Numerical methods initial value problems

Chapter 7 Numerical Solution Methods (Initial Value Problems)... 

Chapter 7 Numerical Solution Methods (Initial Value Problems) uniform throughout. A heat balance on the bath gives... 

Ordinaiy differential Eqs. (13-149) to (13-151) for rates of change of hquid-phase mole fractious are uouhuear because the coefficients of Xi j change with time. Therefore, numerical methods of integration with respect to time must be enmloyed. Furthermore, the equations may be difficult to integrate rapidly and accurately because they may constitute a so-called stiff system as considered by Gear Numerical Initial Value Problems in Ordinaiy Differential Equations, Prentice Hall, Englewood Cliffs, N.J., 1971). The choice of time... 

The forward shooting method seems straightforward but is troublesome to use. What we have done is to convert a two-point boundary value problem into an easier-to-solve initial value problem. Unfortunately, the conversion gives a numerical computation that is ill-conditioned. Extreme precision is needed at the inlet of the tube to get reasonable accuracy at the outlet. The phenomenon is akin to problems that arise in the numerical inversion of matrices and Laplace transforms. 

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. 

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)-... 

In ref. 144 the author presents the construction of a non-standard explicit algorithm for initial-value problems. The order of the developed method is two and also is A-stable. The new proposed method is proven to be suitable for solving different kind of initial-value problems such as non-singular problems, singular problems, stiff problems and singularly perturbed problems. Some numerical experiments are considered in order to check the behaviour of the method when applied to a variety of initial-value problems. 

In ref 148 the author obtain a new kind of trigonometrically fitted explicit Numerov-type method for the numerical integration of second-order initial value problems with... 

In ref 152 the author produces methods based on numerical differentiation some classes of special multistep methods. For these methods the regions of absolute stability are shown. Numerical efficiency of the methods is examined by application of some of Henrici and of some methods obtained in this paper of the same order to a second-order initial value problem. 

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. 

Kamel Al-Khaled and M. Naim Anwar, Numerical comparison of methods for solving second-order ordinary initial value problems. Applied Mathematical Modelling, 2007, 31, 292-301. 

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