scholarly journals P-Stable Higher Derivative Methods with Minimal Phase-Lag for Solving Second Order Differential Equations

2011 ◽  
Vol 2011 ◽  
pp. 1-15 ◽  
Author(s):  
Fatheah A. Hendi
Keyword(s):  
Oscillatory Behavior ◽  
Multistep Method ◽  
Second Order ◽  
Phase Lag ◽  
Stability Properties ◽  
Higher Derivative ◽  
Orbital Problems ◽  
Algebraic Order ◽  
The Stability ◽  
Physical Phenomena

Some new higher algebraic order symmetric various-step methods are introduced. For these methods a direct formula for the computation of the phase-lag is given. Basing on this formula, calculation of free parameters is performed to minimize the phase-lag. An explicit symmetric multistep method is presented. This method is of higher algebraic order and is fitted both exponentially and trigonometrically. Such methods are needed in various branches of natural science, particularly in physics, since a lot of physical phenomena exhibit a pronounced oscillatory behavior. Many exponentially-fitted symmetric multistepmethods for the second-order differential equation are already developed. The stability properties of several existing methods are analyzed, and a newP-stable method is proposed, to establish the existence of methods to which our definition applies and to demonstrate its relevance to stiff oscillatory problems. The work is mainly concerned with two-stepmethods but extensions tomethods of larger step-number are also considered. To have an idea about its accuracy, we examine their phase properties. The efficiency of the proposed method is demonstrated by its application to well-known periodic orbital problems. The new methods showed better stability properties than the previous ones.

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

2016 ◽  
Vol 27 (05) ◽  
pp. 1650049 ◽  
Author(s):  
Junyan Ma ◽  
T. E. Simos
Keyword(s):  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Truncation Error ◽  
Resonance Problem ◽  
Phase Lag ◽  
Step Method ◽  
Algebraic Order ◽  
Test Equation ◽  
The Stability ◽  
Scalar Test Equation

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.

Variable-step-size second-order-derivative multistep method for solving first-order ordinary differential equations in system simulation

2020 ◽  
Vol 11 (01) ◽  
pp. 2050001
Author(s):  
Lei Zhang ◽  
Chaofeng Zhang ◽  
Mengya Liu
Keyword(s):  
Differential Equations ◽  
Numerical Method ◽  
Multistep Method ◽  
Second Order ◽  
Order Derivative ◽  
Variable Step Size ◽  
Variable Order ◽  
Step Size ◽  
Variable Step ◽  
The Stability

According to the relationship between truncation error and step size of two implicit second-order-derivative multistep formulas based on Hermite interpolation polynomial, a variable-order and variable-step-size numerical method for solving differential equations is designed. The stability properties of the formulas are discussed and the stability regions are analyzed. The deduced methods are applied to a simulation problem. The results show that the numerical method can satisfy calculation accuracy, reduce the number of calculation steps and accelerate calculation speed.

A new efficient implicit four-step method with vanished phase-lag and some of its derivatives for the numerical solution of the radial Schr¨odinger equation

2017 ◽  
Vol 8 (1-2) ◽  
pp. 77 ◽  
Author(s):  
Ali Shokri ◽  
Morteza Tahmourasi
Keyword(s):  
Approximate Solution ◽  
Stability Analysis ◽  
Multistep Method ◽  
Phase Lag ◽  
Order Method ◽  
Step Method ◽  
One Dimensional ◽  
First Derivative ◽  
Algebraic Order ◽  
The One

A new four-step implicit linear sixth algebraic order method with vanished phase-lag and its first derivative is constructed in this paper. The purpose of this paper is to develop an efficient algorithm for the approximate solution of the one-dimensional radial Schr¨odinger equation and related problems. In order to produce an efficient multistep method the phase-lag property and its derivatives are used. An error analysis and a stability analysis is also investigated and a comparison with other methods is also studied. The efficiency of the new methodology is proved via theoretical analysis and numerical applications.

scholarly journals Unconditional Stability of a Numerical Method for the Dual-Phase-Lag Equation

2017 ◽  
Vol 2017 ◽  
pp. 1-5
Author(s):  
M. A. Castro ◽  
J. A. Martín ◽  
F. Rodríguez
Keyword(s):  
Numerical Method ◽  
Numerical Solutions ◽  
Unconditional Stability ◽  
Bioheat Transfer ◽  
Phase Lag ◽  
Dual Phase ◽  
Stability Properties ◽  
The Stability ◽  
Transfer Problems ◽  
Uniformly Bounded

The stability properties of a numerical method for the dual-phase-lag (DPL) equation are analyzed. The DPL equation has been increasingly used to model micro- and nanoscale heat conduction in engineering and bioheat transfer problems. A discretization method for the DPL equation that could yield efficient numerical solutions of 3D problems has been previously proposed, but its stability properties were only suggested by numerical experiments. In this work, the amplification matrix of the method is analyzed, and it is shown that its powers are uniformly bounded. As a result, the unconditional stability of the method is established.

A new high algebraic order efficient finite difference method for the solution of the Schrödinger equation

Filomat ◽  
2017 ◽  
Vol 31 (15) ◽  
pp. 4999-5012 ◽  
Author(s):  
Ming Dong ◽  
Theodore Simos
Keyword(s):  
Truncation Error ◽  
Interval Of Periodicity ◽  
Phase Lag ◽  
Local Truncation Error ◽  
Step Method ◽  
Dinger Equation ◽  
Algebraic Order ◽  
Test Equation ◽  
The Stability ◽  
Scalar Test Equation

The development of a new five-stages symmetric two-step method of fourteenth algebraic order with vanished phase-lag and its first, second, third and fourth derivatives is analyzed in this paper. More specifically: (1) we will present the development of the new method, (2) we will determine the local truncation error (LTE) of the new proposed method, (3) we will analyze the local truncation error based on the radial time independent Schr?dinger equation, (4) we will study the stability and the interval of periodicity of the new proposed method based on a scalar test equation with frequency different than the frequency of the scalar test equation used for the phase-lag analysis, (5) we will test the efficiency of the new obtained method based on its application on the coupled differential equations arising from the Schr?dinger equation.

A NEW SYMMETRIC EIGHT-STEP PREDICTOR-CORRECTOR METHOD FOR THE NUMERICAL SOLUTION OF THE RADIAL SCHRÖDINGER EQUATION AND RELATED ORBITAL PROBLEMS

2011 ◽  
Vol 22 (02) ◽  
pp. 133-153 ◽  
Author(s):  
G. A. PANOPOULOS ◽  
Z. A. ANASTASSI ◽  
T. E. SIMOS
Keyword(s):  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
New Method ◽  
Phase Lag ◽  
Oscillating Solutions ◽  
Orbital Problems ◽  
Radial Schrödinger Equation ◽  
Algebraic Order ◽  
Predictor Corrector ◽  
Minimal Phase

A new general multistep predictor-corrector (PC) pair form is introduced for the numerical integration of second-order initial-value problems. Using this form, a new symmetric eight-step predictor-corrector method with minimal phase-lag and algebraic order ten is also constructed. The new method is based on the multistep symmetric method of Quinlan–Tremaine,1 with eight steps and 8th algebraic order and is constructed to solve numerically the radial time-independent Schrödinger equation. It can also be used to integrate related IVPs with oscillating solutions such as orbital problems. We compare the new method to some recently constructed optimized methods from the literature. We measure the efficiency of the methods and conclude that the new method with minimal phase-lag is the most efficient of all the compared methods and for all the problems solved.

On the Stability of Some Linear Nonautonomous Random Systems

1968 ◽  
Vol 35 (1) ◽  
pp. 7-12 ◽  
Author(s):  
E. F. Infante
Keyword(s):  
Random Systems ◽  
Second Order ◽  
Stability Conditions ◽  
Almost Sure Stability ◽  
Stability Properties ◽  
The Stability ◽  
Second Order Equations

A theorem and two corollaries for the almost sure stability of linear nonautonomous random systems are presented. These results are applied to the study of the stability properties of some often encountered second-order equations and the obtained stability conditions are compared to previously known criteria.

scholarly journals Direct Two-Point Block One-Step Method for Solving General Second-Order Ordinary Differential Equations

2012 ◽  
Vol 2012 ◽  
pp. 1-16 ◽  
Author(s):  
Zanariah Abdul Majid ◽  
Nur Zahidah Mokhtar ◽  
Mohamed Suleiman
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Multistep Method ◽  
Second Order ◽  
Variable Step Size ◽  
Step Size ◽  
Step Method ◽  
One Step ◽  
The Stability ◽  
The One

A direct two-point block one-step method for solving general second-order ordinary differential equations (ODEs) directly is presented in this paper. The one-step block method will solve the second-order ODEs without reducing to first-order equations. The direct solutions of the general second-order ODEs will be calculated at two points simultaneously using variable step size. The method is formulated using the linear multistep method, but the new method possesses the desirable feature of the one-step method. The implementation is based on the predictor and corrector formulas in thePE(CE)mmode. The stability and precision of this method will also be analyzed and deliberated. Numerical results are given to show the efficiency of the proposed method and will be compared with the existing method.

scholarly journals A New Algorithm for the Approximation of the Schrödinger Equation

Open Physics ◽  
2016 ◽  
Vol 14 (1) ◽  
pp. 628-642 ◽  
Author(s):  
Rong-an LIN ◽  
Theodore E. Simos
Keyword(s):  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Computational Cost ◽  
Control Technique ◽  
Phase Lag ◽  
Step Method ◽  
Periodicity Analysis ◽  
Algebraic Order ◽  
The Stability ◽  
First Time

AbstractIn this paper a four stages twelfth algebraic order symmetric two-step method with vanished phase-lag and its first, second, third, fourth and fifth derivatives is developed for the first time in the literature. For the new proposed method: (1) we will study the phase-lag analysis, (2) we will present the development of the new method, (3) the local truncation error (LTE) analysis will be studied. The analysis is based on a test problem which is the radial time independent Schrödinger equation, (4) the stability and the interval of periodicity analysis will be presented, (5) stepsize control technique will also be presented, (6) the examination of the accuracy and computational cost of the proposed algorithm which is based on the approximation of the Schrödinger equation.

scholarly journals A Family of Functionally-Fitted Third Derivative Block Falkner Methods for Solving Second-Order Initial-Value Problems with Oscillating Solutions

Mathematics ◽  
2021 ◽  
Vol 9 (7) ◽  
pp. 713
Author(s):  
Higinio Ramos ◽  
Ridwanulahi Abdulganiy ◽  
Ruth Olowe ◽  
Samuel Jator
Keyword(s):  
Initial Value Problems ◽  
Second Order ◽  
Initial Value ◽  
Oscillatory Solutions ◽  
Oscillating Solutions ◽  
Algebraic Order ◽  
Direct Numerical Solution ◽  
The Stability ◽  
Direct Numerical Integration ◽  
Third Derivative

One of the well-known schemes for the direct numerical integration of second-order initial-value problems is due to Falkner (Falkner, 1936. Phil. Mag. S. 7, 621). This paper focuses on the construction of a family of adapted block Falkner methods which are frequency dependent for the direct numerical solution of second-order initial value problems with oscillatory solutions. The techniques of collocation and interpolation are adopted here to derive the new methods. The study of the properties of the proposed adapted block Falkner methods reveals that they are consistent and zero-stable, and thus, convergent. Furthermore, the stability analysis and the algebraic order conditions of the proposed methods are established. As may be seen from the numerical results, the resulting family is efficient and competitive compared to some recent methods in the literature.

Sign in / Sign up

Export Citation Format

Share Document