Oscillatory Initial Value Problems Research Articles

Numerical Approximation of Oscillatory Initial Value Problem Using the Homotopy Analysis Algorithm

In this paper, we present numerical approximation for oscillatory initial value problems (IVPs) using the homotopy analysis algorithm. The convergence of the method is discussed and numerical experiments are presented to illustrate the computational effectiveness of the algorithm. The results obtained are in good agreement with the exact solutions and Adomian decomposition method (ADM). These results show that the algorithm introduced here is accurate and easy to apply without linearization.

NUMERICAL APPROXIMATION OF OSCILLATORY INITIAL VALUE PROBLEMS USING THE HOMOTOPY ANALYSIS ALGORITHM

In this paper, we present numerical approximation for oscillatory initial value problems (IVPs) using the homotopy analysis algorithm. The convergence of the method is discussed and numerical experiments are presented to illustrate the computational effectiveness of the algorithm. The results obtained are in good agreement with the exact solutions and Adomian decomposition method (ADM). These results show that the algorithm introduced here is easy to apply without linearization.

A second-derivative functionally fitted method of maximal order for oscillatory initial value problems

A second-derivative functionally fitted method of maximal order for oscillatory initial value problems

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 explicit six-step singularly P-stable Obrechkoff method for the numerical solution of second-order oscillatory initial value problems

In this paper, we present an explicit six-step singularly P-stable Obrechkoff method of tenth algebraic order for solving second-order linear periodic and oscillatory initial value problems of ordinary differential equations. The advantage of this new singularly P-stable Obrechkoff method is that it is a high-order explicit method, and thus does not require additional predictor stages. The numerical stability and phase properties of the new method is analyzed. Four numerical examples show that the new explicit method is more accurate than Obrechkoff schemes of the same order when applied to the numerical solution of second-order initial value problems with highly oscillatory solutions.

Spectral Rectangular Collocation Formula: An Approach for Solving Oscillatory Initial Value Problems or/and Boundary Value Problems in Ordinary Differential Equations

The idea of rectangularization of a differentiation matrix through collocation was recently suggested to be more useful in discretization process, especially when the traditional row replacement approach fails. In this regard, we employ the state-of-the-art technique of rectangularization to discretize some oscillatory initial value problems (IVPs) and also extended the new approach to a non-linear boundary value problem (BVP). The numerical implementation was composed using some few lines of executable Python codes. Our findings are instructive and quite revealing and supported by evidence from our numerical experiments and simulations.

Construction of L stable second derivative trigonometrically fitted block backward differentiation formula for the solution of oscillatory initial value problems

A second derivative trigonometrically fitted block backward differentiation formula (SDTFBBDF) based on the collocation technique is proposed in this paper. The method is specifically considered for the numerical solution of oscillatory problems. The SDTFBBDF which has one main method and one complementary method as by-products of the continuous second derivative backward differentiation formula (CSDBDF) whose coefficients depend on the frequency and step size. The stability and the convergence of SDTFBBDF are established and the performance of the proposed method is verified on some numerical examples to show its accuracy and efficiency.

A new four-step P-stable Obrechkoff method with vanished phase-lag and some of its derivatives for the numerical solution of radial Schrödinger equation

In this paper, we present a new four-step implicit P-stable Obrechkoff method of tenth algebraic order for solving one-dimensional second-order linear periodic and oscillatory initial value problems of ordinary differential equations such as reasonable Schrödinger equation. Numerical stability and phase properties of the new methods are analyzed. Numerical experiments are carried out to show the efficiency and robustness of our new methods in comparison with the well known codes proposed in the scientific literature.

Trigonometrically fitted three-derivative Runge–Kutta methods for solving oscillatory initial value problems

Trigonometrically fitted three-derivative Runge–Kutta (TFTHDRK) methods for solving numerically oscillatory initial value problems are proposed and developed. TFTHDRK methods improve three-derivative Runge–Kutta (THDRK) methods [Numer. Algor. 74: 247–265, 2017] and integrate exactly the problem whose solutions can be expressed as the linear combinations of functions from the set of {exp(iwt),exp(−iwt)} or equivalently the set {cos (wt), sin (wt)}, where w approximate the main frequency of the problem. The order conditions are deduced by the theory of rooted trees and B-series and two new explicit special TFTHDRK methods with order five and seven, respectively, are constructed. Linear stability of TFTHDRK methods is examined. Numerical results show the superiority of the new methods over other methods from the scientific literature.

Eighth-order explicit two-step hybrid methods with symmetric nodes and weights for solving orbital and oscillatory IVPs

The construction of new two-step hybrid (TSH) methods of explicit type with symmetric nodes and weights for the numerical integration of orbital and oscillatory second-order initial value problems (IVPs) is analyzed. These methods attain algebraic order eight with a computational cost of six or eight function evaluations per step (it is one of the lowest costs that we know in the literature) and they are optimal among the TSH methods in the sense that they reach a certain order of accuracy with minimal cost per step. The new TSH schemes also have high dispersion and dissipation orders (greater than 8) in order to be adapted to the solution of IVPs with oscillatory solutions. The numerical experiments carried out with several orbital and oscillatory problems show that the new eighth-order explicit TSH methods are more efficient than other standard TSH or Numerov-type methods proposed in the scientific literature.

Trigonometrically fitted multi-step RKN methods for second-order oscillatory initial value problems

Trigonometrically fitted multi-step Runge–Kutta–Nyström (TFMSRKN) methods for solving numerically oscillatory special second-order initial value problems are introduced. TFMSRKN methods integrate exactly the differential system whose solutions can be expressed as the linear combinations of functions from the set {exp(iwt),exp(−iwt)} or equivalently the set {cos (wt), sin (wt)}, where w represents an approximation of the main frequency of the problem. The corresponding order conditions are given and two explicit four-stage TFMSRKN methods with order five are constructed. Stability of the new methods is examined and the regions of stability are depicted. Numerical results show that our new methods are more efficient in comparison with other well-known high quality methods proposed in the scientific literature.

Trigonometrically fitted multi-step Runge-Kutta methods for solving oscillatory initial value problems

In this paper, trigonometrically fitted multi-step Runge-Kutta (TFMSRK) methods for the numerical integration of oscillatory initial value problems are proposed and studied. TFMSRK methods inherit the frame of multi-step Runge-Kutta (MSRK) methods and integrate exactly the problem whose solutions can be expressed as the linear combinations of functions from the set of $\{\exp (\mathrm {i}wt),\exp (-\mathrm {i}wt)\},$ or equivalently the set $\{\cos (wt),\sin (wt)\}$ , where w represents an approximation of the main frequency of the problem. The general order conditions are given and four new explicit TFMSRK methods with order three and four, respectively, are constructed. Stability of the new methods is examined and the corresponding regions of stability are depicted. Numerical results show that our new methods are more efficient in comparison with other well-known high quality methods proposed in the scientific literature.

A 5(4) Embedded Pair of Explicit Trigonometrically-Fitted Runge–Kutta–Nyström Methods for the Numerical Solution of Oscillatory Initial Value Problems

A 5(4) pair of embedded explicit trigonometrically-fitted Runge–Kutta–Nyström (EETFRKN) methods especially designed for the numerical integration of second order initial value problems with oscillatory solutions is presented in this paper. Algebraic order analysis and the interval of absolute stability for the new method are also discussed. The new method is capable of integrating the test equation y ″ = − w 2 y . The new method is much more efficient than the other existing Runge–Kutta and Runge–Kutta–Nyström methods.

On a family of trigonometrically fitted extended backward differentiation formulas for stiff and oscillatory initial value problems

A Class of Continuous Extended Backward Differentiation Formula using Trigonometric Basis functions (CEBDTB) with one superfuture point is constructed and used to generate a family of Trigonometrically Fitted Extended Backward Differentiation Formula (TFEBDF) and other discrete methods as by-products. This family of discrete schemes together with the TFEBDF are simultaneously applied as numerical integrators by assembling them into a Block Trigonometrically Fitted Extended Backward Differentiation Method (BTFEBDM). The error analysis, stability properties, and implementation of the BTFEBDM are discussed, and the class of methods is shown to be suitable for oscillatory and/or stiff Initial Value Problems (IVPs). The performance of the method is demonstrated on some numerical examples to show its accuracy and computational efficiency.

Hybrid high algebraic order two-step method with vanished phase-lag and its first, second, third, fourth and fifth derivatives

A hybrid tenth algebraic order two-step method with vanished phase-lag and its first, second, third, fourth and fifth derivatives are obtained in this paper. We will investigate • the construction of the method • the local truncation error (LTE) of the newly obtained method. We will also compare the lte of the newly developed method with other methods in the literature (this is called the comparative LTE analysis) • the stability (interval of periodicity) of the produced method using frequency for the scalar test equation different from the frequency used in the scalar test equation for phase-lag analysis (this is called stability analysis) • the application of the newly obtained method to the resonance problem of the Schrödinger equation. We will compare its effectiveness with the efficiency of other known methods in the literature. It will be proved that the developed method is effective for the approximate solution of the Schrödinger equation and related periodical or oscillatory initial value or boundary value problems.

A FOURTH ORDER TRIGONOMETRICALLY FITTED METHOD WITH THE BLOCK UNIFICATION IMPLEMENTATION APPROACH FOR OSCILLATORY INITIAL VALUE PROBLEMS

Abstract: This paper is concerned with the construction and implementation of a continuous fourth order trigonometrically fitted method on oscillatory initial value problems (IVPs). The continuous scheme is used to generate four discrete methods as by products. The four discrete schemes are weighted thesame and used via the Block Unification Approach to obtain approximate solutions to first order IVPs with oscillating solutions. The convergence of the method is established and two test problems are given to show the accuracy and computational efficiency of the scheme.

Optimized Hybrid Methods for Solving Oscillatory Second Order Initial Value Problems

Two-step optimized hybrid methods of order five and order six are developed for the integration of second order oscillatory initial value problems. The optimized hybrid method (OHMs) are based on the existing nonzero dissipative hybrid methods. Phase-lag, dissipation or amplification error, and the differentiation of the phase-lag relations are required to obtain the methods. Phase-fitted methods based on the same nonzero dissipative hybrid methods are also constructed. Numerical results show that OHMs are more accurate compared to the phase-fitted methods and some well-known methods appeared in the scientific literature in solving oscillating second order initial value problems. It is also found that the nonzero dissipative hybrid methods are more suitable to be optimized than phase-fitted methods.

A Family of Trigonometrically Fitted Enright Second Derivative Methods for Stiff and Oscillatory Initial Value Problems

A family of Enright’s second derivative formulas with trigonometric basis functions is derived using multistep collocation method. The continuous schemes obtained are used to generate complementary methods. The stability properties of the methods are discussed. The methods which can be applied in predictor-corrector form are implemented in block form as simultaneous numerical integrators over nonoverlapping intervals. Numerical results obtained using the proposed block form reveal that the new methods are efficient and highly competitive with existing methods in the literature.

Zero-dissipative semi-implicit hybrid method for solving oscillatory or periodic problems

In this paper, a new semi-implicit two-step hybrid method with fifth algebraic order is derived for the integration of second-order oscillatory initial value problems. The new method possesses dispersion of order eight and dissipation of order infinity. Numerical experiment reveals the superiority of the new method for solving oscillatory or periodic problems over several methods of the same algebraic order in the literature.

Block hybrid method using trigonometric basis for initial value problems with oscillating solutions

A continuous hybrid method using trigonometric basis (CHMTB) with one `off-step' point is developed and used to produce two discrete hybrid methods which are simultaneously applied as numerical integrators by assembling them into a block hybrid method with trigonometric basis (BHMTB) for solving oscillatory initial value problems (IVPs). The stability property of the BHMTB is discussed and the performance of the method is demonstrated on some numerical examples to show accuracy and efficiency advantages.