Jichun Li's Computational Partial Differential Equations Using MATLAB PDF

By Jichun Li

Short assessment of Partial Differential Equations The parabolic equations The wave equations The elliptic equations Differential equations in broader areasA speedy assessment of numerical tools for PDEsFinite distinction equipment for Parabolic Equations advent Theoretical concerns: balance, consistence, and convergence 1-D parabolic equations2-D and 3D parabolic equationsNumerical examples with MATLAB codesFinite Read more...

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providing an advent to complicated numerical equipment, this textbook introduces numerous significant numerical equipment for fixing numerous partial differential equations (PDEs) in technological know-how and Read more...

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108] or [3, p. 237]): n+ 1 1 n t (uj + unj+1 ) − [f (unj+1 ) − f (unj )], 2 2 x t n+ 1 n+ 1 [f (uj+ 12 ) − f (uj− 12 )]. 26) can be easily constructed: un+1 − un−1 f (unj+1 ) − f (unj−1 ) j j + = 0. 31) Finally, we want to mention that similar schemes can be developed for conservation laws in two and three space dimensions (cf. [7] or [3]). 34) 46 Computational Partial Differential Equations Using MATLAB where the constant a > 0. Let ζ = x + at, η = x − at. 35) where φ(ζ, η) = u(x, t). 35) twice gives u(x, t) = ψ1 (x + at) + ψ2 (x − at), where ψ1 and ψ2 are arbitrary twice differentiable functions.

40) whose truncation error is O(( t)2 + ( x)2 ). As for stability, using the von Neumann technique, we obtain λ−2+ or 1 4 1 1 1 = μ2 [−4λ sin2 k x − sin2 k x] λ 2 2 λ 2 1 1 (1 + 2μ2 sin2 k x)λ2 − 2λ + (1 + 2μ2 sin2 k x) = 0. 40) is unconditionally stable. 45) which holds true for any (x, y, t) ∈ (0, 1)2 × (0, tF ). 45). 46) where for simplicity we assume that x = y = h. Denote μ = ht . By von Neumann stability analysis, the amplification factor λ satisfies the equation 1 1 λ2 − 2λ + 1 = μ2 [−4λ sin2 kx x − 4λ sin2 ky y] 2 2 or 1 1 λ2 − 2[1 − 2μ2 (sin2 kx x + sin2 ky y)]λ + 1 = 0.

Xj = j x, 0 ≤ j ≤ J, x = 1/J, tn = n t, 0 ≤ n ≤ N, x = tF /N. First let us consider a central difference scheme: un+1 − unj unj+1 − unj−1 j +a = 0. 7) is unconditionally unstable. Hence we have to be very careful in choosing proper difference approximations. , when a > 0, we approximate ux |nj by backward difference. , in this case by forward difference. To check its stability, let us denote μ = a unj = λn eik(j t x. 8), we see that the amplification factor λ(k) = 1 − μ(1 − e−ik x ) = 1 − μ + μ cos k x − iμ sin k x.

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