applications of partial differential equations in engineering

... fundamental feature of the Poisson process that due to the independence of its increments the location of the set of points in the time interval [0, 1], see Fig 1.1.6, can be intuitively interpreted ... their increments over a period (s, t] are independent of As for t ≥ 0, s ∈ [0, t] 1.2 Supermartingales and Martingales Martingales As we will see later in the context of asset pricing and investing ... intensity appears to be large enough for realistic modeling of the dynamics of quantities in finance Here continuous trading noise and a few single events model the typical sources of uncertainty...

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Ngày tải lên: 29/10/2015, 14:19

... 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 Dirichlet ... write this system of linear equations in matrix form we need to make a vector out of u Let us number the two dimensions of grid points in a single one-dimensional sequence by defining i ≡ j(L + 1) ... circles) All grid points must be maintained in memory given at some initial time t0 for all x, then the equations describe how u(x, t) propagates itself forward in time In other words, equations (19.0.1)...

Ngày tải lên: 28/10/2013, 22:15

... are used in determining a new point (shown connected by dashed lines) A differencing scheme is Courant stable if the differencing domain of dependency is larger than that of the PDEs, as in (a), ... depends on information within some domain of dependency to the past, shown here shaded The differencing scheme (19.1.15) has its own domain of dependency determined by the choice of points on one ... improving the accuracy of first-order upwind differencing In the continuum equation, material originally a distance v∆t away Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING...

Ngày tải lên: 07/11/2013, 19:15

... 710 Chapter 16 Integration of Ordinary Differential Equations CITED REFERENCES AND FURTHER READING: Gear, C.W 1971, Numerical Initial Value Problems in Ordinary Differential Equations (Englewood ... description of each of these types follows Runge-Kutta methods propagate a solution over an interval by combining the information from several Euler-style steps (each involving one evaluation of the ... routines; rkqs, bsstep, stiff, and stifbs are steppers; rkdumb and odeint are drivers Section 16.6 of this chapter treats the subject of stiff equations, relevant both to ordinary differential equations...

Ngày tải lên: 15/12/2013, 04:15

... discussion of the pitfalls in constructing a good Runge-Kutta code is given in [3] Here is the routine for carrying out one classical Runge-Kutta step on a set of n differential equations You input ... step in a sequence of steps in identical manner Prior behavior of a solution is not used in its propagation This is mathematically proper, since any point along the trajectory of an ordinary differential ... Figure 16.1.2 Midpoint method Second-order accuracy is obtained by using the initial derivative at each step to find a point halfway across the interval, then using the midpoint derivative across...

Ngày tải lên: 15/12/2013, 04:15

... 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 a differencing scheme ... tridiagonal form again and in practice usually retains the stability advantages of fully implicit differencing Schrodinger Equation ¨ Sometimes the physical problem being solved imposes constraints on the ... not to be interested in We want those scales to something stable, “innocuous,” and perhaps not too physically unreasonable We want to build this innocuous behavior into our differencing scheme...

Ngày tải lên: 15/12/2013, 04:15

... underlying PDEs, perhaps allowing second-order spatial differencing for first-order -in- space PDEs When you increase the order of a differencing method to greater than the order of the original PDEs, ... sin kx ∆ − αx sin ky ∆)2 856 Chapter 19 Partial Differential Equations (19.3.13) Called the alternating-direction implicit method (ADI), this embodies the powerful concept of operator splitting ... are using is known to be extremely stable, we not recommend anything higher than second-order in time (for sets of first-order equations) For spatial differencing, we recommend the order of the...

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 ... y-values on these x-lines Then fill in the intermediate x-lines as in the original CR algorithm The trick is to choose the number of levels of CR so as to minimize the total number of arithmetic operations ... boundary Instead of the expansion (19.4.2), we now need an expansion in sine waves: 860 Chapter 19 Partial Differential Equations If f(y = l∆) ≡ fl , then we get An from the inverse formula An = sinh...

Ngày tải lên: 15/12/2013, 04:15

... we make use of updated values of u on the right-hand side of (19.5.5) as soon as they become available In other words, the averaging is done in place” instead of being “copied” from an earlier ... the version of SOR implemented below, we shall adopt odd-even ordering The last practical point is that in practice the asymptotic rate of convergence in SOR is not attained until of order J iterations ... solving the elliptic equation u=ρ (19.5.32) In either case, the operator splitting is of the form L = Lx + Ly (19.5.33) where Lx represents the differencing in x and Ly that in y For example, in...

Ngày tải lên: 15/12/2013, 04:15

... Ordinary Differential Equations } nrerror("Too many steps in routine odeint"); } CITED REFERENCES AND FURTHER READING: Gear, C.W 1971, Numerical Initial Value Problems in Ordinary Differential Equations ... free_vector(ak2,1,n); } Noting that the above routines are all in single precision, don’t be too greedy in specifying eps The punishment for excessive greediness is interesting and worthy of Gilbert and ... bound the error of each step individually In some applications you may be unusually sensitive about a “global” accumulation of errors, from beginning to end of the integration and in the worst possible...

Ngày tải lên: 24/12/2013, 12:16