solve coupled differential equations mathematica
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...
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Coupled continuum equations - the dynamics of sandpile surfaces
... 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 ... Figs. 11.12and 11.14.The mean-field equations corresponding to Case C (which turn out to be identicalto the so-called BCRE equations [214] have also been solved numerically [96]; fromFig. 11.16 ... 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...
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Partial Differential Equations part 2
... implicit schemes, which require us to solve implicit equations couplingthe un+1jfor various j. (Explicit and implicit methods for ordinary differential equations were discussed in Đ16.6.) The ... 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 ... the continuum equation, material originally a distance v∆t away 840Chapter 19. Partial Differential Equations Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN...
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Tài liệu Integration of Ordinary Differential Equations part 1 doc
... 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 ... 1971,Numerical Initial Value Problems in Ordinary Differential Equations (EnglewoodCliffs, NJ: Prentice-Hall), Chapter 2. [2]Shampine, L.F., and Watts, H.A. 1977, inMathematical Software III, J.R. Rice, ... 710Chapter 16. Integration of Ordinary Differential Equations Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN...
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Tài liệu Partial Differential Equations part 3 pptx
... time, by contrast with FTCS (which is called fullyexplicit). To solve equation (19.2.8) one has to solve a set of simultaneous linear equations at each timestep for the un+1j. Fortunately, this ... 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) allows ... generalized to N +1dimensions.However, the computing power necessary to solve the resulting equations is enor-mous. If you have solved a one-dimensional problem with 100 spatial grid points,solving...
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Tài liệu Partial Differential Equations part 4 ppt
... generalized to N +1dimensions.However, the computing power necessary to solve the resulting equations is enor-mous. If you have solved a one-dimensional problem with 100 spatial grid points,solving ... 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...
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Tài liệu Partial Differential Equations part 5 ppt
... methods that is somewhatmore physical. Suppose we wish to solve the elliptic equationLu = ρ (19.5.1) 858Chapter 19. Partial Differential Equations Sample page from NUMERICAL RECIPES IN C: THE ... 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 ... because they are knownboundary values. Equation (19.4.34) can be solved for uJ/2by the standardtridiagonal algorithm. The two equations at level f − 1 involve uJ/4and u3J/4.Theequation...
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Tài liệu Partial Differential Equations part 6 doc
... Thesemethods can solve elliptic PDEs discretized on N grid points in O(N ) operations.The rapid direct elliptic solvers discussed in Đ19.4 solve special kinds of elliptic equations in O(N log ... 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 ... 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 RECIPES IN C:...
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Tài liệu Integration of Ordinary Differential Equations part 3 doc
... 1971,Numerical Initial Value Problems in Ordinary Differential Equations (EnglewoodCliffs, NJ: Prentice-Hall). [1]Cash, J.R., and Karp, A.H. 1990,ACM Transactions on Mathematical Software, vol. 16, pp. ... 1971,Numerical Initial Value Problems in Ordinary Differential Equations (EnglewoodCliffs, NJ: Prentice-Hall), Chapter 2. [2]Shampine, L.F., and Watts, H.A. 1977, inMathematical Software III, J.R. Rice, ... ,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...
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Tài liệu Integration of Ordinary Differential Equations part 4 ppt
... 1971,Numerical Initial Value Problems in Ordinary Differential Equations (EnglewoodCliffs, NJ: Prentice-Hall). [1]Cash, J.R., and Karp, A.H. 1990,ACM Transactions on Mathematical Software, vol. 16, pp. ... 722Chapter 16. Integration of Ordinary Differential Equations Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN ... on Mathematical Software, vol. 16, pp. 201–222. [2]Shampine, L.F., and Watts, H.A. 1977, inMathematical Software III, J.R. Rice, ed. (New York:Academic Press), pp. 257–275; 1979,Applied...
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Tài liệu Integration of Ordinary Differential Equations part 5 pdf
... 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 Đ16.7.) 726Chapter 16. Integration of Ordinary Differential...
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Tài liệu Integration of Ordinary Differential Equations part 6 pdf
... 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 rst-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...
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Tài liệu Integration of Ordinary Differential Equations part 7 pptx
... Ordinary Differential Equations (EnglewoodCliffs, NJ: Prentice-Hall). [1]Kaps, P., and Rentrop, P. 1979,Numerische Mathematik, vol. 33, pp. 55–68. [2]Shampine, L.F. 1982,ACM Transactions on Mathematical ... g2.g2[i]=dydx[i]+h*C2X*dfdx[i]+C21*g1[i]/h;lubksb(a,n,indx,g2); Solve for g2.for (i=1;i<=n;i++) Compute intermediate values of y and x.y[i]=ysav[i]+A31*g1[i]+A32*g2[i]; 736Chapter 16. Integration of Ordinary Differential Equations Sample ... hf(yn+1)(16.6.15)In general thisissome nasty set of nonlinear equationsthathasto be solvediterativelyat each step. Suppose we try linearizing the equations, as in Newton’s method:yn+1= yn+ hf(yn)+∂f∂yyn·(yn+1−...
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