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Article

A Domain Decomposition Method for Solving Singularly Perturbed Two Point Boundary Value Problems via Exponential Splines

1Department of Mathematics, Prasad V Potluri Siddhartha Institute of Technology, Vijayawada, Andhra Pradesh, INDIA

2Department of Mathematics, National Institute of Technology, Warangal, INDIA


American Journal of Numerical Analysis. 2014, 2(4), 128-135
DOI: 10.12691/ajna-2-4-5
Copyright © 2014 Science and Education Publishing

Cite this paper:
P. Padmaja, Y.N. Reddy. A Domain Decomposition Method for Solving Singularly Perturbed Two Point Boundary Value Problems via Exponential Splines. American Journal of Numerical Analysis. 2014; 2(4):128-135. doi: 10.12691/ajna-2-4-5.

Correspondence to: Y.N.  Reddy, Department of Mathematics, National Institute of Technology, Warangal, INDIA. Email: ynreddy_nitw@yahoo.com

Abstract

In this paper, we presented a domain decomposition method via exponential splines for solving singularly perturbed two-point boundary value problems with the boundary layer at one end (left or right) point. The method is distinguished by the following fact: The original singularly perturbed two-point boundary value problem is divided into two problems, namely inner and outer region problems. The terminal boundary condition is obtained from the solution of the reduced problem. Using stretching transformation, a modified inner region problem is constructed. Then, the inner region problem is solved as two-point boundary value problems by employing exponential splines. Several linear and nonlinear problems are solved to demonstrate the applicability of the method.

Keywords

References

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[11]  Reddy, Y.N. & Pramod Chakravarthy, P. (2003), Method of Reduction of Order for Solving Singularly Perturbed Two-Point Boundary Value Problems, Applied Mathematics and Computation, 136, 27-45.
 
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Article

Optimal Quadrature Formulas with Derivative in the Space L2(m)(0,1)

1Institute of Mathematics, National University of Uzbekistan, Do‘rmon yo‘li str., Tashkent, Uzbekistan


American Journal of Numerical Analysis. 2014, 2(4), 115-127
DOI: 10.12691/ajna-2-4-4
Copyright © 2014 Science and Education Publishing

Cite this paper:
Abdullo R. Hayotov, Farhod A. Nuraliev, Kholmat M. Shadimetov. Optimal Quadrature Formulas with Derivative in the Space L2(m)(0,1). American Journal of Numerical Analysis. 2014; 2(4):115-127. doi: 10.12691/ajna-2-4-4.

Correspondence to: Abdullo  R. Hayotov, Institute of Mathematics, National University of Uzbekistan, Do‘rmon yo‘li str., Tashkent, Uzbekistan. Email: hayotov@mail.ru

Abstract

This paper studies the problem of construction of optimal quadrature formulas in the sense of Sard in the space . In this paper the quadrature sum consists of values of the integrand and its first derivative at nodes. The coefficients of optimal quadrature formulas are found and the norm of the optimal error functional is calculated for arbitrary natural number and for any using S.L. Sobolev method which is based on discrete analogue of the differential operator. In particular, for m=2,3 optimality of the classical Euler-Maclaurin quadrature formula is obtained. Starting from m=4 new optimal quadrature formulas are obtained.

Keywords

References

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Article

Interpolation Splines Minimizing Semi-Norm in K2(P2) Space

1Institute of Mathematics, National University of Uzbekistan, Tashkent, Uzbekistan

2Tashkent Institute of Railway Engineers, Tashkent, Uzbekistan


American Journal of Numerical Analysis. 2014, 2(4), 107-114
DOI: 10.12691/ajna-2-4-3
Copyright © 2014 Science and Education Publishing

Cite this paper:
Kholmat M. Shadimetov, Abdullo R. Hayotov, Azamov S. Siroj. Interpolation Splines Minimizing Semi-Norm in K2(P2) Space. American Journal of Numerical Analysis. 2014; 2(4):107-114. doi: 10.12691/ajna-2-4-3.

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

Abstract

In the present paper using S.L. Sobolev’s method interpolation splines minimizing the semi-norm in K2(P2) space are constructed. Explicit formulas for coefficients of interpolation splines are obtained. The obtained interpolation spline is exact for the functions and . Also we give some numerical results where we showed connection between optimal quadrature formula and obtained interpolation spline in the space K2(P2).

Keywords

References

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Article

Variational Iteration Method for a Singular Perturbation Boundary Value Problems

1Department of Mathematics, Jaypee University of Engineering & Technology, Guna (M.P), India


American Journal of Numerical Analysis. 2014, 2(4), 102-106
DOI: 10.12691/ajna-2-4-2
Copyright © 2014 Science and Education Publishing

Cite this paper:
Hradyesh Kumar Mishra, Sonali saini. Variational Iteration Method for a Singular Perturbation Boundary Value Problems. American Journal of Numerical Analysis. 2014; 2(4):102-106. doi: 10.12691/ajna-2-4-2.

Correspondence to: Sonali  saini, Department of Mathematics, Jaypee University of Engineering & Technology, Guna (M.P), India. Email: hk.mishra@juet.ac.in, sonali.saini1386@gmail.com

Abstract

In this paper, the author used He’s variational iteration method for solving singularly perturbed two-point boundary value problems. Few examples are solved to demonstrate the applicability of the method. It is observed that a good choice of the freely selected initial approximation in the variational iteration method leads to closed form solutions by using only one or two iterations. It is also observed that the variational iteration method can be easily applied to the initial and boundary value problems. Graphs are also plotted for the numerical examples.

Keywords

References

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Article

A Modification of Newton Method with Third-Order Convergence

1Faculty of Mathematics and Physics Engineering Polytechnic University of Tirana, Albania


American Journal of Numerical Analysis. 2014, 2(4), 98-101
DOI: 10.12691/ajna-2-4-1
Copyright © 2014 Science and Education Publishing

Cite this paper:
Gentian Zavalani. A Modification of Newton Method with Third-Order Convergence. American Journal of Numerical Analysis. 2014; 2(4):98-101. doi: 10.12691/ajna-2-4-1.

Correspondence to: Gentian  Zavalani, Faculty of Mathematics and Physics Engineering Polytechnic University of Tirana, Albania. Email: zavalanigentian@hotmail.com

Abstract

In this paper, we present a new modification of Newton method for solving non-linear equations. Analysis of convergence shows that the new method is cubically convergent. Per iteration the new method requires two evaluations of the function and one evaluation of its first derivative. Thus, the new method is preferable if the computational costs of the first derivative are equal or more than those of the function itself. Finally, we give some numerical examples to demonstrate our method is more efficient than other classical iterative methods.

Keywords

References

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