Order IVPs Research Articles

A New Methodology for the Development of Efficient Multistep Methods for First–Order IVPs with Oscillating Solutions IV: The Case of the Backward Differentiation Formulae

A theory for the calculation of the phase–lag and amplification–factor for explicit and implicit multistep techniques for first–order differential equations was recently established by the author. His presentation also covered how the approaches’ efficacy is affected by the elimination of the phase–lag and amplification–factor derivatives. This paper will apply the theory for computing the phase–lag and amplification–factor, originally developed for implicit multistep methods, to a subset of implicit methods, called backward differentiation formulae (BDF), and will examine the impact of the phase–lag and amplification–factor derivatives on the efficiency of these strategies. Next, we will show you the stability zones of these brand-new approaches. Lastly, we will discuss the results of numerical experiments and draw some conclusions about the established approaches.

Application of artificial bee colony algorithm as numerical solution for first order IVPs and industrial robot arm control problem

Application of artificial bee colony algorithm as numerical solution for first order IVPs and industrial robot arm control problem

A new order m – 1 rational integrator with an inhomogeneuous constant

In this research work we derived a new order m – 1, rational integrator with an inhomogeneous constant for the solution of ordinary differential equations. We establish the integrator formula with an interpolants of order m – 1, to determine the convergence and consistency of the method with the use of MATLAB softwares. We compared our integrators with “Numerical Strategies for the System of First Order IVPs using Block Hybrid Extended Trapezoidal Multistep of Second Kind for Stiff ODEs” and “A quartic based denominator of order six rational integrator in ordinary differential Equations” to solve real-life problems which ascertain the convergence and consistency of our scheme. Our results show good improvement over the existing methods compared with.

Development of an Efficient Diagonally Implicit Runge–Kutta–Nyström 5(4) Pair for Special Second Order IVPs

In this work, a new pair of diagonally implicit Runge–Kutta–Nyström methods with four stages is constructed. The proposed method has been derived to solve initial value problems of special second-order ordinary-differential equations. The principal local truncation error of the new method is obtained, and the main characteristics of the new method are analyzed. Some numerical experiments are performed, which demonstrate the robustness and efficiency of the new embedded pair.

A Model for Solving First, Second and Third Order IVPs Directly

A Model for Solving First, Second and Third Order IVPs Directly

Application of the regular perturbation method for the solution of first-order initial value problems

Several analytical and numerical methods are globally used in the solution of linear and non-linear ordinary differential equations. In recent times, an analytical method called the perturbation method was proposed by JiHuan He. This technique involves the use of the perturbation parameter, ε. The types of perturbation method which are usually used in Mathematics are the regular perturbation method and the singular perturbation method. In this research, various numerical examples of first-order IVPs are solved using the regular perturbation method. This method gives an approximate analytical solution to the first order IVPs and are discovered to be efficient and easy to adopt.

The new class of multistep multiderivative hybrid methods for the numerical solution of chemical stiff systems of first order IVPs

In this paper, we present a general form of Nth derivative multistep methods. In these hybrid multistep multiderivative methods, additional stage points (or off-step points) have been used in the first derivative of the solution to improve the absolute stability regions. The accuracy and stability properties of these methods are investigated. We apply the new methods for the numerical integration of some famous stiff chemical problems such as Belousov–Zhabotinskii reaction, the Chapman atmosphere, chemical Akzo-Nobel problem, ROBER problem (suggested by Robertson) and some others which are widely used in numerical studies.

A New Explicit Singularly P-Stable Four-Step Method for the Numerical Solution of Second Order IVPs

In this paper, we introduce a new symmetric explicit four-step method with variable coefficients for the numerical solution of second-order linear periodic and oscillatory initial value problems of ordinary differential equations. For the first time in the literature, we generate an explicit method with the most important singularly P-stability property. The method is multiderivative and has algebraic order eight and infinite order of phase-lag. The numerical results for some chemical (e.g. orbit problems of Stiefel and Bettis) as well as quantum chemistry problems (i.e. systems of coupled differential equations) indicated that the new method is superior, efficient, accurate and stable.

An Accurate Continuous Hybrid Block Method for the Direct Solution of General Second Initial Value Problem

In this research, a new hybrid block method of order 5 for solving second order Initial Value Problems (IVPs) is determined. One step with 3 off step points is used to generate the coefficients of the new method. This method approximates the solutions of the second order IVPS at xn+1/6, xn+2/6, xn+3/6, xn+1, concurrently by using interpolation and collocation approach with fixed step size. The developed method don’t need to reduce the IVPs to its equivalent system of first order IVPs. The numerical results obtained show that the new method outperforms the existing methods in terms of error.

The Treatment of Second Order Ordinary Differential Equations Using Equidistant One-step Block Hybrid

A general one-step hybrid block method with equidistant of order 6 has been successfully developed for the direct solution of second order IVPs in this article. Numerical analysis shows that the developed method is consistent and zero-stable which implies its convergence. The analysis of the new method is examined on two highly and mildly stiff second-order initial value problems to illustrate the efficiency of the method. It is obvious that our method performs better than the existing method within which we compare our result with. Hence, the approach is an adequate one for solving special second order IVPs.

Highly Stable Super-Implicit Hybrid Methods for Special Second Order IVPs

The idea of symmetric super-implicit linear multi-step methods (SSILMMs) necessitates the use of not just past and present solution values of the ordinary differential equations (ODEs), but also, future values of the solution. Such methods have been proposed recently for the numerical solution of second-order ODEs. One technique to obtain more accurate integration process is to construct linear multi-step methods with hybrid points employing future solution values. In this regard, we construct families of Stӧrmer-Cowell type hybrid SSILMMs having higher order than that of the symmetric super-implicit method recently proposed for the same step number using the Taylors series approach. The newly derived hybrid SSILMMs are p-stable with accurate results when tested on some special second order IVPs.

Numerical Strategies for the System of First Order IVPs Using Block Hybrid Extended Trapezoidal Multistep Method of Second Kind for Stiff ODEs

A Modified Three Step Block Hybrid Extended Trapezoidal Multistep Method of Second Kind (BHETR₂s) with two off-grid points, one at interpolation and another at collocation point yielding uniform order six (6, 6, 6, 6, 6)^T for the Numerical Integration of initial value problems of stiff Ordinary Differential Equations was developed. The main method and additional equations were obtained from the same continuous formulation through interpolation and collocation procedures. The stability properties of the method was discussed and from the stability region obtained, the method is suitable for the solution Stiff Ordinary Differential Equations. Three numerical examples were considered to illustrate the efficiency and accuracy.

Improved Extended Runge-Kutta-like Method for Solving First Order IVPs.

In this research, we proposed a family of improved extended Runge-Kutta-like methods which incorporate the function as well as the derivative of the function for the numerical integration of autonomous and non-autonomous ordinary differential equations. The proposed methods are more accurate than the existing methods in the literature and acquire bigger regions of stability. Numerical examples illustrating the computational accuracy are presented and the stability regions are also shown.

Optimization of explicit two-step hybrid methods for solving orbital and oscillatory problems

The construction of optimized explicit two-step hybrid methods for solving orbital problems and oscillatory second order IVPs is analyzed. These methods have variable coefficients depending on the parameter ν=ωh, where h is the integration step-size and ω represents an approximation of the main frequency of the problem. Some optimized explicit two-step hybrid schemes with orders four and six are derived and their stability and phase properties are analyzed. The new methods have the property of being zero dissipative for all the values of the parameter ω whereas their dispersion errors (phase-lag) are optimized in terms of the relative error committed in the approximation of the main frequency of the problem. The numerical experiments carried out with several orbital and oscillatory problems show that the new optimized two-step schemes are more efficient than other methods recently proposed in the scientific literature for solving this class of problems.

One Leg hybrid P-stable substitution LMM for oscilatory IVPs in ODEs.

This presents P-stable successive substitution one-leg hybrid LMM for the numerical solution of oscillatory second order IVPs in ODEs without explicitly defined first order derivative. These problems occurs amongst others, in orbital mechanics where the methods to be presented finds ready applications and need not any a priori knowledge of the period of the solution of the defining ODE. Journal of the Nigerian Association of Mathematical Physics Vol. 10 2006: pp. 345-354

NUCLEAR FUEL DEPLETION ANALYSIS USING MATLAB SOFTWARE

Coupled first order IVPs are frequently used in many parts of engineering and sciences. In this article, we presented a code including three computer programs which are joint with the Matlab software to solve and plot the solutions of the first order coupled stiff or non-stiff IVPs. Some engineering and scientific problems related to IVPs are given and fuel depletion (production of the 239 Pu isotope) in a Pressurized Water Nuclear Reactor (PWR) are computed by the present code.

A two-step explicit P-stable method of high phase-lag order for second order IVPs

This paper presents a two-step explicit P-stable method of high phase-lag order for solving second order ordinary differential equations. Basing on a special vector operation with respect to a smooth function, the method can be extended to be vector-applicable in multi-dimensional space.

Zero Dissipative, Explicit Numerov-Type Methods for Second Order IVPs with Oscillating Solutions

New explicit, zero dissipative, hybrid Numerov type methods are presented in this paper. We derive these methods using an alternative which avoids the use of costly high accuracy interpolatory nodes. We only need the Taylor expansion at some internal points then. The method is of sixth algebraic order at a cost of seven stages per step while their phase lag order is fourteen. The zero dissipation condition is satisfied, so the methods possess an non empty interval of periodicity. Numerical results over some well known problems in physics and mechanics indicate the superiority of the new method.

Three and four step least squares finite element schemes in the time domain

The paper presents a formulation and analysis of three and four step least squares algorithms for first order IVPs. The three step algorithm is derived using cubic Lagrangian interpolation, and is found to be third order accurate but only conditionally stable. Fourth order Lagrangian interpolation is used to obtain a four step least squares scheme which is A 0 -stable but inconsistent.

A class of block methods for second order IVPs∗:

A systematic approach for generating r-point /c-step methods of order p for positive integers r, k, p is proposed. A method of order p, step number k and generating r values at a time is classified as method (r, k, p). To every method of order p is associated a ‘free’ method of (order p + 1) which is exploited to control the step-size. The two methods are then classified as method (r,k,p/p+ 1). Fatunla (1991, 1993) respecti-vely gave the case for methods (2,1,3/4) and (3,1,5/6). In this paper, we present methods (2,3,5/6) and (4,1,7/8) which are zero-stable.