Special Second-order Initial Value Problems Research Articles

Trigonometric-Fitted Explicit Numerov-Type Method with Vanishing Phase-Lag and Its First and Second Derivatives

An effectively four-stage, sixth-order, hybrid explicit Numerov-type method is presented for the solution of the special second-order initial value problem. The new method uses variable step, is trigonometric fitted and the phase-lag is nullified along with its first and second derivative. Extensive numerical tests illustrate the superiority of our proposal over similar methods found in the relevant 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 hybrid methods for oscillatory special second-order initial value problems

ABSTRACTIn this paper, trigonometrically fitted multi-step hybrid (TFMSH) methods for the numerical integration of oscillatory special second-order initial value problems are proposed and studied. TFMSH methods inherit the frame of multi-step hybrid (MSH) methods and integrate exactly the differential system whose solutions can be expressed as the linear combinations of functions from the set or equivalently the set , where w represents an approximation of the main frequency of the problem. The corresponding order conditions are given and two explicit TFMSH methods with order six and seven, 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.

Multi-step Runge–Kutta–Nyström methods for special second-order initial value problems

In this paper, multi-step Runge–Kutta–Nyström methods for the numerical integration of special second-order initial value problems are proposed and studied. These methods include classical Runge–Kutta–Nyström methods as special cases. General order conditions are derived by using the theory of B-series based on the set of special Nyström-trees, and two explicit methods with order five and six, respectively, are constructed. Numerical results show that our new methods are more efficient in comparison with classical Runge–Kutta–Nyström methods and other well-known high quality methods proposed in the scientific literature.

Embedded explicit two-step Runge-Kutta-Nystrom for solving special second-order IVPs

A third-order three-stage explicit Two-step Runge-Kutta-Nystr¨om (TSRKN) method embedded into fourth-order three-stage TSRKN method is developed to solve special second-order initial value problems (IVPs) directly. The stability of the method is investigated. Numerical results are obtained by solving a standard set of test problems, which then reduced to first-order system when solved using Runge-Kutta (RK) method and comparison are made with existing RK method with same order using variable step-size. The results clearly showed the advantage and the efficiency of the new method.

Review of explicit Falkner methods and its modifications for solving special second-order I.V.P.s

The unusual implementation of explicit Falkner methods for solving special second-order initial-value problems increases the order of the method as will be shown by means of numerical examples. In this paper we made an analysis of the propagation of the truncation errors in both the usual and the unusual implementations, thus justifying the numerical results obtained. By that, some error bounds for the global truncation errors on the solution and on the derivative are provided. Some numerical examples confirm that the bounds are realistic. Stability analysis is also addressed, and intervals of stability are presented.

Embedded diagonally implicit Runge–Kutta–Nystrom 4(3) pair for solving special second-order IVPs

In this paper, third-order 3-stage diagonally implicit Runge–Kutta–Nystrom method embedded in fourth-order 4-stage for solving special second-order initial value problems is constructed. The method has the property of minimized local truncation error as well as the last row of the coefficient matrix is equal to the vector output. The stability of the method is investigated and a standard set of test problems are tested upon and comparisons on the numerical results are made when the same set of test problems are reduced to first-order system and solved using existing Runge–Kutta method. The results clearly shown the advantage and the efficiency of the new method.

New P-stable high-order methods with minimal phase-lag for the numerical integration of the radial Schrödinger equation

A new family of P-stable two-step high-order methods with minimal phaselag are developed for the numerical integration of the special second-order initial value problem. An application to the one-dimensional Schrödinger equation, indicates that these new methods are generally more accurate than other previously developed finite difference methods.

Some new Numerov-type methods with minimal phase lag for the numerical integration of the radial Schrödinger equation

A new family of P-stable two-step methods with minimal phase lag is developed for the numerical integration of the special second-order initial value problem. An application to the one-dimensional Schrödinger equation indicates that these new methods are generally more accurate than previously developed finite difference methods.

Some new variable-step methods with minimal phase lag for the numerical integration of special second-order initial-value problem

Two new variable step methods with minimal phase lag are developed for the numerical integration of the special second-order initial-value problem. An application to the one- dimensional Schrödinger equation on the phase-shift problem indicates that these new methods are generally more accurate than other previously developed finite difference methods, especially in the case of the high energies.

A new variable-step method for the numerical integration of special second-order initial value problems and their application to the one-dimensional Schrödinger equation

A new variable-step method is developed for the numerical integration of special second-order initial value problems. An application to the one-dimensional Schrödinger equation on the phase-shift problem, indicates that this new method is generally more accurate than other previously developed finite difference methods, especially in the case of high energies.

High-order methods with minimal phase-lag for the numerical integration of the special second-order initial value problem and their application to the one-dimensional Schrödinger equation

Two two-step sixth-order methods with phase-lag of order eight and ten are developed for the numerical integration of the special second-order initial value problem. One of these methods is P-stable and the other has an interval of periodicity larger than the Numerov method. An application to the one-dimensional Schrödinger equation on the resonance problem, indicates that these new methods are generally more accurate than methods developed by Chawla and Rao. We note that the new methods introduce a new approach for the numerical integration of the Schrödinger equation.

Explicit two-step methods with minimal phase-lag for the numerical integration of special second-order initial-value problems and their application to the one-dimensional Schrödinger equation

Two-step sixth-order methods with phase-lag of order eight, ten and twelve are developed for the numerical integration of the special second-order initial-value sproblem. An application to the one-dimensional Schrödinger equation on the resonance problem indicates that these new methods are generally more accurate than methods developed by Chawla, Rao and Neta (this journal, 1986, 1987).

Stability of collocation-based Runge-Kutta-Nyström methods

We analyse the attainable order and the stability of Runge-Kutta-Nystrom (RKN) methods for special second-order initial-value problems derived by collocation techniques. Like collocation methods for first-order equations the step point order ofs-stage methods can be raised to 2s for alls. The attainable stage order is one higher and equalss+1. However, the stability results derived in this paper show that we have to pay a high price for the increased stage order.

Runge-Kutta-Nystrom triples

The embedded Runge-Kutta-Nystrom process for the numerical solution of the special second-order initial-value problem can be extended by the addition of “dense” formulae for both y and y′ which yield solutions at points within the normal step intervals. As for Runge-Kutta methods two modes of implementation are possible and it is shown that these are equivalent under certain conditions. The cases of “dense” formulae applied to RKN4(3) and RKN6(4) families are considered and test results are presented.

Orbitally stable multistep methods

Lambert and Watson (1976) examine the family of symmetric linear multistep methods for the special second-order initial value problem, and connect the property of symmetry with a property of periodicity. The problems of celestial mechanics may be formulated as second-order initial value problems, but these frequently incorporate the first derivative explicity. It is common for such equations to be reduced to a system of first-order equations. Thus motivated, we utilize ideas from the aforementioned paper to determine the family of linear multistep methods for first-order initial value problems that possess an analogous property of periodicity. This family of orbitally stable methods is illustrated by examining the regularized equations of motion of an artificial earth satellite in an oblate atmosphere.