Partial Differential Equations

... (19 .0.3) we substitute a finite-difference representation (see Figure 19 .0.2), uj +1, l − 2uj,l + uj 1, l uj,l +1 − 2uj,l + uj,l 1 + = ρj,l ∆2 ∆2 (19 .0.5) or equivalently uj +1, l + uj 1, l + uj,l +1 ... 1) + l for j = 0, 1, , J, l = 0, 1, , L (19 .0.7) In other words, i increases most rapidly along the columns representing y values Equation (19 .0.6) now becomes ui+L +1 + ui−(L +1)...

Ngày tải lên: 28/10/2013, 22:15

... differencing of equation (19.1 .20 ) is n+1 n rj+1 /2 − rj+1 /2 n+1 /2 sj n+1 /2 = ∆t n−1 /2 − sj =v ∆t n+1 /2 − sj ∆x n n rj+1 /2 − rj−1 /2 sj+1 (19.1.35) ∆x If you substitute equation (19.1 .22 ) in equation (19.1.35), ... (19.1.40) Then ξ = − iα sin k∆x − 2 (1 − cos k∆x) (19.1.41) so |ξ |2 = − 2 (1 − 2 )(1 − cos k∆x )2 (19.1. 42) The stability criterion |ξ |2 ≤ is therefore 2...

Ngày tải lên: 07/11/2013, 19:15

... Dividing by α, we see that the difference equations are just the finite-difference form of the equilibrium equation 850 Chapter 19 Partial Differential Equations t or n (a) x or j Fully Implicit ... (1 + iH∆t)−1 ψj ψj (19.2 .34 ) These are both first-order accurate in time, as can be seen by expanding equation (19.2 .32 ) However, neither operator in (19.2 .33 ) or (19.2 .34 ) is unitar...

Ngày tải lên: 15/12/2013, 04:15

... 8 54 Chapter 19 Partial Differential Equations Lax Method for a Flux-Conservative Equation As an example, we show how ... 1-800-872- 742 3 (North America only),or send email to trade@cup.cam.ac.uk (outside North America) − (cos kx∆ − cos ky ∆)2 − (αy sin kx ∆ − αx sin ky ∆)2 856 Chapter 19 Partial Differential Equations ... REFERENCES AND FURTHER READING: Ames, W.F 1977, Numerica...

Ngày tải lên: 15/12/2013, 04:15

... 858 Chapter 19 Partial Differential Equations Fourier Transform Method The discrete inverse Fourier transform in both ... waves: 860 Chapter 19 Partial Differential Equations If f(y = l∆) ≡ fl , then we get An from the inverse formula An = sinh πn L−1 fl sin l=1 πnl L (19.4.14) u = ujl + uH jl (19.4. 15) By adding appropriate ... North America) ∂u = g(y) ∂x 862 Chapter 19 Partial...

Ngày tải lên: 15/12/2013, 04:15

... problem for which ρJacobi is given by equation (19.5.11) Then equations (19.5.19) and (19.5.20) give 868 Chapter 19 Partial Differential Equations Consider a general second-order elliptic equation ... Chapter 19 Partial Differential Equations ADI (Alternating-Direction Implicit) Method The ADI method of §19.3 for diffusion equations can be turned into a relaxation method for e...

Ngày tải lên: 15/12/2013, 04:15

... http://www.nr.com or call 1-800- 872 -74 23 (North America only),or send email to trade@cup.cam.ac.uk (outside North America) S E 876 Chapter 19 Partial Differential Equations Its symbol is therefore ... fill0(double **u, int n); double disc,fact,h=0.5; 888 Chapter 19 Partial Differential Equations CITED REFERENCES AND FURTHER READING: Brandt, A 1 977 , Mathematics of Computation,...

Ngày tải lên: 24/12/2013, 12:16

... [W] ✷ Lemma (i) E is locally strictly convex in (S, V ) (ii) F is locally strictly concave in T , locally strictly convex in V (iii) H is locally strictly concave in P , locally strictly convex ... differential form of Harnack’s inequality Clausius inequality a Cycles b Heating c Almost reversible cycles V Conservation laws and kinetic equations A Some physical PDE Compressible Euler eq...

Ngày tải lên: 17/03/2014, 14:29

... type, which we prove to be Lp-bounded for < p < and invertible The Lp-boundedness is a consequence of the Lp-boundedness of the Cauchy integral (Coifman, McIntosh and Meyer) Z fw Tf (z) = w (? ... implication since a harmonic function u which satis es ( ) has non-tangential limits a.e on @ Rn+1, the limit-function u0 = u( ; 0) Lp(Rn) and + u(x; y) = py u0(x) Sketch of a proof Assume u...

Ngày tải lên: 31/03/2014, 15:16

... problems are well-posed These include boundary value problems for (stationary) elliptic partial di erential equations and initial-boundary value problems for (time-dependent) equations of parabolic, ... gives a Hilbert space in which C0 (G) is a dense subspace 16 CHAPTER I ELEMENTS OF HILBERT SPACE Suppose V ( ) is a scalar product space and let B k k denote the completio...

Ngày tải lên: 31/03/2014, 15:56