scholarly journals A Two-Step Modified Explicit Hybrid Method with Step-Size-Dependent Parameters for Oscillatory Problems

2020 ◽  
Vol 2020 ◽  
pp. 1-7
Author(s):  
Faieza Samat ◽  
Eddie Shahril Ismail
Keyword(s):  
Differential Equations ◽  
Hybrid Method ◽  
Absolute Stability ◽  
Numerical Results ◽  
Sixth Order ◽  
New Method ◽  
Step Size ◽  
Linear Combinations ◽  
Size Dependent ◽  
Modified Formula

A new two-step modified explicit hybrid method with parameters depending on the step-size is constructed. This method is derived using the coefficients from a sixth-order explicit hybrid method with extended interval of absolute stability and then imposed each stage of the modified formula to exactly integrate the differential equations with solutions that can be expressed as linear combinations of sinwx and coswx, where w is the known frequency. Numerical results show the advantage of the new method for solving oscillatory problems.

scholarly journals A NEW SUPERCLASS OF BLOCK BACKWARD DIFFERENTIATION FORMULA FOR STIFF ORDINARY DIFFERENTIAL EQUATIONS

2014 ◽  
Vol 07 (01) ◽  
pp. 1350034 ◽  
Author(s):  
M. B. Suleiman ◽  
H. Musa ◽  
F. Ismail ◽  
N. Senu ◽  
Z. B. Ibrahim
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Numerical Results ◽  
New Method ◽  
Differentiation Formula ◽  
Stiff Ordinary Differential Equations ◽  
Backward Differentiation Formula ◽  
Error Constant

A superclass of block backward differentiation formula (BBDF) suitable for solving stiff ordinary differential equations is developed. The method is of order 3, with smaller error constant than the conventional BBDF. It is A-stable and generates two points at each step of the integration. A comparison is made between the new method, the 2-point block backward differentiation formula (2BBDF) and 1-point backward differentiation formula (1BDF). The numerical results show that the method developed outperformed the 2BBDF and 1BDF methods in terms of accuracy. It also reduces the integration steps when compared with the 1BDF method.

A Multiparameter Perturbation Approach for the Vibration of Plates Subject to Arbitrary, Independent, In-Plane Loads

1974 ◽  
Vol 41 (4) ◽  
pp. 1081-1086
Author(s):  
S. F. Bassily ◽  
S. M. Dickinson
Keyword(s):  
Power Series ◽  
Differential Equations ◽  
Numerical Results ◽  
Perturbation Approach ◽  
Rectangular Plates ◽  
Linear Combinations

A multiparameter perturbation approach for the solution of a class of differential equations is developed and is applied to the study of the vibration of rectangular plates under arbitrary in-plane loading fields describable by linear combinations of two or more independent loading parameters. The solution takes the form of a power series involving all the independent loading parameters and is valid over whole ranges of those parameters. Numerical results for fully clamped plates subject to uniform, combined, direct, and shear in-plane loads are presented in order to demonstrate the utility of the approach.

scholarly journals A New Two Derivative FSAL Runge-Kutta Method of Order Five in Four Stages

2020 ◽  
Vol 17 (1) ◽  
pp. 0166
Author(s):  
Hussain Et al.
Keyword(s):  
Differential Equations ◽  
Numerical Solution ◽  
Ordinary Differential Equations ◽  
Numerical Results ◽  
New Method ◽  
Kutta Method ◽  
First Order ◽  
Runge Kutta Method ◽  
Runge Kutta ◽  
The Stability

A new efficient Two Derivative Runge-Kutta method (TDRK) of order five is developed for the numerical solution of the special first order ordinary differential equations (ODEs). The new method is derived using the property of First Same As Last (FSAL). We analyzed the stability of our method. The numerical results are presented to illustrate the efficiency of the new method in comparison with some well-known RK methods.

scholarly journals A Four-Stage Fifth-Order Trigonometrically Fitted Semi-Implicit Hybrid Method for Solving Second-Order Delay Differential Equations

2016 ◽  
Vol 2016 ◽  
pp. 1-7
Author(s):  
Sufia Zulfa Ahmad ◽  
Fudziah Ismail ◽  
Norazak Senu
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Hybrid Method ◽  
Delay Differential Equations ◽  
Scientific Literature ◽  
Second Order ◽  
New Method ◽  
Implicit Methods ◽  
Delay Differential ◽  
Fifth Order

We derived a two-step, four-stage, and fifth-order semi-implicit hybrid method which can be used for solving special second-order ordinary differential equations. The method is then trigonometrically fitted so that it is suitable for solving problems which are oscillatory in nature. The methods are then used for solving oscillatory delay differential equations. Numerical results clearly show the efficiency of the new method when compared to the existing explicit and implicit methods in the scientific literature.

scholarly journals Solving Second-Order Delay Differential Equations by Direct Adams-Moulton Method

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Hoo Yann Seong ◽  
Zanariah Abdul Majid ◽  
Fudziah Ismail
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Delay Differential Equations ◽  
Direct Method ◽  
Second Order ◽  
Numerical Results ◽  
Step Size ◽  
First Order ◽  
Delay Differential ◽  
Fifth Order

This paper will consider the implementation of fifth-order direct method in the form of Adams-Moulton method for solving directly second-order delay differential equations (DDEs). The proposed direct method approximates the solutions using constant step size. The delay differential equations will be treated in their original forms without being reduced to systems of first-order ordinary differential equations (ODEs). Numerical results are presented to show that the proposed direct method is suitable for solving second-order delay differential equations.

scholarly journals Block Milne’s Implementation For Solving Fourth Order Ordinary Differential Equations

2018 ◽  
Vol 8 (3) ◽  
pp. 2943-2948
Author(s):  
J. G. Oghonyon ◽  
S. A. Okunuga ◽  
K. S. Eke ◽  
O. A. Odetunmibi
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Fourth Order ◽  
Interpolation Polynomial ◽  
Numerical Results ◽  
Convergence Criteria ◽  
Step Size ◽  
Stiff Ordinary Differential Equations ◽  
Grid Points ◽  
Predictor Corrector

Block predictor-corrector method for solving non-stiff ordinary differential equations (ODEs) started with Milne’s device. Milne’s device is an extension of the block predictor-corrector method providing further benefits and better results. This study considers Milne’s devise for solving fourth order ODEs. A combination of Newton’s backward difference interpolation polynomial and numerical integration method are applied and integrated at some selected grid points to formulate the block predictor-corrector method. Moreover, Milne’s devise advances the computational efficiency by applying the principal local truncation error (PLTE) of the block predictor-corrector method after establishing the order. The numerical results were exhibited to attest the functioning of Milne’s devise in solving fourth order ODEs. The complete results were obtained with the aid of Mathematica 9 kernel for Microsoft Windows. Numerical results showcase that Milne’s device is more effective than existent methods in terms of design new step size, determining the convergence criteria and maximizing errors at all examined convergence levels.

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

2011 ◽  
Vol 2011 ◽  
pp. 1-17 ◽  
Author(s):  
F. Samat ◽  
F. Ismail ◽  
M. B. Suleiman
Keyword(s):  
Hybrid Method ◽  
Numerical Experiments ◽  
Variable Step Size ◽  
Step Size ◽  
Linear Combinations ◽  
Variable Step ◽  
Three Stages ◽  
Last Stage ◽  
Exponentially Fitted ◽  
The Stability

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.

A PHASE-FITTED AND AMPLIFICATION-FITTED EXPLICIT TWO-STEP HYBRID METHOD FOR SECOND-ORDER PERIODIC INITIAL VALUE PROBLEMS

2006 ◽  
Vol 17 (05) ◽  
pp. 663-675 ◽  
Author(s):  
HANS VAN DE VYVER
Keyword(s):  
Hybrid Method ◽  
Initial Value Problems ◽  
Numerical Results ◽  
New Method ◽  
Test Problems ◽  
Phase Lag ◽  
Initial Value ◽  
Periodic Initial Value Problems ◽  
Free Parameters ◽  
Fifth Order

In this paper a phase-fitted and amplification-fitted explicit two-step hybrid method is introduced. The construction is based on a modification of a fifth-order dissipative method recently developed by Franco.19 Two free parameters are added in order to nullify the phase-lag and the amplification. Numerical results obtained for well-known test problems show the efficiency of the new method when it is compared with other existing codes.

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

Symmetry ◽  
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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