power series solution of partial differential equations
Partial Differential Equations part 1
... 19. Partial Differential Equations 19.0 IntroductionThe numerical treatment of partial differential equations is, by itself, a vastsubject. Partial differential equations are at the heart of ... entiresecondvolume of Numerical Recipes dealing with partial differential equations alone. (Thereferences[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...
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Partial Differential Equations part 2
... of thedifference equations are so slowly varying as to be considered constant in spaceand 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 analyticsolutionforthe evolutionof two uniformstates of a fluid separated by a discontinuity,the ... by alarge number of cells of uniform states, and piece them together using the Riemann solution. There have been many generalizations of Godunov’s approach, of whichthe most powerful is probably...
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Tài liệu Integration of Ordinary Differential Equations part 1 doc
... Integration of Ordinary Differential Equations 16.0 IntroductionProblems involving ordinary differential equations (ODEs) can always bereduced to the study of sets of first-order differential equations. ... auxiliary variables.The generic problem in ordinary differential equations is thus reduced to thestudy of a set of N coupled first-order differential equations for the functionsyi,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 (NewYork: Academic Press).16.1...
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Tài liệu Integration of Ordinary Differential Equations part 2 pptx
... 1973,Computational Methods in Ordinary Differential Equations (New York: Wiley).Lapidus, L., and Seinfeld, J. 1971,Numerical Solution of Ordinary Differential Equations (NewYork: Academic Press).16.1 ... 710Chapter 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 verifyby expansion in power series) that the step’s error is only one power of h smallerthan the correction, i.e O(h2) added to (16.1.1).There...
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Tài liệu Partial Differential Equations part 3 pptx
... −∂2∂x2+ V (x)(19.2.31)The formal solution of equation (19.2.30) isψ(x, t)=e−iHtψ(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 850Chapter 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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Tài liệu Partial Differential Equations part 4 ppt
... 856Chapter 19. Partial Differential Equations Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5)Copyright (C) ... δ2yun+1j,l(19.3.16)The advantage of this method is that each substep requires only the solution of asimple tridiagonal system.Operator Splitting Methods GenerallyThe basic idea of operator splitting, ... ∆t)···un+1= Um(un+(m−1)/m, ∆t)(19.3.20) 854Chapter 19. Partial Differential Equations Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5)Copyright (C)...
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Tài liệu Solution of Linear Algebraic Equations part 1 docx
... as many equations as unknowns, and there is a goodchance of solving for a unique solution set of xj’s. Analytically, there can fail tobe 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 (ATÃ A) Ã x =(ATÃb)(2.0.4)where ATdenotes the transpose of the matrix A. Equations (2.0.4) are called thenormal 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 areeffectively fewer equations...
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Tài liệu Solution of Linear Algebraic Equations part 2 ppt
... of N ì N matrices, with M sets of right-handside vectors, in completely analogous fashion. The routine implemented belowis, of course, general. 38Chapter 2. Solution of Linear Algebraic Equations Sample ... rows of A and the corresponding rows of the b’sand of 1, does not change (or scramble in any way) the solution x’s andY. Rather, it just corresponds to writing the same set of linear equations in ... columnwe are about to eliminate. Partial pivoting is easier than full pivoting, because wedon’t have to keep track of the permutation of the solution vector. Partial pivotingmakes available...
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Tài liệu Solution of Linear Algebraic Equations part 11 ppt
... (2.10.4) of the algorithm is needed, so we separate it off into its own routine rsolv. 98Chapter 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 willmeet special cases where QR is the method of choice. 100Chapter 2. Solution of Linear Algebraic Equations Sample page from NUMERICAL ... America).x[i]=sum/p[i];}}A typicaluseof choldcand cholslis in theinversionof covariancematrices describingthe fit of data to a model; see, e.g., Đ15.6. In this, and many other applications,one often needsL−1....
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Tài liệu Partial Differential Equations part 5 ppt
... 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, wecan repeat the process. Taking the number of ... 862Chapter 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]. 860Chapter 19. Partial Differential Equations Sample...
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Tài liệu Partial Differential Equations part 6 doc
... ease of programming outweighs expense of computertime. 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 algorithmwithin this framework to solve your specific problem. We can only give a brief 868Chapter 19. Partial Differential Equations Sample page from NUMERICAL ... America).The beauty of Chebyshev acceleration is that the norm of the error always decreaseswith each iteration. (This is the norm of the actual error in uj,l. The norm of the residual ξj,lneed...
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Tài liệu Integration of Ordinary Differential Equations part 3 doc
... generally useful stepperroutine 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].Letus require ... ,n−1y(x+H)≈yn≡12[zn+zn−1+hf(x + H, zn)](16.3.2) 714Chapter 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 accuracymay be demanded not directly in the solution itself, but in...
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Tài liệu Integration of Ordinary Differential Equations part 4 ppt
... 722Chapter 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 turnsout that the error of (16.3.2), expressed as a power ... power series in h, the stepsize, containsonly even powers of h,yn− y(x + H)=∞i=1αih2i(16.3.3)where H is held constant, but h changes by varying n in (16.3.1). The importance of this...
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Tài liệu Integration of Ordinary Differential Equations part 5 pdf
... extrapolate eachcomponent of a vector of quantities. 728Chapter 16. Integration of Ordinary Differential Equations Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN ... methoda degree of robustness for problems with discontinuities.Let us remind you once again that scaling of the variables is often crucial forsuccessful integration of differential equations. The ... method does an excellent job of feeling its way through rocky or discontinuousterrain. 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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