Download Computational Partial Differential Equations Using MATLAB by Jichun Li PDF

By Jichun Li

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

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proposing an creation to complicated numerical tools, this textbook introduces numerous significant numerical equipment for fixing quite a few partial differential equations (PDEs) in technology and Read more...

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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.

Solving such a system is quite laborious if we consider that first it is not easy to form the coefficient matrix, and second it is quite expensive to store and solve the matrix directly. More details will be discussed when we come to the elliptic problems in a later chapter. A simple and efficient method, the so-called Alternate Direction Implicit (ADI) method, for solving 2-D parabolic problems was first proposed by 26 Computational Partial Differential Equations Using MATLAB Peaceman and Rachford in 1955 [11].

L=⎢ ⎥ ⎥, U = ⎢ ⎢ ⎢ ⎥ ⎥ . . . ⎥ ⎢ ⎢ ⎥ ⎣ ⎣ ⎦ 0 αn−1 cn−1 ⎦ βn−1 1 0 0 βn 1 0 0 αn Prove that the αi and βi can be computed by the relations α1 = a1 , βi = bi /αi−1 , αi = ai − βi ci−1 , i = 2, · · · , n. , y1 = f1 , yi = fi − βi yi−1 , i = 2, · · · , n, xn = yn /αn , xi = (yi − ci xi+1 )/αi , i = n − 1, · · · , 1. Prove that the total number of multiplication/division is 5n − 4. 3. 1. 1 on a 31 × 51 grid for (x, t) domain. Compare the numerical solution to its analytical solution: u(x, t) = 1 − 2 ∞ k=1 2 2 4 e−(2k−1) π t cos(2k − 1)πx.

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