ISSN (Print): 2372-2118

ISSN (Online): 2372-2126

Editor-in-Chief: Emanuele Galligani

Website: http://www.sciepub.com/journal/AJNA

   

Article

Numerical Simulation of Time-fractional Fourth Order Differential Equations via Homotopy Analysis Fractional Sumudu Transform Method

1Department of Mathematics, Jaypee University of Engineering and Technology, Guna, INDIA


American Journal of Numerical Analysis. 2015, 3(3), 52-64
doi: 10.12691/ajna-3-3-1
Copyright © 2015 Science and Education Publishing

Cite this paper:
Rishi Kumar Pandey, Hradyesh Kumar Mishra. Numerical Simulation of Time-fractional Fourth Order Differential Equations via Homotopy Analysis Fractional Sumudu Transform Method. American Journal of Numerical Analysis. 2015; 3(3):52-64. doi: 10.12691/ajna-3-3-1.

Correspondence to: Hradyesh  Kumar Mishra, Department of Mathematics, Jaypee University of Engineering and Technology, Guna, INDIA. Email: hk.mishra@juet.ac.in

Abstract

The work provides an incipient analytical technique called the Homotopy Analysis Fractional Sumudu Transform Method (HAFSTM) for solving time-fractional fourth order differential equations with variable coefficients. The HAFSTM is the cumulation of the homotopy analysis method (HAM) and sumudu transform method (STM). The numerical simulation of the proposed method has the sundry applications, it can solve linear and nonlinear boundary value quandaries without utilizing Adomian polynomial, and He’s polynomial, which can be considered a clear advantage of this incipient algorithm. The solutions obtained by proposing technique are very lucid and less computationally implementable.

Keywords

References

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Article

Unification of Well-known Numeric Methods for Solving Nonlinear Equations

1Sharif University of Thechnology, Iran


American Journal of Numerical Analysis. 2015, 3(3), 65-76
doi: 10.12691/ajna-3-3-2
Copyright © 2015 Science and Education Publishing

Cite this paper:
Amin Najafi Amin. Unification of Well-known Numeric Methods for Solving Nonlinear Equations. American Journal of Numerical Analysis. 2015; 3(3):65-76. doi: 10.12691/ajna-3-3-2.

Correspondence to: Amin  Najafi Amin, Sharif University of Thechnology, Iran. Email: Amin_NajafiAmin@yahoo.com

Abstract

This article is a fairly comprehensive document on the numerical solution of nonlinear equations. The aim of this paper is to unify all numerical methods for solving nonlinear equations and complete the Najafi-Nikkhah method [1,2] and generalize the famous methods for solving systems of nonlinear equations. So, the available methods in this field, are being investigated and it will be indicated that how these techniques, despite the apparent dispersion, all are obtained from a unified idea, and this unified pattern would help find new techniques in a systematic way. All current methods require that the initial starting point or points to be close to the solution appropriately, but for the equations with complicated appearance finding the initial guess would not be easy. So, this article intends to provide an appropriate response for this fundamental issue for the first time. An algorithm is proposed to complete the Najafi-Nikkhah technique [1,2], and declares the procedure to make the initial guess become closer to the solution, even if they are far away from each other. Then, the procedure could be completed by one of the common methods available in this field. Dispersion of the current methods causes the confusion in the case of using them. Therefore, these methods should be compared with some criteria to determine the use of them in the practical applications. In the next section of this article, these criteria together with some comparisons including a series of tables and diagram would be provided. Finally, in the last section, generalization of the current methods for solving the nonlinear equation with a singular unknown to the methods for solving the systems of nonlinear equations would be investigated. Then, the generalization of the modified Najafi-Nikkhah method for the systems of nonlinear equations will be presented which can reduce the dependency of the initial guess to the solution, significantly.

Keywords

References

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Article

Construction of Optimal Quadrature Formula for Numerical Calculation of Fourier Coefficients in Sobolev Space L2(1)

1Department of Comptational Methods, Institute of Mathematics, National University of Uzbekistan, Tashkent, Uzbekistan


American Journal of Numerical Analysis. 2016, 4(1), 1-7
doi: 10.12691/ajna-4-1-1
Copyright © 2016 Science and Education Publishing

Cite this paper:
Nurali D. Boltaev, Abdullo R. Hayotov, Kholmat M. Shadimetov. Construction of Optimal Quadrature Formula for Numerical Calculation of Fourier Coefficients in Sobolev Space L2(1). American Journal of Numerical Analysis. 2016; 4(1):1-7. doi: 10.12691/ajna-4-1-1.

Correspondence to: Abdullo  R. Hayotov, Department of Comptational Methods, Institute of Mathematics, National University of Uzbekistan, Tashkent, Uzbekistan. Email: hayotov@mail.ru

Abstract

In the present paper the optimal quadrature formula for approximate evaluation of Fourier coefficients is constructed for functions of the space . At the same time the explicit formulas for optimal coefficients, which are very useful in applications, are obtained. The obtained formula is exact for constant. In particular, as consequences of the main result the new optimal quadrature formulas for approximate evaluation of integrals and are obtained. Furthermore, the order of convergence of the constructed optimal quadrature formula is studied.

Keywords

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