Initial value problems 177 explicit methods

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 146 the authors present a non-standard (nonlinear) two-step explicit P-stable method of fourth algebraic order and 12th phase-lag order for solving second-order linear periodic initial value problems of ordinary differential equations. The proposed method can be extended to be vector-applicable for multi-dimensional problem based on a special vector arithmetic with respect to an analytic function. 

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 this paper the authors produced a class of non-linear explicit second-order methods for solving one-dimensional periodic initial value problems (IVPs). These methods are P-stable (based on the definition given by Lambert and Watson ) and they have phase-lag of high-order. The authors introduce a special vector operation such that the obtained methods to be extended to be vector-applicable directly. With this extension the produced methods can be applied to multi-dimensional problems. 

M. M. Chawla and P. S. Rao, A Numerov-type method with minimal phase-lag for the integration of second order periodic initial-value problems. II Explicit method, J. Comput. Appl. Math., 1986, 15, 329—337. 

G. Psihoyios and T. E. Simos, Exponentially and trigonometrically fitted explicit advanced step-point (EAS) methods for initial value problems with oscillating solutions, International Journal of Modern Physics C, 2003(2), 175-184. 

H. Van de Vyver, A phase-fitted and amplification-fitted explicit two-step hybrid method for second-order periodic initial value problems, Internat. J. Modern Phys. C, 2006, 17, 663-675. 

Numerical solution of the ODEs describing initial value problems is possible by explicit and implicit techniques, which are described in the Sections 11.1.1 and 11.1.2, respectively. It is worth noting that the techniques are formulated for the solution of a single equation (Equation 11.1), but they can be used for solving multiple ODEs as well. Theoretical background of these methods as well as their stabilities are described elsewhere [1,2] and will not be discussed here. 

In [161] the author produces a new explicit Numerov-type method for the approximate solution of second-order linear initial value problems with oscillating solutions. The computational cost of the proposed method is six function evaluations per step. The characteristic of the new proposed method are ... 

T. E. Simos, An Explicit 4-Step Phase-Fitted Method for the Numerical-Integration of 2nd-order Initial-Value Problems, Journal of Computational and Applied Mathematics, 1994, 55(2), 125-133. 

G. Saldanha and D. J. Saldanha, A class of explicit two-step superstable methods for second-order linear initial value problems, International Journal of Computer Mathematics, 2009, 86(8), 1424-1432. 

The modern methods for numerical solution of the initial-value problem for systems of ordinary differential equations (ODE) suppose usually the explicit dependence of the derivative of the solution [7]... 

In ref. 152 the authors develop s-stage explicit two- and three-step peer methods of order p = 2s and p = 3s for the numerical approximation of second-order initial value problems where the right-hand side does not depend on y. More specifically, they consider two-step peer methods of the form ... 

In this section, we discuss the stability of finite difference approximations using the well-known von Neumann procedure. This method introduces an initial error represented by a finite Fourier series and examines how this error propagates during the solution. The von Neumann method applies to initial-value problems for this reason it is used to analyze the stability of the explicit method for parabolic equations developed in Sec. 6.4.2 and the explicit method for hyperbolic equations developed in Sec. 6.4.3. 

There are many numerical approaches one can use to approximate the solution to the initial and boundary value problem presented by a parabolic partial differential equation. However, our discussion will focus on three approaches an explicit finite difference method, an implicit finite difference method, and the so-called numerical method of lines. These approaches, as well as other numerical methods for aU types of partial differential equations, can be found in the literature [5,9,18,22,25,28-33]. 

The same numerical methods as those used to solve the homogeneous reactor models (PFR, BR, and stirred tank reactor) as well as the heterogeneous catalytic packed bed reactor models are used for gas-Uquid reactor problems. For the solution of a countercurrent column reactor, an iterative procedure must be applied in case the initial value solvers are used (Adams-Moulton, BD, explicit, or semi-implicit Runge-Kutta). A better alternative is to solve the problem as a true boundary value problem and to take advantage of a suitable method such as orthogonal collocation. If it is impossible to obtain an analytical solution for the liquid film diffusion Equation 7.52, it can be solved numerically as a boundary value problem. This increases the numerical complexity considerably. For coupled reactions, it is known that no analytical solutions exist for Equation 7.52 and, therefore, the bulk-phase mass balances and Equation 7.52 must be solved numerically. 

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