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### Article

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

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

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.

### References

 [1] Natesan, S., Vigo-Aguiar, J. & Ramanujam, N. (2003). A numerical algorithm for singular perturbation problems exhibiting weak boundary layers, Comput. Math. Appl. 45, 469-479. [2] Robert, S. M. (1982). A Boundary-Value Technique for Singular Perturbation Problems, Journal of Mathematical Analysis and Applications, 87, 489-508. [3] Bender, C.M. & Orszag, S. A. (1978). Advanced Mathematical Methods for Scientists and Engineers, Mc. Graw-Hill, New York. [4] Kevorkian, J. & Cole, J. D. (1981). Perturbation Methods in Applied Mathematics, Springer-Verlag, New York. [5] O’ Malley, R. E. (1974). Introduction to Singular Perturbations, Academic Press, New York.
 [6] Nayfeh, A. H. (1973). Perturbation Methods, Wiley, New York. [7] Smith, D. R. (1985). Singular-Perturbation Theory an Introduction with Applications, Cambridge University Press, Cambridge. [8] Hu, X.C., Manteuffel, T.A., Mccormick, S. & Russell, T.F. (1995). Accurate discretization for singular perturbations: the one-dimensional case, SIAM Journal off Numerical Analysis, 32 (1), 83-109. [9] Kadalbajoo, M. K. & Reddy, Y. N. (1987). Initial-Value Technique for a Class of Nonlinear Singular Perturbation Problems, Journal of Optimization Theory and Applications, 53, 395-406. [10] Kadalbajoo M.K. & Devendra Kumar (2008). A non –linear single step explicit scheme for non-linear two point singularly perturbed boundary value problems via initial value technique, Applied Mathematics and Computation, 202, 738-746. [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. [12] Van Niekerk, F.D. (1987), Non linear one step methods for initial value problems, Comput. Math. Appl., 13, 367-371. [13] Higinio Ramos (2007). A non standard explicit integration scheme for initial value problems, Applied Mathematics and Computation, 189, 710-718. [14] Zahra, W.K., Exponential spline solutions for a class of two point boundary value problems over a semu infinite range. Numer. Algor. 53, 561-573, 2009. [15] Zahra, W.K., Finite difference technique based on exponential splines for solution of obstacle problems. Int. J. Computer Math. 88 (14), 3046-3060, 2011. [16] Zahra, W.K., A smooth approximation based on exponential spline solutions for non linear fourth order two point boundary value problems, Appl. Math. Comput. 217, 8447-8457, 2011.
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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

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.

### References

 [1] J.H. Ahlberg, E.N. Nilson, J.L. Walsh, The Theory of Splines and Their Applications, Academic Press, New York – London (1967). [2] I. Babuška, Optimal quadrature formulas, Dokladi Akad. Nauk SSSR. 149 (1963) 227-229. (in Russian). [3] P. Blaga, Gh. Coman, Some problems on optimal quadrature, Stud. Univ. Babe-Bolyai Math. 52, no. 4 (2007) 21-44. [4] B. Bojanov, Optimal quadrature formulas, Uspekhi Mat. Nauk. 60, no. 6(366) (2005) 33-52. (in Russian). [5] T. Catina S, Gh. Coman, Optimal quadrature formulas based on the -function method, Stud. Univ. Babe-Bolyai Math. 51, no. 1 (2006) 49-64.
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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

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).

### References

 [1] Ahlberg, J.H., Nilson, E.N., Walsh, J.L.: The theory of splines and their applications, Mathematics in Science and Engineering, New York: Academic Press, (1967). [2] Arcangeli, R., Lopez de Silanes, M.C., Torrens, J.J.: Multidimensional minimizing splines, Kluwer Academic publishers. Boston, (2004). [3] Attea, M.: Hilbertian kernels and spline functions, Studies in Computational Matematics 4, C. Brezinski and L. Wuytack eds, North-Holland, (1992). [4] Berlinet, A., Thomas-Agnan, C.: Reproducing Kernel Hilbert Spaces in Probability and Statistics, Kluwer Academic Publisher, (2004). [5] Bojanov, B.D., Hakopian, H.A., Sahakian, A.A.: Spline functions and multivariate interpolations, Kluwer, Dordrecht, (1993).
 [6] de Boor, C.: Best approximation properties of spline functions of odd degree, J. Math. Mech. 12, (1963), pp.747-749. [7] de Boor, C.: A practical guide to splines, Springer-Verlag, (1978). [8] Duchon, J.: Splines minimizing rotation-invariant semi-norms in Sobolev spaces, (1977), pp. 85-100. [9] Eubank, R.L.: Spline smoothing and nonparametric regression. Marcel-Dekker, New-York, (1988). [10] Freeden, W.: Spherical spline interpolation-basic theory and computational aspects, Journal of Computational and Applied Mathematics, 11, 367-375, (1984). [11] Freeden, W.: Interpolation by multidimensional periodic splines, Journal of Approximation Theory, 55, 104-117 (1988). [12] Green, P.J., Silverman: Nonparametric regression and generalized linear models. A roughness penalty approach. Chapman and Hall, London, (1994). [13] Golomb, M.: Approximation by periodic spline interpolants on uniform meshes, Journal of Approximation Theory, 1, (1968), pp. 26-65. [14] Shadimetov Kh.M., Azamov S.S.: Construction of discrete analogue of the differential operator d4/dx4+d2/dx2+1 and its properties (Russian). Uzbek Math. Zh. 2010, no 1, 181-188. (2010). [15] Shadimetov, Kh.M., Hayotov, A.R., Azamov, S.S.: Optimal quadrature formula in K2(P2)space, Applied Numerical Mathematics, 62, 1893-1909 (2012). [16] Holladay, J.C.: Smoothest curve approximation, Math. Tables Aids Comput. V.11. (1957) 223-243. [17] Ignatev, M.I., Pevniy, A.B.: Natural splines of many variables, Nauka, Leningrad, (in Russian) (1991). [18] Korneichuk, N.P., Babenko, V.F., Ligun, A.A.: Extremal properties of polynomials and splines, Naukovo dumka, Kiev, (in Russian) (1992). [19] Laurent, P.-J.: Approximation and Optimization, Mir, Moscow, (in Russian) (1975). [20] Mastroianni, G., Milovanovic, G.V.: Interpolation Processes – Basic Theory and Applications, Springer Monographs in Mathematics, Springer – Verlag, Berlin – Heidelberg (2008). [21] Nürnberger, G.: Approximation by Spline Functions, Springer, Berlin (1989). [22] Schoenberg, I.J.: On trigonometric spline interpolation, J. Math. Mech. 13, (1964), pp.795-825. [23] Schumaker, L.: Spline functions: basic theory, J. Wiley, New-York, (1981). [24] Sobolev, S.L.: Formulas of Mechanical Cubature in n- Dimensional Space, in: Selected Works of S.L.Sobolev, Springer, (2006), pp. 445-450. [25] Sobolev, S.L.: On Interpolation of Functions of n Variables, in: Selected Works of S.L.Sobolev, Springer, (2006), pp.451-456. [26] Sobolev, S.L.: The coefficients of optimal quadrature formulas, in: Selected Works of S.L.Sobolev. Springer, (2006), pp.561-566. [27] Sobolev, S.L.: Introduction to the Theory of Cubature Formulas, Nauka, Moscow, (in Russian) (1974). [28] Sobolev, S.L., Vaskevich, V.L.: The Theory of Cubature Formulas. Kluwer Academic Publishers Group, Dordrecht (1997). [29] Stechkin, S.B., Subbotin, Yu.N.: Splines in computational mathematics, Nauka, Moscow, (in Russian) (1976). [30] Vasilenko, V.A.: Spline functions: Theory, Algorithms, Programs, Nauka, Novosibirsk, (in Russian) (1983). [31] Vladimirov, V.S.: Generalized functions in mathematical physics. -M.: Nauka, (in Russian) (1979). [32] Wahba, G.: Spline models for observational data. CBMS 59, SIAM, Philadelphia, (1990).
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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

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.

### References

 [1] Abdul-Majid Wazwaz, The variational iteration method for solving linear and nonlinear systems of PDEs, Computers and Mathematics with Applications, 54 (2007), 895-902. [2] B.Batiha,M.S.M. Noorani, I. Hashim, K. Batiha, Numerical simulations of systems of PDEs by variational iteration method, Physics Letters A, 372, (2008), 822-829. [3] C. M. Bender and S. A. Orszag, “Advanced Mathematical Methods for Scientists and Engineers,” McGraw-Hill, New York, 1978. [4] Davod Khojasteh Salkuyeh, Convergence of the variational iteration method for solving linear systems of ODEs with constant coefficients, Computers and Mathematics with Applications, 56 (2008), 2027-2033. [5] J. Jayakumar, Improvement of numerical solution by boundary value technique for singularly perturbed one-dimensional reaction diffusion problem, Applied Mathematics and Computation, 142 (2003), 417-447.
 [6] J. Kevorkian, J.D. Cole, Perturbation Methods in Applied Mathematics, Springer-Verlag, New York (1981). [7] J. He, A New Approach to Nonlinear Partial Differential Equations, Communication Nonlinear Science & Numerical Simulation, Vo1.2, No.4 (Dec. 1997), 230-235. [8] J.H. He, Approximate analytical solution for seepage flow with fractional derivatives in porous media, Comput. Methods Appl. Mech. Eng. 167 (1998) 57-68. [9] J.H. He, Approximate solution of nonlinear differential equations with convolution product nonlinearities, Comput. Methods Appl. Mech. Eng. 167 (1998) 69-73. [10] J.H. He, Variational iteration method-a kind of non-linear analytical technique: some examples, Internat. J. Nonlinear Mech. 34 (1999) 699-708. [11] J.H. He, Variational iteration method for autonomous ordinary differential systems, Appl. Math. Comput. 114 (2000) 115-123. [12] J. He, Homotopy perturbation method: a new nonlinear analytical technique, Applied Mathematics and Computation 135 (2003) 73-79. [13] J.H. He, The homotopy perturbation method for nonlinear oscillators with discontinuities, Appl. Math. Comput. 151 (2004) 287-292. [14] J.H. He, Variational iteration method—Some recent results and new interpretations, J. Comput. Appl. Math. in press. [15] J.I. Ramos, On the variational iteration method and other iterative techniques for nonlinear differential equations, Applied Mathematics and Computation, 199 (2008), 39-69. [16] Laila M.B. Assas, Variational iteration method for solving coupled-KdV equations, Chaos, Solitons and Fractals, 38 (2008), 1225-1228. [17] Liu Jinbo, Tang Jiang, Variational iteration method for solving an inverse parabolic equation, Physics Letters A., 372 (2008), 3569-3572. [18] M. Javidi, Y. Jalilian, Exact solitary wave solution of Boussinesq equation by VIM, Chaos, Solitons and Fractals, 36 (2008), 1256-1260. [19] Mehdi Dehghan, Fatemeh Shakeri, Application of He’s variational iteration method for solving the Cauchy reaction- diffusion problem, Journal of Computational and Applied Mathematics., 214 (2008), 435-446. [20] H.K.Mishra, A comparative study of variational iteration method and He Laplace method, Applied Mathematics, 2012, 3, pp. 1193-1201. [21] H.K. Mishra., Initial value technique for singular perturbation boundary value problems, American Journal of Numerical Analysis, 2 (2014), 49-54. [22] Hradyesh Kumar Mishra, Sonali Saini, Numerical Solution of Singularly Perturbed Two-Point Boundary Value Problem via Liouville-Green Transform, American Journal of Computational Mathematics, 3(2013)1-5. [23] Mustafa Inc, The approximate and exact solutions of the space- and time fractional Burgers equations with initial conditions by variational iteration method, J. Math. Anal. Appl., 345 (2008), 476-484. [24] R.E. O’Malley, Introduction to Singular Perturbations, Academic Press, New York (1974). [25] S. Das, Analytical solution of a fractional diffusion equation by variational iteration method, Computers and Mathematics with Applications, 57 (2009), 483-487. [26] Y.N. Reddy, P. Pramod Chakravarthy, An initial-value approach for solving singularly perturbed two-point boundary value problems, Applied Mathematics and Computation, 155, No. 1 (2004), 95-110.
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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

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.

### References

 [1] Burden, R. L., & Douglas Faires, J. (2001).Numerical analysis. Boston: PWS publishing company. [2] Noor, M. A. (2006 b). New iterative methods for nonlinear equations. Journal of applied mathematical and computation. [3] H.H.H. Homeier, On Newton-type methods with cubic convergence, J. Comput. Appl. Math. 176 (2005) 425-432. [4] M. Frontini, E. Sormani, Third-order methods from quadrature formulae for solving systems of nonlinear equations, Appl. Math. Comput. 149 (2004) 771-782. [5] Chun, C., 2005. Iterative methods improving Newton’s method by the decomposition method, Computers Math. Appl., 50: 1559-1568
 [6] S. Weerakoon, T.G.I. Fernando, A variant of Newton’s method with accelerated third- order convergence, Appl. Math. Lett. 13 (2000) 87-93. [7] F. A. Potra, V. Pta ´k, Nondiscrete induction and iterative processes, Research Notes in Mathematics, vol. 103, Pitman, Boston, 1984. [8] L. F. Shampine, R. C. Allen, S. Pruess, Fundamentals of Numerical Computing, John Wiley and Sons, New York, 1997. [9] F. Freudensten, B. Roth, Numerical solution of systems of nonlinear equations, J. ACM 10 (1963) 550-556. [10] Soheili, A. R., Ahmadian, S. A., & Naghipoor, J. (2008). A Family of Predictor-Corrector Methods Based on Weight Combination of Quadratures for Solving Nonlinear equations. International journal of nonlinear science, 6, 29-33. [11] W. Gautschi, Numerical Analysis: An Introduction, Birkha ¨user, 1997 [12] Rostam K. Saeed and Fuad W. Khthr, New third order iterative method for solving nonlinear equations, Journal of Applied Sciences Research, Vol. 7, No. 6, p. 916-921, 2011. [13] Rostam K. Saeed and Shno O. Ahmed, Modified Iterative methodfor Solving Nonlinear, equation Journal of Kirkuk University-Scientific Studies, Vol. 7, No. 1, p. 146-152, 2012.
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