields 1 2 ∂t r − νr∂x2 r = rfLR . 2 Integration along Ω gives d xR r 20 + 2ν ∂x r 20 − 2ν r∂x r xL ≤ 2 r 0 fLR 0 . dt The boundary term cancels in the Dirichlet, Neumann and periodic case. For Robin boundary conditions, we get xR xL −2ν r∂x r = 2ν 2 r (·, xL ) + r2 (·, xR ) .

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An Introduction to Numerical Classification by H. T. Clifford


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