coupled differential equations mathematica

Ngày tải lên: 31/03/2014, 15:56

Ngày tải lên: 31/03/2014, 15:56

Ngày tải lên: 21/06/2014, 03:20

... America). Chapter 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 ... Recipes dealing with partial differential equations alone. (The references [1-4] provide, of course, available alternatives.) In most mathematics books, partial differential equations (PDEs) are classified into ... possible choices for ellipticsecond-order equations, but morecomplicated boundaryconditions can also be encountered.) 834 Chapter 19. Partial Differential Equations Sample page from NUMERICAL RECIPES...

Ngày tải lên: 28/10/2013, 22:15

... Eq. (11.4). However, Eq. (11.4) is effectively decoupled from Eq. (11.5), while Eq. (11.5) is manifestly coupled to Eq. (11.4). In order for the coupled Eqs. (11.4) to qualify as a valid model ... specific scenarios. In general, the complexity of sandpile dynamics leads us to equations which are coupled, nonlinear and noisy: these equations present challenges to the theoretical 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 specific 160 Coupled continuum equations: sandpile surfaces −2 0 2 4 6 8 10 12 14 16 −5...

Ngày tải lên: 01/11/2013, 08:20

... schemes, which require us to solve implicit equations coupling the u n+1 j for various j. (Explicit and implicit methods for ordinary differential equations were discussed in §16.6.) The FTCS ... relation F(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 ... In the continuum equation, material originally a distance v∆t away 840 Chapter 19. Partial Differential Equations Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN...

Ngày tải lên: 07/11/2013, 19:15

... 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...

Ngày tải lên: 15/12/2013, 04:15

... 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 ... 1971, Numerical Initial Value Problems in Ordinary Differential Equations (Englewood Cliffs, NJ: Prentice-Hall), Chapter 2. [2] Shampine, L.F., and Watts, H.A. 1977, in Mathematical Software III , J.R. Rice, ... 710 Chapter 16. Integration of Ordinary Differential Equations Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN...

Ngày tải lên: 15/12/2013, 04:15

... 848 Chapter 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 +1leaves us with a nasty set of coupled nonlinear equations to solve at each timestep. Often there is an easier way: If the form of D(u) ... given the initial wavepacket, ψ(x, t =0), together with boundary 852 Chapter 19. Partial Differential Equations Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN...

Ngày tải lên: 15/12/2013, 04:15

... 856 Chapter 19. Partial Differential Equations Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN ... U 2 (u n+(1/m) , ∆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 ... 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...

Ngày tải lên: 15/12/2013, 04:15

... 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 ... with T (1) =21−T 2 g (1) j =∆ 2 (g j−1 −T·g j +g j+1 ) (19.4.33) After one 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 ... 862 Chapter 19. Partial Differential Equations Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN...

Ngày tải lên: 15/12/2013, 04:15

... however, the multigrid methods can solve general elliptic 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 brief 868 Chapter 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 equation Lu = ρ (19.5.1) 864 Chapter 19. Partial Differential Equations Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN...

Ngày tải lên: 15/12/2013, 04:15

... 1971, Numerical Initial Value Problems in Ordinary Differential Equations (Englewood Cliffs, 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 (Englewood Cliffs, NJ: Prentice-Hall), Chapter 2. [2] Shampine, L.F., and Watts, H.A. 1977, in Mathematical Software III , J.R. Rice, ... ,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...

Ngày tải lên: 24/12/2013, 12:16

... 1971, Numerical Initial Value Problems in Ordinary Differential Equations (Englewood Cliffs, NJ: Prentice-Hall). [1] Cash, J.R., and Karp, A.H. 1990, ACM Transactions on Mathematical Software , vol. 16, pp. ... 722 Chapter 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, in Mathematical Software III , J.R. Rice, ed. (New York: Academic Press), pp. 257–275; 1979, Applied...

Ngày tải lên: 24/12/2013, 12:16

... Method The 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 solution of a set of equations. A second warning is that the techniques in this section are not particularly good for differential equations that have singular points inside ... ordinary differential equations with minimal computational effort. (A possible exception, infrequently encountered in practice, is discussed in §16.7.) 726 Chapter 16. Integration of Ordinary Differential...

Ngày tải lên: 24/12/2013, 12:16

... Second-Order Conservative Equations Usually when you have a system of high-order differential equations to solve it is best to 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, the possibility of a stiff set of equations arises. Stiffness occurs in ... is a particular class of equations that occurs quite frequently in practice where you can gain about a factor of two in efficiency by differencing the equations directly. The equations are second-order...

Ngày tải lên: 24/12/2013, 12:16

... Ordinary Differential Equations (Englewood Cliffs, 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 ... 27, pp. 505–535. 16.6 Stiff Sets of Equations As soon as one deals with more than one first-order differential equation, the possibility of a stiff set of equations arises. Stiffness occurs in ... multiplications on the right-hand side of (16.6.22), we rewrite the equations in terms of quantities g i = i−1  j=1 γ ij k j + γk i (16.6.23) The equations then take the form (1/γh − f  ) · g 1 = f(y 0 ) (1/γh...

Ngày tải lên: 24/12/2013, 12:16

Ngày tải lên: 24/12/2013, 12:16

Ngày tải lên: 24/12/2013, 12:16