the application of partial differential equation

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... understanding of the lesson. PART III CONCLUSION 1. Summaries In summary, the study deals with the theories of the role of grammar, students’ motivation, and the application of games in teaching ... win. 3. 81.1% of the students find that the games guided by their teacher are easy to understand, 18.4% of the students sometimes don’t understand the rule of the games, and 0.5% of the students ... tense or the form of verb in each clause of that sentence, guess the meaning and the usage of this condition. The group which gives the clear and correct answer will be the winner. The teacher...

Ngày tải lên: 07/11/2012, 14:44

... In order to safeguard the reliable operation of the steering gear, two motors are to be provided, with any one of them in service and the other in the state of stand-by. The hand steering system ... that, the power supply for the outdoor illumination shall originate from the distribution box fitted in the wheel-house so that it can be easily cut off from the wheel house for the sake of convenient ... automatically if the main network or emergency network fails, or the voltage drops to 40 % of the rated value. On the contrary, the circuit will be open automatically as well when the main network...

Ngày tải lên: 24/10/2013, 01:15

... 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 ... are: • What are the variables? • What equations are satisfied in the interior of the region of interest? • What equations are satisfied by points on the boundary of the region of interest? (Here ... points on the boundary of the spatial region of interest? Examples: Dirichletconditionsspecify the values of the boundarypoints as a function of time; Neumann conditionsspecify the values of the normal...

Ngày tải lên: 28/10/2013, 22:15

... approximate the fluid 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 ... time. In that case, the independent solutions, or eigenmodes, of the difference equations are all of the form u n j = ξ n e ikj∆x (19.1.12) 846 Chapter 19. Partial Differential Equations Sample ... 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 Riemann shock problem....

Ngày tải lên: 07/11/2013, 19:15

... software n. phần mềm 30. program n. chương trình 31. the system software phần mềm hệ thống 32. the application software phần mềm ứng dụng 33. the operation system(OS) hệ điều hành 34. the ... Ngôn ngữ Assemble 36. the data base management system(DBMS) hệ quản lý cơ sở dữ liệu 37. the office automation(OA) tự động hóa văn phòng 38. the graphic processing software phần mềm xử lý ... diệu,thành công rực rỡ 2. as early as ngay từ sớm 3. take the form of dưới hình dạng của 4. let sth. pass cho đi qua 5. vast quantities of lượng lớn của 6. in a word nói tóm lại 7. do one's...

Ngày tải lên: 13/12/2013, 21:15

... 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. ... 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 the general ... and (ii) neither is it very stable (see §16.6 below). Consider, however, the use of a step like (16.1.1) to take a “trial” step to the midpoint of the interval. Then use the value of both x and...

Ngày tải lên: 15/12/2013, 04:15

... and (ii) neither is it very stable (see §16.6 below). Consider, however, the use of a step like (16.1.1) to take a “trial” step to the midpoint of the interval. Then use the value of both x and ... different coefficients of higher-order error terms. Adding up the right combination of these, we can eliminate the error terms order by order. That is the basic idea of the Runge-Kuttamethod. ... obtained by using the initial derivative at each step to find a point halfway across the interval, then using the midpoint derivative across the full width of the interval. In the figure, filled...

Ngày tải lên: 15/12/2013, 04:15

... parabolic equation, the diffusion equation in one space dimension, ∂u ∂t = ∂ ∂x  D ∂u ∂x  (19.2.1) where D is the diffusion coefficient. Actually, this equation is a flux-conservative equation of the ... pieces like the right-hand side of (19.2.1) in many other situations. Consider first the case when D is a constant. Then the equation ∂u ∂t = D ∂ 2 u ∂x 2 (19.2.3) can be differenced in the obvious ... nasty set of coupled nonlinear equations to solve at each timestep. Often there is an easier way: If the form of D(u) allows us to integrate dz = D(u)du (19.2.23) analytically for z(u), then the right-hand...

Ngày tải lên: 15/12/2013, 04:15

... increase the order of a differencing method to greater than the order of the original PDEs, you introduce spurious solutions to the difference equations. This does not create a problem if they all ... ∆t/m) (19.3.22) The timestep for each fractional step in (19.3.22) is now only 1/m of thefulltimestep, because each partial operation acts with all the terms of the original operator. Equation( 19.3.22) ... differencingscheme for the operator L. In fact, as a rule of thumb, it is often sufficient to have stableU i ’s only for the operator pieces having the highest number of spatial derivatives — the other U i ’s...

Ngày tải lên: 15/12/2013, 04:15

... 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 ... dimension and solve the tridiagonal equations by the usual algorithm in the other dimension) gives about a factor of two gain in speed. The optimal FACR with r =2gives another factor of two gain in ... constant in space. The cyclic reduction method is somewhat more general; its applicability is related to the question of whether the equations are separable (in the sense of “separation of variables”)....

Ngày tải lên: 15/12/2013, 04:15

... North 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 ... is the diagonal part of A, L is the lower triangle of A with zeros on the diagonal, and U is the upper triangle of A with zeros on the diagonal. In the Jacobi method we write for the rth step of ... < 1 for the relaxation to work at all! The rate of convergence of the method is set by the rate for the slowest-decaying eigenmode, i.e., the factor with largest modulus. The modulus of this...

Ngày tải lên: 15/12/2013, 04:15

... will be that all equations are within their respective allowed errors. In other words, we will rescale the stepsize according to the needs of the “worst-offender” equation. How is ∆ 0 , the desired ... to fold these considerations into a 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 ... on the other hand, then the equation tells how much we can safely increase the stepsize for the next step. Local extrapolation consists in accepting the fifth order value y n+1 , even though the...

Ngày tải lên: 24/12/2013, 12:16

... evaluation per step h instead of the two required by second-order Runge-Kutta. Perhaps there are applications where the simplicity of (16.3.2), easily coded in-line in some other program, recommends ... use of the modified midpoint method by itself will be dominated by the embedded Runge-Kutta method with adaptive stepsize control, as implemented in the preceding section. The usefulness ofthe ... practice, the method finds its most important application as a part of the more powerful Bulirsch-Stoer technique, treated in §16.4. You can therefore consider this section as a preamble to §16.4. The...

Ngày tải lên: 24/12/2013, 12:16

... integration of differential equations. The scaling “trick” suggested in the discussion following equation (16.2.8) is a good general purpose choice, but not foolproof. Scaling by the maximum values of the ... A k is the work to obtain row k of the extrapolation tableau, so A k+1 is the work to obtain column k. We will assume the work is dominated by the cost of evaluating the functions defining the right-hand ... functions even after the various terms in powers of h all have comparable magnitudes. In other words, h can be so large as to make the whole notion of the “order” of the method meaningless — and the method...

Ngày tải lên: 24/12/2013, 12:16