Initial value problem, solutions NUMERICAL COMPUTER METHODS

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)]. 

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 solution for f(q ) cannot be obtained analytically. Although the similarity transformation has reduced the set of PDEs, (10-64), to a single ODE, (10-75), the latter is still nonlinear. In fact, Blasius originally solved (10-75) by using a numerical method, but with the algebra carried out by hand Fortunately, today accurate numerical solutions can be obtained with a computer. The main difficulty in solving (10 75) numerically is that most methods for solving ODEs are set up for initial-value problems. 

As the next step up in complexity, we consider the case of multiple reactions. Some analytical solutions are available for simple cases with multiple reactions, and Axis provides a comprehensive list [2], but the scope of these is limited. We focus on numerical computation as a general method for these problems. Indeed, we find that even numerical solution of some of these problems is challenging for two reasons. First, steep concentration profiles often occur for realistic parameter values, and we wish to compute these profiles accurately. It is not unusual for species concentrations to change by 10 orders of magnitude within the pellet for realistic reaction and diffusion rates. Second, we are solving boundary-value problems because the boundary conditions are provided at the center and exterior surface of the pellet. Boundary-value problems (BVPs) are generally much more difficult to solve than initial-value problems (IVPs). 

A. Konguetsof and T. E. Simos, An exponentially-fitted and trigonome-trically-litted method for the numerical solution of periodic initial-value problems. Computers and Mathematics with Applications, 2003, 45, 547-554. 

D. F. Papadopoulos, Z. A. Anastassi and T. E. Simos, A phase-fitted Runge-Kutta-Nystrom method for the numerical solution of initial value problems with oscillating solutions. Computer Physics Communications, 2009, 180(10), 1839-1846. 

Abstract. A class (m,k)-methods is discussed for the numerical solution of the initial value problems for impHcit systems of ordinary differential equations. The order conditions and convergence of the numerical solution in the case of implementation of the scheme with the time-lagging of matrices derivatives for systems of index 1 are obtained. At A < 4 the order conditions are studied and schemes optimal computing costs are obtained. 

Long before electronic computers were invented, it was realized that mathematical sophistication could be introduced into numerical integration in order to save computational elTort and improve accuracy. Textbooks of numerical analysis are full of ways to do this. The most popular of them, the Runge-Kutta and predictor-corrector algorithms, once were standard methods for numerical solution of the initial value problems of chemical kinetics. They have been replaced, however, by more suitable methods invented for the specific purpose of dealing with chemical kinetics problems. 

The Runge-Kutta method is widely used as a numerical method to solve differential equations. This method is more accurate than the improved Euler s method. This method computes the solution of the initial value problem. 

The mathematical modelling of the T-H-M phenomena uses initial - boundary value problems for differential or variational equations involving the physical principles. We shall assume that these problems are discretized by the finite element or similar methods. What we want to point out is that the numerical solution can be computationally very expensive due to... 

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