coupled differential equations examples

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Partial Differential Equations part 1

... America).Chapter 19. Partial Differential Equations 19.0 IntroductionThe numerical treatment of partial differential equations is, by itself, a vastsubject. Partial differential equations are at the ... Recipes dealing with partial differential equations alone. (Thereferences[1-4]provide, of course, available alternatives.)In most mathematics books, partial differential equations (PDEs) are classiﬁedinto ... possible choices forellipticsecond-order equations, but morecomplicated boundaryconditionscan also be encountered.)834Chapter 19. Partial Differential Equations Sample page from NUMERICAL RECIPES...

Coupled continuum equations - the dynamics of sandpile surfaces

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... Eq. (11.4).However, Eq. (11.4) is effectively decoupled from Eq. (11.5), while Eq. (11.5) ismanifestly coupled to Eq. (11.4). In order for the coupled Eqs. (11.4) to qualify asa valid model ... speciﬁcscenarios. In general, the complexity of sandpile dynamics leads us to equations which are coupled, nonlinear and noisy: these equations present challenges to thetheoretical physicist in more ways ... has been set forth in the formal equations (11.4),(11.7) and (11.12) with varying forms of the transfer term T (h,ρ) to model speciﬁc160 Coupled continuum equations: sandpile surfaces−20246810121416−5...

Partial Differential Equations part 2

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... schemes, which require us to solve implicit equations couplingthe un+1jfor various j. (Explicit and implicit methods for ordinary differential equations were discussed in §16.6.) The FTCS ... relationF(u)=0 −v−v 0·u (19.1.5)(The physicist-reader may recognize equations (19.1.3) as analogous to Maxwell’s equations for one-dimensional propagation of electromagnetic waves.)We will ... stabilityrequirement that ever turns up in PDEs. It keeps doing so in our model examples just because those examples are so simple in form. The method of analysis is,however, general.Except...

Tài liệu Integration of Ordinary Differential Equations part 1 doc

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... 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...

Tài liệu Integration of Ordinary Differential Equations part 2 pptx

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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 (NewYork: Academic Press).16.1 ... REFERENCES AND FURTHER READING:Gear, C.W. 1971,Numerical Initial Value Problems in Ordinary Differential Equations (EnglewoodCliffs, NJ: Prentice-Hall).Acton, F.S. 1970,Numerical Methods That ... 710Chapter 16. Integration of Ordinary Differential Equations Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN...

Tài liệu Partial Differential Equations part 3 pptx

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... 848Chapter 19. Partial Differential Equations Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN ... are not as easy. The replacement (19.2.22) with n → n +1leavesus with a nasty set of coupled nonlinear equations to solve at each timestep. Oftenthere is an easier way: If the form of D(u) ... given the initial wavepacket, ψ(x, t =0), together with boundary852Chapter 19. Partial Differential Equations Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN...

Tài liệu Partial Differential Equations part 4 ppt

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... 856Chapter 19. Partial Differential Equations Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN ... U2(un+(1/m), ∆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 ... again!CITED REFERENCES AND FURTHER READING:Ames, W.F. 1977,Numerical Methods for Partial Differential Equations , 2nd ed. (New York:Academic Press), Chapter 2.Goldberg, A., Schey, H.M., and...

Tài liệu Partial Differential Equations part 5 ppt

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... 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 ... withT(1)=21−T2g(1)j=∆2(gj−1−T·gj+gj+1)(19.4.33)After one level of CR, we have reduced the number of equations by a factor oftwo. Since the resulting equations are of the same form as the original equation, wecan repeat ... 862Chapter 19. Partial Differential Equations Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN...

Tài liệu Partial Differential Equations part 6 doc

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... however, the multigrid methods can solve generalelliptic equations with nonconstant coefﬁcients with hardly any loss in efﬁciency.Even nonlinear equations can be solved with comparable speed.Unfortunately ... framework to solve your speciﬁc problem. We can only give a brief868Chapter 19. Partial Differential Equations Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN ... physical. Suppose we wish to solve the elliptic equationLu = ρ (19.5.1)864Chapter 19. Partial Differential Equations Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN...

Tài liệu Integration of Ordinary Differential Equations part 3 doc

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... 722Chapter 16. Integration of Ordinary Differential Equations Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN ... REFERENCES AND FURTHER READING:Gear, C.W. 1971,Numerical Initial Value Problems in Ordinary Differential Equations (EnglewoodCliffs, NJ: Prentice-Hall). [1]Cash, J.R., and Karp, A.H. 1990,ACM ... ,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...

Tài liệu Integration of Ordinary Differential Equations part 4 ppt

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... 722Chapter 16. Integration of Ordinary Differential Equations Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN ... REFERENCES AND FURTHER READING:Gear, C.W. 1971,Numerical Initial Value Problems in Ordinary Differential Equations (EnglewoodCliffs, NJ: Prentice-Hall). [1]Cash, J.R., and Karp, A.H. 1990,ACM ... modiﬁed midpointmethod tothe Bulirsch-Stoertechnique(§16.4) 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...

Tài liệu Integration of Ordinary Differential Equations part 5 pdf

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... MethodThe techniques described in this section are not for differential equations containing nonsmooth functions. For example, you might have a differential equation whose right-handside involves a ... quick-and-dirty, low-accuracy solutionof a set of equations. A second warning is that the techniques in this section arenot particularly good for differential equations that have singular points inside ... ordinary differential equations with minimal computational effort. (A possibleexception, infrequently encountered in practice, is discussed in §16.7.)726Chapter 16. Integration of Ordinary Differential...

Tài liệu Integration of Ordinary Differential Equations part 6 pdf

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... Second-Order Conservative Equations Usually when you have a system of high-order differential equations to solve it is bestto reformulate them as a system of ﬁrst-order equations, as discussed ... 27, pp. 505–535.16.6 Stiff Sets of Equations As soon as one deals with more than one ﬁrst-order differential equation, thepossibility of a stiff set of equations arises. Stiffness occurs in ... isa particular class of equations that occurs quite frequently in practice where you can gainabout a factor of two in efﬁciency by differencing the equations directly. The equations aresecond-order...

Tài liệu Integration of Ordinary Differential Equations part 7 pptx

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... 27, pp. 505–535.16.6 Stiff Sets of Equations As soon as one deals with more than one ﬁrst-order differential equation, thepossibility of a stiff set of equations arises. Stiffness occurs in ... rule:yn+1− yn−1=2hfyn+1+ yn−12(16.6.29)738Chapter 16. Integration of Ordinary Differential Equations Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN ... multiplications on the right-hand side of (16.6.22), werewrite the equations in terms of quantitiesgi=i−1j=1γijkj+ γki(16.6.23)The equations then take the form(1/γh − f) · g1= f(y0)(1/γh...