power series solution of partial differential equations

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... 19. Partial Differential Equations 19.0 Introduction The numerical treatment of partial differential equations is, by itself, a vast subject. Partial differential equations are at the heart of ... entiresecond volume of Numerical Recipes dealing with partial differential equations alone. (The references [1-4] provide, of course, available alternatives.) In most mathematics books, partial differential equations ... What are the variables? ã What equations are satised in the interior of the region of interest? ã What equations are satised by points on the boundary of the region of interest? (Here Dirichlet...

Ngày tải lên: 28/10/2013, 22:15

... of the difference equations are so slowly varying as to be considered constant in space and time. In that case, the independent solutions, or eigenmodes, of the difference equations are all of ... differencing based on Taylor series and includes the nonlinearity of the equations explicitly. There is an analytic solutionforthe evolutionof two uniformstates of a fluid separated by a discontinuity, the ... by a large number of cells of uniform states, and piece them together using the Riemann solution. There have been many generalizations of Godunov’s approach, of which the most powerful is probably...

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... Integration of Ordinary Differential Equations 16.0 Introduction Problems involving ordinary differential equations (ODEs) can always be reduced to the study of sets of first-order differential equations. ... auxiliary variables. The generic problem in ordinary differential equations is thus reduced to the study of a set of N coupled first-order differential equations for the functions y i ,i=1,2, ,N, having ... 1973, Computational Methods in Ordinary Differential Equations (New York: Wiley). Lapidus, L., and Seinfeld, J. 1971, Numerical Solution of Ordinary Differential Equations (New York: Academic Press). 16.1...

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... 1973, Computational Methods in Ordinary Differential Equations (New York: Wiley). Lapidus, L., and Seinfeld, J. 1971, Numerical Solution of Ordinary Differential Equations (New York: Academic Press). 16.1 ... 710 Chapter 16. Integration of Ordinary Differential Equations Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5) Copyright ... (see Figure 16.1.1). That means (and you can verify by expansion in power series) that the step’s error is only one power of h smaller than the correction, i.e O(h 2 ) added to (16.1.1). There...

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... − ∂ 2 ∂x 2 + V (x)(19.2.31) The formal solution of equation (19.2.30) is ψ(x, t)=e −iHt ψ(x, 0) (19.2.32) where the exponential of the operator is defined by its power series expansion. The unstable explicit ... Unless the solution you are looking for is known to be smooth, and the high-order method you 850 Chapter 19. Partial Differential Equations Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC ... amplitudes, so that the evolution of the larger-scale features of interest takes place superposed with a kind of “frozen in” (though fluctuating) background of small-scale stuff. This answer gives...

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... 856 Chapter 19. Partial Differential Equations Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5) Copyright (C) ... δ 2 y u n+1 j,l  (19.3.16) The advantage of this method is that each substep requires only the solution of a simple tridiagonal system. Operator Splitting Methods Generally The basic idea of operator splitting, ... ∆t) ··· u n+1 = U m (u n+(m−1)/m , ∆t) (19.3.20) 854 Chapter 19. Partial Differential Equations Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5) Copyright (C)...

Ngày tải lên: 15/12/2013, 04:15

... as many equations as unknowns, and there is a good chance of solving for a unique solution set of x j ’s. Analytically, there can fail to be a unique solution if one or more of the M equations ... set of equationsto be solved can bewritten as the N ìN set of equations (A T Ã A) Ã x =(A T Ãb)(2.0.4) where A T denotes the transpose of the matrix A. Equations (2.0.4) are called the normal equations ... columns of the matrix inverse of A (Đ2.1 and Đ2.3). ã Calculation of the determinant of a square matrix A (Đ2.3). If M<N,orifM=Nbut the equations are degenerate, then there are effectively fewer equations...

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... of N ì N matrices, with M sets of right-hand side vectors, in completely analogous fashion. The routine implemented below is, of course, general. 38 Chapter 2. Solution of Linear Algebraic Equations Sample ... rows of A and the corresponding rows of the b’s and of 1, does not change (or scramble in any way) the solution x’s and Y. Rather, it just corresponds to writing the same set of linear equations in ... column we are about to eliminate. Partial pivoting is easier than full pivoting, because we don’t have to keep track of the permutation of the solution vector. Partial pivoting makes available...

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... (2.10.4) of the algorithm is needed, so we separate it off into its own routine rsolv. 98 Chapter 2. Solution of Linear Algebraic Equations Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC ... not used for typical systems of linear equations. However, we will meet special cases where QR is the method of choice. 100 Chapter 2. Solution of Linear Algebraic Equations Sample page from NUMERICAL ... America). x[i]=sum/p[i]; } } A typicaluseof choldcand cholslis in theinversionof covariancematrices describing the fit of data to a model; see, e.g., Đ15.6. In this, and many other applications,one often needs L −1 ....

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... level of CR, we have reduced the number of equations by a factor of two. Since the resulting equations are of the same form as the original equation, we can repeat the process. Taking the number of ... 862 Chapter 19. Partial Differential Equations Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5) Copyright (C) ... systems. In practice, equations (19.4.33) should be rewritten to avoid numerical instabil- ity. For these and other practical details, refer to [2] . 860 Chapter 19. Partial Differential Equations Sample...

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... ease of programming outweighs expense of computer time. Occasionally, the sparse matrix methods of Đ2.7 are useful for solving a set of difference equations directly. For production solution of ... adjust the various components of the algorithm within this framework to solve your specific problem. We can only give a brief 868 Chapter 19. Partial Differential Equations Sample page from NUMERICAL ... America). The beauty of Chebyshev acceleration is that the norm of the error always decreases with each iteration. (This is the norm of the actual error in u j,l . The norm of the residual ξ j,l need...

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... generally useful stepper routine is this: One of the arguments of the routine will of course be the vector of dependent variables at the beginning of a proposed step. Call that y[1 n].Let us require ... ,n−1 y(x+H)≈y n ≡ 1 2 [z n +z n−1 +hf(x + H, z n )] (16.3.2) 714 Chapter 16. Integration of Ordinary Differential Equations Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5) Copyright ... efficiency are not mere tens of percents or factors of two; they can sometimes be factors of ten, a hundred, or more. Sometimes accuracy may be demanded not directly in the solution itself, but in...

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... 722 Chapter 16. Integration of Ordinary Differential Equations Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5) Copyright ... usefulness ofthe modied midpointmethod tothe Bulirsch-Stoertechnique (Đ16.4) derives from a deep result about equations (16.3.2), due to Gragg. It turns out that the error of (16.3.2), expressed as a power ... power series in h, the stepsize, contains only even powers of h, y n − y(x + H)= ∞  i=1 α i h 2i (16.3.3) where H is held constant, but h changes by varying n in (16.3.1). The importance of this...

Ngày tải lên: 24/12/2013, 12:16

... extrapolate each component of a vector of quantities. 728 Chapter 16. Integration of Ordinary Differential Equations Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN ... method a degree of robustness for problems with discontinuities. Let us remind you once again that scaling of the variables is often crucial for successful integration of differential equations. The ... method does an excellent job of feeling its way through rocky or discontinuous terrain. It is also an excellent choice for quick-and-dirty, low-accuracy solution of a set of equations. A second warning...

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