Initial value, problems, numerical solution

Equations of the first land are very sensitive to solution errors so that they present severe numerical problems. Volterra equations are similar to initial value problems. 

NUMERICAL SOLUTION OF ORDINARY DIFFERENTIAL EQUATIONS AS INITIAL VALUE PROBLEMS... 

The above two examples were chosen so as to point out the similarity between a physical experiment and a simple numerical experiment (Initial Value Problem). In both cases, after the initial transients die out, we can only observe attractors (i.e. stable solutions). In both of the above examples however, a simple observation of the attractors does not provide information about the nature of the instabilities involved, or even about the nature of the observed solution. In both of these examples it is necessary to compute unstable solutions and their stable and/or unstable manifolds in order to track and analyze the hidden structure, and its implications for the observable system dynamics. 

This equation must be solved for yn +l. The Newton-Raphson method can be used, and if convergence is not achieved within a few iterations, the time step can be reduced and the step repeated. In actuality, the higher-order backward-difference Gear methods are used in DASSL [Ascher, U. M., and L. R. Petzold, Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations, SIAM, Philadelphia (1998) and Brenan, K. E., S. L. Campbell, and L. R. Petzold, Numerical Solution of Initial-Value Problems in Differential-Algebraic Equations, North Holland Elsevier (1989)]. 

Other methods can be used in space, such as the finite element method, the orthogonal collocation method, or the method of orthogonal collocation on finite elements. One simply combines the methods for ordinary differential equations (see Ordinary Differential Equations—Boundary Value Problems ) with the methods for initial-value problems (see Numerical Solution of Ordinary Differential Equations as Initial Value Problems ). Fast Fourier transforms can also be used on regular grids (see Fast Fourier Transform ). 

Absorption columns can be modeled in a plate-to-plate fashion (even if it is a packed bed) or as a packed bed. The former model is a set of nonlinear algebraic equations, and the latter model is an ordinary differential equation. Since streams enter at both ends, the differential equation is a two-point boundary value problem, and numerical methods are used (see Numerical Solution of Ordinary Differential Equations as Initial-Value Problems ). 

Dynamic simulations are also possible, and these require solving differential equations, sometimes with algebraic constraints. If some parts of the process change extremely quickly when there is a disturbance, that part of the process may be modeled in the steady state for the disturbance at any instant. Such situations are called stiff, and the methods for them are discussed in Numerical Solution of Ordinary Differential Equations as Initial-Value Problems. It must be realized, though, that a dynamic calculation can also be time-consuming and sometimes the allowable units are lumped-parameter models that are simplifications of the equations used for the steady-state analysis. Thus, as always, the assumptions need to be examined critically before accepting the computer results. 

Numerical Solution Equations 6.40 and 6.41 represent a nonlinear, coupled, boundary-value system. The system is coupled since u and V appear in both equations. The system is nonlinear since there are products of u and V. Numerical solutions can be accomplished with a straightforward finite-difference procedure. Note that Eq. 6.41 is a second-order boundary-value problem with values of V known at each boundary. Equation 6.40 is a first-order initial-value problem, with the initial value u known at z = 0. 

K.E. Brenan, S.L. Campbell, and L.R. Petzold. Numerical Solution of Initial-Value Problems in Differential Algebraic Equations. SIAM, Philadelphia, PA, second edition, 1996. 

Numerical Solution of the Resulting Initial Value Problem... 

Due to these inner iterations via IVP solvers and due to the need to solve an associated nonlinear systems of equations to match the local solutions globally, boundary value problems are generally much harder to solve and take considerably more time than initial value problems. Typically there are between 30 and 120 I VPs to solve numerous times in each successful run of a numerical BVP solver. 

The essence of solving the problem is shown in Fig. 4. There are two ways in which the basic equations can be solved by numerical means and by analytical procedures. In general, the PDEs or ODEs that describe actual situations are nonlinear and must be solved numerically using a computer. Each PDE is transformed into a set of ODEs by the method of lines. The ODEs are reduced to the solution of initial value problems,... 

The outlet concentration from a maximum mixedness reactor is found by evaluating the solution to Equation 9-34 at X = 0 CAout = CA(0). The analytical solution to Equation 9-34 is rather complex for reaction order n > 1, the (-rA) term is usually non-linear. Using numerical methods, Equation 9-34 can be treated as an initial value problem. Choose a value for CAout = CA(0) and integrate Equation 9-34. If CA(X) achieves a steady state value, the correct value for CA(0) was guessed. Once Equation 9-34 has been solved subject to the appropriate boundary conditions, the conversion may be calculated from CAout = Ca(0)-... 

Brenan, K. E., Campbell, S. L., and Petzold, L. R., Numerical Solution of Initial-value Problems in Differential-algebraic Equations . SIAM, Philadelphia (1996). 

The disadvantage of including axial dispersion is that an exit boundary condition must be specified, and in cases where an analytical solution is not available, a numerical boundary-value problem must be solved in the axial direction, rather than an initial-value problem. 

This section concerns the Cauchy problem or initial value problem, where initial data at time t = 0 are given. It was noticed by Rutkevitch [6,7], and systematized by Joseph et al. [8], Joseph and Saut [9], and Dupret and Marchal [10] that Maxwell type models can present Hadamard instabilities, that is, instabilities to short waves. (See [11] for a recent discussion of more general models.) Then, the Cauchy problem is not well-posed in any good class but analytic. Highly oscillatory initial data will grow exponentially in space at any prescribed time. An ill-posed problem leads to catastrophic instabilities in numerical simulations. For example, even if one initiates the solution in a stable region, one could get arbitrarily close to an unstable one. 

The last decades much work has been done on exponential fitting and the numerical solution of periodic initial value problems (see refs. 1,2,6-114,118 and references therein). 

In ref. 142 the authors are studied the Numerov-type ODE solvers for the numerical solution of second-order initial value problems. They present a powerful and efficient symbolic code in MATHEMATICA for the derivation of their order conditions and principal truncation error terms. They also present the relative tree theory for such order conditions along with the elements of combinatorial mathematics, partitions of integer numbers and computer algebra which are the basis of the implementation of the S5unbolic code. We must that one of the authors is an expert on this specific field. 

In ref 156 the author studies the stability properties of a family of exponentially fitted Runge-Kutta-Nystrom methods. More specifically the author investigates the P-stability which is a very important property usually required for the numerical solution of stiff oscillatory second-order initial value problems. In this paper P-stable exponentially fitted Runge-Kutta-Nystrom methods with arbitrary high order are developed. The results of this paper are proved based on a S5unmetry argument. 

In ref 171 new Runge-Kutta-Nystrom methods for the numerical solution of periodic initial value problems are obtained. These methods are able to integrate exactly the harmonic oscillator. In this paper the analysis of the production of an embedded 5(3) RKN pair with four stages is presented. 

Yonglei Fang and Xinyuan Wu, A trigonometrically fitted explicit Numerov-type method for second-order initial value problems with oscillating solutions, Applied Numerical Mathematics, in press. 

We tested the efficiency of our newly obtained scheme against well known methods, with excellent results. The numerical illustration showed that our method is considerably more efficient compared to well known methods used for the numerical solution of initial value problems with oscillating solutions. 

Numerical illustrations show that the procedure of trigonometric fitting is an efficient way to produce numerical methods for the solution of second-order linear initial value problems (IVPs) with oscillating solutions. 

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