• Journal of applied mathematics & informatics
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• v.26 no.3_4
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• pp.689-706
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• 2008

In this paper, a fifth order numerical method is presented for solving singularly perturbed two point boundary value problems with a boundary layer at one end point. The two point boundary value problem is transformed into general first order ordinary differential equation system. A discrete approximation of a fifth order compact difference scheme is presented for the first order system. An asymptotically equivalent first order equation of the original singularly perturbed two point boundary value problem is obtained from the theory of singular perturbations. It is used in the fifth order compact difference scheme to get a two term recurrence relation and is solved. Several linear and non-linear singular perturbation problems have been solved and the numerical results are presented to support the theory. It is observed that the present method approximates the exact solution very well.

• Bulletin of the Korean Chemical Society
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• v.24 no.8
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• pp.1102-1106
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• 2003

In our previous work we developed the Finite Field method in order to calculate the fifth-order Raman response. The method was applied to calculate various polarization components of the two-dimensional response of liquid $CS_2$. So far, all calculations relied on the dipole-induced dipole. Accurate time-dependent density functional theory calculations have shown that this model has big discrepancies, when molecules are close together as in the liquid. We now report results of investigations on the importance of multipole and electron overlap effects on the polarizability and the fifth-order Raman response. It is shown that these collision effects, especially the induced multipoles, are crucial in the description of the fifth-order response. The impact is found to be especially pronounced for the χ_{mmzzzz}^{(5)}$response that is solely due to interaction induced effects. The calculated response will be compared with various experimental results.

• Journal of applied mathematics & informatics
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• v.27 no.1_2
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• pp.441-452
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• 2009

In this paper, a fifth order numerical method is presented for solving singularly perturbed differential-difference equations with negative shift. In recent papers the term negative shift has been using for delay. Similar boundary value problems are associated with expected first exit time problem of the membrane, potential in models for neuron and in variational problems in control theory. In the numerical treatment for such type of boundary value problems, first we use Taylor approximation to tackle terms containing small shifts which converts it to a boundary value problem for singularly perturbed differential equation. The two point boundary value problem is transformed into general first order ordinary differential equation system. A discrete approximation of a fifth order compact difference scheme is presented for the first order system and is solved using the boundary conditions. Several numerical examples are solved and compared with exact solution. It is observed that present method approximates the exact solution very well.

• Journal of applied mathematics & informatics
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• v.33 no.1_2
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• pp.61-76
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• 2015

In this paper, a fifth order extension of Potra's third order iterative method is proposed for solving nonlinear equations. A convergence theorem along with the error bounds is established. The method takes three functions and one derivative evaluations giving its efficiency index equals to 1.495. Some numerical examples are also solved and the results obtained are compared with some other existing fifth order methods. Next, the interval extension of both third and fifth order Potra's method are developed by using the concepts of interval analysis. Convergence analysis of these methods are discussed to establish their third and fifth orders respectively. A number of numerical examples are worked out using INTLAB in order to demonstrate the efficacy of the methods. The results of the proposed methods are compared with the results of the interval Newton method.

• Journal of the Chungcheong Mathematical Society
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• v.24 no.3
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• pp.437-447
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• 2011

In this paper we present a new variant of the Euler-Chebyshev method for solving nonlinear equations. Analysis of convergence is given to show that the presented methods are at least fifth-order convergent. Several numerical examples are given to illustrate that newly presented methods can be competitive to other known fifth-order methods and the Newton method in the efficiency and performance.

• Journal of applied mathematics & informatics
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• v.31 no.1_2
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• pp.229-239
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• 2013

This paper studies the generalized fifth-order KdV equation with variable coefficients using Lie symmetry methods.Lie group classification with respect to the time dependent coefficients is performed. Then we get the similarity reductions using the symmetry and give some exact solutions.

• Journal of applied mathematics & informatics
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• v.35 no.1_2
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• pp.191-204
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• 2017

A new fifth-order weighted Runge-Kutta algorithm based on heronian mean for solving initial value problem in ordinary differential equations is considered in this paper. Comparisons in terms of numerical accuracy and size of the stability region between new proposed Runge-Kutta(5,5) algorithm, Runge-Kutta (5,5) based on Harmonic Mean, Runge-Kutta(5,5) based on Contra Harmonic Mean and Runge-Kutta(5,5) based on Geometric Mean are carried out as well. The problems, methods and comparison criteria are specified very carefully. Numerical experiments show that the new algorithm performs better than other three methods in solving variety of initial value problems. The error analysis is discussed and stability polynomials and regions have also been presented.

• Journal of Korean Society of Coastal and Ocean Engineers
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• v.4 no.4
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• pp.187-200
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• 1992

A general scheme is developed to determine the Langrangian motions of water particles by the Eulerian velocity at their mean positions by using Taylor's theorem. Utilizing the Stokes finite-amplitude wave theory, the orbital motions and the mass transport velocity including the effects of higher-order wave components are determined. The fifth-order approximation of orbital motion gives very good predictions of actual water particle motion in Stokes fifth-order wave theory except near the free-surface. The fifth-order theory predicts the mass transport velocity less than that given by the existing second-order theory over the whole water depth.

• International journal of advanced smart convergence
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• v.2 no.1
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• pp.18-20
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• 2013

The significance, is to introduce a novel way to employ the improved Runge-Kutta fifth order five stage method, here after called as Modified IRK(5,5) method, for system of second order robot arm problem and variations in angles at the joints in which parameters governing with two degrees of freedom which requires lesser number of function evaluations per time step as compared to the existing ones, in order to save time and spaceAn ultimate aim of this present paper is to solve application problem such as robot arm and initial value problems by applying Runge-Kutta fifth order five stage numerical techniques. The calculated output for robot arm coincides with exact solution which is found to be better, suitable and feasible for solving real time problems.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.27 no.4
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• pp.207-231
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• 2023

This paper aims to improve the alternative formulation of the fifth- and sixth-order accurate weighted essentially non-oscillatory (AWENO) finite difference schemes. The first is to derive the AWENO scheme with sixth-order accuracy in the smooth region of the solution. Second, a new weighted polynomial functions combining the perturbed forms with conserved variable to the AWENO is constructed; the new form of tunable functions are invented to maintain non-oscillatory property. Detailed numerical experiments are presented to illustrate the behavior of the new perturbational AWENO schemes. The performance of the present scheme is evaluated in terms of accuracy and resolution of discontinuities using a variety of one and two-dimensional test cases. We show that the resulted perturbational AWENO schemes can achieve fifth- and sixth-order accuracy in smooth regions while reducing numerical dissipation significantly near singularities.