A Variable Step-Size Exponentially Fitted Explicit Hybrid Method for Solving Oscillatory Problems

An exponentially fitted explicit hybrid method for solving oscillatory problems is obtained. This method has four stages. The first three stages of the method integrate exactly differential systems whose solutions can be expressed as linear combinations of{1,x,exp(μx),exp(−μx)},μ∈C, while the last stage of this method integrates exactly systems whose solutions are linear combinations of{1,x,x2,x3,x4,exp(μx),exp(−μx)}. This method is implemented in variable step-size code basing on an embedding approach. The stability analysis is given. Numerical experiments that have been carried out show the efficiency of our method.

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A Class of Methods with Optimal Stability Properties for the Numerical Solution of IVPs: Construction and Implementation

International Journal of Computational Methods ◽

10.1142/s0219876217500074 ◽

2017 ◽

Vol 14 (01) ◽

pp. 1750007

Author(s):

Masoumeh Hosseini Nasab ◽

Gholamreza Hojjati ◽

Ali Abdi

Keyword(s):

Numerical Solution ◽

Numerical Experiments ◽

Point Of View ◽

Variable Step Size ◽

Second Derivative ◽

Step Size ◽

Stability Properties ◽

Variable Step ◽

The Stability ◽

Size Mode

Considering the methods with future points technique from second derivative general linear methods (SGLMs) point of view, makes it possible to improve their stability properties. In this paper, we extend the stability regions of a modified version of E2BD formulas to optimal one and show its effectiveness by numerical verifications. Also, implementation issues, with numerical experiments, of these methods are investigated in a variable step-size mode.

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Variable Step Exponentially Fitted Explicit Sixth-Order Hybrid Method with Four Stages for Spring-Mass and Other Oscillatory Problems

Symmetry ◽

10.3390/sym12030387 ◽

2020 ◽

Vol 12 (3) ◽

pp. 387

Author(s):

Faieza Samat ◽

Eddie Shahril Ismail

Keyword(s):

Hybrid Method ◽

Hybrid Methods ◽

Variable Coefficients ◽

Sixth Order ◽

Variable Step Size ◽

Local Error ◽

Step Size ◽

Oscillatory Solutions ◽

Variable Step ◽

Exponentially Fitted

For the numerical integration of differential equations with oscillatory solutions an exponentially fitted explicit sixth-order hybrid method with four stages is presented. This method is implemented using variable step-size while its derivation is accomplished by imposing each stage of the formula to integrate exactly { 1 , t , t 2 , … , t k , exp ( ± μ t ) } where the frequency μ is imaginary. The local error that is employed in the step-size selection procedure is approximated using an exponentially fitted explicit fourth-order hybrid method. Numerical comparisons of the new and existing hybrid methods for the spring-mass and other oscillatory problems are tabulated and discussed. The results show that the variable step exponentially fitted explicit sixth-order hybrid method outperforms the existing hybrid methods with variable coefficients for solving several problems with oscillatory solutions.

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

International Journal of Modeling Simulation and Scientific Computing ◽

10.1142/s1793962320500014 ◽

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.

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Embedded Exponentially-Fitted Explicit Runge-Kutta-Nyström Methods for Solving Periodic Problems

Computation ◽

10.3390/computation8020032 ◽

2020 ◽

Vol 8 (2) ◽

pp. 32

Author(s):

Musa Demba ◽

Poom Kumam ◽

Wiboonsak Watthayu ◽

Pawicha Phairatchatniyom

Keyword(s):

Error Estimation ◽

Variable Step Size ◽

Local Error ◽

Step Size ◽

Local Error Estimation ◽

Nyström Methods ◽

Periodic Problems ◽

Variable Step ◽

Exponentially Fitted ◽

Runge Kutta

In this work, a pair of embedded explicit exponentially-fitted Runge–Kutta–Nyström methods is formulated for solving special second-order ordinary differential equations (ODEs) with periodic solutions. A variable step-size technique is used for the derivation of the 5(3) embedded pair, which provides a cheap local error estimation. The numerical results obtained signify that the new adapted method is more efficient and accurate compared with the existing methods.

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The stability of variable step-size LMS algorithms

IEEE Transactions on Signal Processing ◽

10.1109/78.806072 ◽

1999 ◽

Vol 47 (12) ◽

pp. 3277-3288 ◽

Cited By ~ 38

Author(s):

S.B. Gelfand ◽

Yongbin Wei ◽

J.V. Krogmeier

Keyword(s):

Variable Step Size ◽

Step Size ◽

Variable Step ◽

The Stability

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Variable Step Size Adams Methods for BSDEs

Journal of Mathematics ◽

10.1155/2021/9799627 ◽

2021 ◽

Vol 2021 ◽

pp. 1-13

Author(s):

Qiang Han

Keyword(s):

High Order ◽

Variable Step Size ◽

Necessary And Sufficient Condition ◽

Step Size ◽

Truncation Errors ◽

Adams Methods ◽

Variable Step ◽

High Order Convergence ◽

The Stability ◽

Necessary And Sufficient

For backward stochastic differential equations (BSDEs), we construct variable step size Adams methods by means of Itô–Taylor expansion, and these schemes are nonlinear multistep schemes. It is deduced that the conditions of local truncation errors with respect to Y and Z reach high order. The coefficients in the numerical methods are inferred and bounded under appropriate conditions. A necessary and sufficient condition is given to judge the stability of our numerical schemes. Moreover, the high-order convergence of the schemes is rigorously proved. The numerical illustrations are provided.

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An Accurate Block Solver for Stiff Initial Value Problems

ISRN Applied Mathematics ◽

10.1155/2013/567451 ◽

2013 ◽

Vol 2013 ◽

pp. 1-10

Author(s):

H. Musa ◽

M. B. Suleiman ◽

F. Ismail ◽

N. Senu ◽

Z. B. Ibrahim

Keyword(s):

Initial Value Problems ◽

Variable Step Size ◽

Initial Value ◽

Step Size ◽

Differentiation Formula ◽

Stiff Initial Value Problems ◽

Variable Step ◽

Compute Solution ◽

The Stability ◽

Size Mode

New implicit block formulae that compute solution of stiff initial value problems at two points simultaneously are derived and implemented in a variable step size mode. The strategy for changing the step size for optimum performance involves halving, increasing by a multiple of 1.7, or maintaining the current step size. The stability analysis of the methods indicates their suitability for solving stiff problems. Numerical results are given and compared with some existing backward differentiation formula algorithms. The results indicate an improvement in terms of accuracy.

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Adaptive feedforward control for crosswind landing with variable step size

Proceedings of the Institution of Mechanical Engineers Part G Journal of Aerospace Engineering ◽

10.1177/09544100211005429 ◽

2021 ◽

pp. 095441002110054

Author(s):

Qi Bian ◽

Menghan Zhang ◽

Jian Ma

Keyword(s):

Control System ◽

Flight Control ◽

Feedforward Control ◽

Variable Step Size ◽

Test Bed ◽

Step Size ◽

Aircraft Landing ◽

Variable Step ◽

The Stability ◽

Crosswind Landing

In this article, a modified Steiglitz–McBride (SM) structure with variable step size is proposed for aircraft feedforward control to reduce lateral landing deviation in the presence of crosswind disturbance. Currently, most of the aircraft landing studies focus on the trajectory tracking problems in the longitudinal plane and seldom focus on lateral disturbances. This article develops a modified SM structure–based feedforward control system to online estimate both the primary path and the control path (secondary path), where a fuzzy logic–based variable step size strategy is implemented to ensure fast convergence rate and strong robustness under complex crosswind scenarios. As a consequence, the lateral deviation during crosswind landing could be largely reduced as fast as possible while the stability of the flight control system is maintained. A Boeing 747 model is used as a test bed, and the simulations are carried out on two different crosswind conditions to demonstrate the feasibility and effectiveness of the proposed method.

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