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American Journal of Numerical Analysis

## Article

# Optimal Quadrature Formulas with Derivative in the Space L_{2}^{(m)}(0,1)

^{1}Institute 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 L

_{2}

^{(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

## Keywords

## 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. | ||

[6] | M.A. Chakhkiev, Linear differential operators with real spectrum, and optimal quadrature formulas, Izv. Akad. Nauk SSSR Ser. Mat. 48, no. 5 (1984) 1078-1108. (in Russian). | ||

[7] | Gh. Coman, Quadrature formulas of Sard type (Romanian), Studia Univ. Babes-Bolyai Ser. Math.-Mech. 17, no. 2 (1972) 73-77. | ||

[8] | Gh. Coman, Monosplines and optimal quadrature formulae in L_{p}, Rend. Mat. (6) 5 (1972) 567-577. | ||

[9] | A.O. Gelfond, Calculus of Finite Differences, Nauka, Moscow, 1967. (in Russian). | ||

[10] | A. Ghizzetti, A. Ossicini, Quadrature Formulae, Akademie Verlag, Berlin, 1970. | ||

[11] | R.W. Hamming, Numerical methods for Scientists and Engineers, McGraw Bill Book Company, Inc., USA, 1962. | ||

[12] | A.R. Hayotov, G.V. Milovanović, Kh.M. Shadimetov, On an optimal quadrature formula in the sense of Sard. Numerical Algorithms, v.57, no. 4, (2011) 487-510. | ||

[13] | P. Köhler, On the weights of Sard’s quadrature formulas, Calcolo, 25 (1988) 169-186. | ||

[14] | F. Lanzara, On optimal quadrature formulae, J. Ineq. Appl. 5 (2000) 201-225. | ||

[15] | A.A. Maljukov, I.I. Orlov, Construction of coefficients of the best quadrature formula for the class W_{L2}^{(2)}(M;ON) with equally spaced nodes, Optimization methods and operations research, applied mathematics, pp. 174-177, 191. Akad. Nauk SSSR Sibirsk. Otdel. Sibirsk. Ènerget. Inst., Irkutsk (1976). (in Russian). | ||

[16] | L.F. Meyers, A. Sard, Best approximate integration formulas, J. Math. Physics, 29 (1950) 118-123. | ||

[17] | S.M. Nikol’skii, To question about estimation of approximation by quadrature formulas, Uspekhi Matem. Nauk, 5:2 (36) (1950) 165-177. (in Russian). | ||

[18] | S.M. Nikol’skii, Quadrature Formulas, Nauka, Moscow, 1988. (in Russian). | ||

[19] | A. Sard, Best approximate integration formulas; best approximation formulas, Amer. J. Math. 71 (1949) 80-91. | ||

[20] | A. Sard, Linear approximation, AMS, 1963. | ||

[21] | I.J. Schoenberg, On monosplines of least deviation and best quadrature formulae, J. Soc. Indust. Appl. Math. Ser. B Numer. Anal. 2 (1965) 144-170. | ||

[22] | I.J. Schoenberg, On monosplines of least square deviation and best quadrature formulae II. SIAM J. Numer. Anal. v.3, no. 2 (1966) 321-328. | ||

[23] | I.J. Schoenberg, S.D. Silliman, On semicardinal quadrature formulae. Math. Comp. v.126 (1974) 483-497. | ||

[24] | Kh.M. Shadimetov, Optimal formulas of approximate integration for differentiable functions, Candidate dissertation, Novosibirsk, 1983, p. 140. arXiv:1005.0163v1 [NA.math]. | ||

[25] | Kh.M. Shadimetov, Optimal quadrature formulas in L_{2}^{m}(Ω) and L_{2}^{m}(R^{1}), Dokl. Akad. Nauk UzSSR. no.3 (1983) 5-8. (in Russian). | ||

[26] | Kh.M. Shadimetov. The discrete analogue of the differential operator d^{2m}/dx^{2m} and its construction, Questions of Computations and Applied Mathematics. Tashkent, (1985) 22-35. ArXiv: 1001.0556.v1 [math.NA] Jan. 2010. | ||

[27] | Kh.M. Shadimetov, Optimal Lattice Quadrature and Cubature Formulas, Doklady Mathematics, v.63, no. 1 (2001) 92-94. | ||

[28] | Kh.M. Shadimetov, Construction of weight optimal quadrature formulas in the space L_{2}^{(m)} (0,N), Siberian J. Comput. Math. 5, no. 3, 275-293 (2002). (in Russian). | ||

[29] | Kh.M. Shadimetov, A.R. Hayotov, Optimal quadrature formulas with positive coefficients in L_{2}^{(m)} (0,1) space, J. Comput. Appl. Math. 235, 1114-1128 (2011). | ||

[30] | Shadimetov, Kh.M., Hayotov, A.R.: Optimal quadrature formulas in the sense of Sard in W_{2}^{(m,m-1)} space. Calcolo. | ||

[31] | Shadimetov, Kh.M., Hayotov, A.R., Azamov, S.S.: Optimal quadrature formula in K_{2}(P_{2}). Applied Numerical Mathematics. 62, no.12, 1893-1909 (2012). | ||

[32] | Shadimetov, Kh.M., Hayotov, A.R., Nuraliev, F.A.: On an optimal quadrature formula in Sobolev space L_{2}^{(m)} (0,1). J. Comput. Appl. Math. 243, 91-112 (2013). | ||

[33] | S.L. Sobolev, Introduction to the Theory of Cubature Formulas, Nauka, Moscow, 1974. (in Russian). | ||

[34] | S.L. Sobolev, The coefficients of optimal quadrature formulas, Selected Works of S.L. Sobolev, Springer, (2006) 561-566. | ||

[35] | S.L. Sobolev, V.L. Vaskevich, The Theory of Cubature Formulas, Kluwer Academic Publishers Group, Dordrecht, 1997. | ||

[36] | F.Ya. Zagirova, On construction of optimal quadrature formulas with equal spaced nodes. Novosibirsk (1982), 28 p. (Preprint No. 25, Institute of Mathematics SD of AS of USSR). (in Russian). | ||

[37] | Z.Zh. Zhamalov, Kh.M. Shadimetov, About optimal quadrature formulas (Russian), Dokl. Akademii Nauk UzSSR, 7 (1980) 3-5. (in Russian). | ||

[38] | A.A. Zhensikbaev, Monosplines of minimal norm and the best quadrature formulas (Russian), Uspekhi Matem. Nauk, 36 (1981) 107-159. (in Russian). | ||

## Article

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

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

^{2}Department 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

## Keywords

## 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. | ||

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[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. | ||

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[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. | ||

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## Article

# Fitted Second Order Scheme for Singularly Perturbed Differential-difference Equations

^{1}Department of Mathematics, National Institute of Technology, WARANGAL, India

*American Journal of Numerical Analysis*.

**2014**, 2(5), 136-143

**DOI:**10.12691/ajna-2-5-1

**Copyright © 2014 Science and Education Publishing**

**Cite this paper:**

Lakshmi Sirisha, Y.N. Reddy. Fitted Second Order Scheme for Singularly Perturbed Differential-difference Equations.

*American Journal of Numerical Analysis*. 2014; 2(5):136-143. doi: 10.12691/ajna-2-5-1.

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

## Abstract

## Keywords

## References

[1] | R. Bellman, and K. L. Cooke, Differential-Difference Equations. Academic Press, New York, 1963. | ||

[2] | M.W. Derstine, F.A.H.H.M. Gibbs, D.L. Kaplan, Bifurcation gap in a hybrid optical system, Phys. Rev. A, 26 1982, 3720-3722. | ||

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[12] | M. K. Kadalbajoo and K. K. Sharma, -Uniform fitted mesh method for Singularly Perturbed Differential-Difference Equations: Mixed type of shifts with layer behaviour, International Journal of Computation Mathematics, 81 2004, 49-62. | ||

[13] | M. K. Kadalbajoo and K. K. Sharma, Numerical Treatment of mathematical model arising from a model of neuronal variability, Journal of Math. Anal. Appl., 307 2005 606-627. | ||

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[20] | R. K. Mohanty and N. Jha, A class of variable mesh spline in compression methods for singularly perturbed two point singular boundary value problems, Applied Mathematics and Computation, 168, 2005, 704-716. | ||

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## Article

# Optimal Quadrature Formulas with Polynomial Weight in Sobolev Space

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

*American Journal of Numerical Analysis*.

**2014**, 2(5), 144-151

**DOI:**10.12691/ajna-2-5-2

**Copyright © 2014 Science and Education Publishing**

**Cite this paper:**

Kholmat M. Shadimetov, Abdullo R. Hayotov, Sardor I. Ismoilov. Optimal Quadrature Formulas with Polynomial Weight in Sobolev Space.

*American Journal of Numerical Analysis*. 2014; 2(5):144-151. doi: 10.12691/ajna-2-5-2.

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

## Abstract

_{2}

^{(m)}(0,1). Using S.L. Sobolev’s method we obtain new optimal quadrature formula of such type and give explicit expressions for the corresponding optimal coefficients. Also, we include a few numerical examples in order to illustrate the application of the obtained optimal quadrature formula.

## Keywords

## References

[1] | Babuška, I.: Optimal quadrature formulas (Russian). Dokl. Akad. Nauk SSSR 149, 227-229 (1963). | ||

[2] | Blaga, P., Coman, Gh.: Some problems on optimal quadrature. Stud. Univ. Babeş-Bolyai Math. 52(4), 21-44 (2007). | ||

[3] | A.K.Boltaev, A.R.Hayotov, Kh.M.Shadimetov. About coefficients and order of convergence of the optimal quadrature formula. American Journal of Numerical Analysis, Science and Education Publishing, Vol. 2, no2, 35-48, (2014). | ||

[4] | Catinaş, T., Coman, Gh.: Optimal quadrature formulas based on the φ-function method. Stud. Univ. Babeş-Bolyai Math. 51(1), 49-64 (2006). | ||

[5] | Coman, Gh.: Quadrature formulas of Sard type. Studia Univ. Babeё s-Bolyai Ser. Math.-Mech. 17(2), 73-77 (1972) (in Romanian). | ||

[6] | Coman, Gh.: Monosplines and optimal quadrature formulae in Lp. Rend. Mat. (6) 5, 567-577 (1972). | ||

[7] | Ghizzetti, A., Ossicini, A.: Quadrature Formulae. Akademie Verlag, Berlin (1970). | ||

[8] | A.R.Hayotov, G.V.Milovanovic, Kh.M.Shadimetov. On one optimal quadrature formula in the sense of Sard, Numerical Algorithms, 57, 487-510 (2011). | ||

[9] | A.R.Hayotov, G.V.Milovanovic, Kh.M.Shadimetov, Optimal quadrature formula in the sense of Sard in K_{2}(P_{3}) space, Publications De L’Institute Mathematique, 95 (109), 29-47 (2014) | ||

[10] | A.R.Hayotov, F.A.Nuraliev, Kh.M.Shadimetov. Optimal quadrature formulas with derivative in the space L_{2}^{(m)}(0,1). American Journal of Numerical Analysis, Science and Education Publishing, Vol. 2, no 4, 115-127, (2014) | ||

[11] | Ismoilov S.I. Coefficients of the weight quadrature formulas in the space L_{2}^{(2)}(0,1). Uzbek Mathematical Journal, Tashkent, 2013, no. 2, pp. 30-35. | ||

[12] | Ismoilov S.I. Optimal quadrature formulas with polinomial weight. Uzbek Mathematical Journal, Tashkent, 2014, no.1, pp. 19-29. | ||

[13] | Kӧhler, P.: On the weights of Sard’s quadrature formulas. Calcolo 25, 169-186 (1988). | ||

[14] | Lanzara, F.: On optimal quadrature formulae. J. Inequal. Appl. 5, 201-225 (2000). | ||

[15] | Maljukov, A.A., Orlov, I.I.: Construction of coefficients of the best quadrature formula for the class W_{L1}^{(2)}(M;ON) with equally spaced nodes. In: Optimization Methods and Operations Research, Applied Mathematics, pp. 174-177, 191. Akad. Nauk SSSR Sibirsk. Otdel. Sibirsk. Ѐnerget. Inst., Irkutsk (1976) (in Russian). | ||

[16] | Meyers, L.F., Sard, A.: Best approximate integration formulas. J. Math. Phys. 29, 118-123 (1950). | ||

[17] | Sard, A.: Best approximate integration formulas; best approximation formulas. Am. J. Math. 71, 80-91 (1949). | ||

[18] | Shadimetov, Kh.M.: Optimal formulas of approximate integration for differentiable functions, Candidate dissertation, Novosibirsk, 1983, p. 140. arXiv:1005.0163v1 [NA.math]. | ||

[19] | Shadimetov, Kh.M.: Optimal quadrature formulas in L_{2}^{m}(Ω) and L_{2}^{m}(R^{1}). Dokl. Akad. Nauk UzSSR 1983(3), 5-8 (1983) (in Russian) | ||

[20] | Shadimetov, Kh.M.: The discrete analogue of the differential operator d^{2m}/dx^{2m} and its construction, Questions of Computations and Applied Mathematics. Tashkent, (1985) 22-35. ArXiv:1001.0556.v1 [math.NA] Jan. 2010. | ||

[21] | Shadimetov, Kh.M.: Construction of weight optimal quadrature formulas in the space L_{2}^{(m)}(0,N). Sib. J. Comput. Math. 5(3), 275–293 (2002) (in Russian) | ||

[22] | Shadimetov, Kh.M., Hayotov, A.R.: Optimal quadrature formulas with positive coefficients in L_{2}^{(m)}(0,1) space. J. Comput. Appl. Math. 235, 1114-1128 (2011) | ||

[23] | Kh.M.Shadimetov, A.R.Hayotov, S.S.Azamov. Optimal quadrature formula in K_{2}(P_{2}) space, Appl. Numer. Math., 62, 1893-1909 (2012). | ||

[24] | Kh.M.Shadimetov, A.R.Hayotov, Optimal quadrature formulas with positive coefficients in L_{2}^{(m)}(0,1) space, Journal of Computational and Applied Mathematics, 235: 1114-1128 (2011). | ||

[25] | Kh.M.Shadimetov, A.R.Hayotov, F.A.Nuraliev. On an optimal quadrature formula in the Sobolev space, Journal of Comp. Appl. Math., 243, 91-112 (2013). | ||

[26] | Kh.M.Shadimetov, A.R.Hayotov, D.M.Akhmedov. Optimal quadrature formulas for the Cauchy type singular integral in the Sobolev space L_{2}^{(2)}(-1,-1). American Journal of Numerical Analysis, Science and Education Publishing, Vol.1, no1, 22-31, (2013). | ||

[27] | Kh.M.Shadimetov, A.R.Hayotov. Optimal quadrature formulas in the sense of Sard in W_{2}^{(m,m-1)} space, Calcolo, 51, no.2, 211-243, (2014). | ||

[28] | Schoenberg, I.J.: On monosplines of least deviation and best quadrature formulae. J. Soc. Indust. Appl. Math. Ser. B Numer. Anal. 2, 144-170 (1965). | ||

[29] | Schoenberg, I.J.: On monosplines of least square deviation and best quadrature formulae II. SIAM J. Numer. Anal. 3, 321–328 (1966). | ||

[30] | I.J. Schoenberg, S.D. Silliman, On semicardinal quadrature formulae, Math. Comp. 126 (1974) 483-497. | ||

[31] | Schoenberg, I.J., Silliman, S.D.: On semicardinal quadrature formulae. Math. Comput. 28, 483-497 (1974). | ||

[32] | obolev, S.L.: Introduction to the Theory of Cubature Formulas. Nauka, Moscow (1974) (in Russian). | ||

[33] | Sobolev, S.L., Vaskevich, V.L.: The Theory of Cubature Formulas. Kluwer Academic Publishers Group, Dordrecht (1997). | ||

[34] | Sobolev, S.L.: The coefficients of optimal quadrature formulas. In: Selected Works of S.L. Sobolev, pp. 561–566, Springer (2006). | ||

[35] | Zagirova, F.Ya.: On construction of optimal quadrature formulas with equal spaced nodes, 28 p. Novosibirsk (1982, Preprint No. 25, Institute of Mathematics SD of AS of USSR) (in Russian). | ||

[36] | Zhamalov, Z.Zh., Shadimetov, Kh.M.: About optimal quadrature formulas. Dokl. Akad. Nauk UzSSR 7, 3-5 (1980) (in Russian). | ||

## Article

# The Immersed Interface Method for Elliptic and Parabolic Problems with Discontinuous Coefficients

^{1,}

^{1}Department of Mathematics King Abduall-Aziz University

*American Journal of Numerical Analysis*.

**2014**, 2(5), 152-166

**DOI:**10.12691/ajna-2-5-3

**Copyright © 2014 Science and Education Publishing**

**Cite this paper:**

Noufe Aljahdaly. The Immersed Interface Method for Elliptic and Parabolic Problems with Discontinuous Coefficients.

*American Journal of Numerical Analysis*. 2014; 2(5):152-166. doi: 10.12691/ajna-2-5-3.

Correspondence to: Noufe Aljahdaly, Department of Mathematics King Abduall-Aziz University. Email: nhaljahdaly@kau.edu.sa

## Abstract

## Keywords

## References

[1] | X. Feng and Z. Li, Simplified Immersed Interface Methods for Elliptic Interface Problems with Straight Interfaces, Numerical Methods for Partial Differential Equations, 28 (2012), pp. 188-203. | ||

[2] | R. J. LeVeque, Finite difference methods for ordinary and partial differential equations: Steady-state and time-dependent problems, (2007). | ||

[3] | Zhilin Li, The immersed Interface method: A numerical approach for partial differential equation with interface, PhD thesis, Univerdity of Washington, 1994. | ||

[4] | Z. Li and K. Ito, The immersed interface method: numerical solutions of PDEs involving interfaces and irregular domains, vol. 33, Society for Industrial Mathematics, 2006. | ||

## Article

# Numerical Treatments for the Fractional Fokker-Planck Equation

^{1,}

^{1}Department of Mathematics, Umm Al-Qura University, Makkah, Kingdom of Saudi Arabia

*American Journal of Numerical Analysis*.

**2014**, 2(6), 167-176

**DOI:**10.12691/ajna-2-6-1

**Copyright © 2014 Science and Education Publishing**

**Cite this paper:**

Kholod M. Abualnaja. Numerical Treatments for the Fractional Fokker-Planck Equation.

*American Journal of Numerical Analysis*. 2014; 2(6):167-176. doi: 10.12691/ajna-2-6-1.

Correspondence to: Kholod M. Abualnaja, Department of Mathematics, Umm Al-Qura University, Makkah, Kingdom of Saudi Arabia. Email: kmaboualnaja@uqu.edu.sa

## Abstract

## Keywords

## References

[1] | I.Podlubny, Fractional Differential Equations, Academic press, San Diego, (1999). | ||

[2] | G.Samko, A.A.Kibas, O.I.Marichev, Fractional Integrals and Derivatives: Theory and Applactions, Gordon and Breach, Yverdon, (1993). | ||

[3] | K.B.Oldham, J.Spanier, The Fractional Calculus, Academic Press, NewYork, (1974). | ||

[4] | Y.Luchko, R.Gorenflo, The Initial Value Problem for Some Fractional Equations With Caputo Derivative, Preprint Series A08-98, Frachbreich Mathematic In Formatik, Freicumiver Siat Berlin, (1998). | ||

[5] | G.Adpmian, Nonlinear Stochastic Systems Theory and Applications to Physics, Kluwer Academic Publishers, Dordrecht, (1989). | ||

[6] | Mehdi Dehghan, Mehdi Tatari, The Use of Adomian Decomposition Method for Solving Problems in Calculus of Variational, Mathematical Problems in Engineering, (2006). | ||

[7] | Sennur Somali, Guzin Gokmen, Adomian Decomposition Method For Nonlinear Strum-Liouville Problems, 2, pp 11-20, (2007). | ||

[8] | Zaid Odibat, Shaher Momani, Numerical Methods for Nonlinear Partial Differential Equation of Fractional Order, Applied Mathematical Modeling, 32, pp 28-39, (2008). | ||

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## Article

# Numerical Solution of Singularly Perturbed Two-Point Singular Boundary Value Problems Using Differential Quadrature Method

^{1}Department of Mathematics, National Institute of Technology, Jamshedpur, INDIA

^{2}Department Mathematics, National Institute of Technology, Warangal, INDIA

*American Journal of Numerical Analysis*.

**2014**, 2(6), 177-183

**DOI:**10.12691/ajna-2-6-2

**Copyright © 2014 Science and Education Publishing**

**Cite this paper:**

H.S. Prasad, Y.N. Reddy. Numerical Solution of Singularly Perturbed Two-Point Singular Boundary Value Problems Using Differential Quadrature Method.

*American Journal of Numerical Analysis*. 2014; 2(6):177-183. doi: 10.12691/ajna-2-6-2.

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

## Abstract

## Keywords

## References

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## Article

# A Fitted Second Order Finite Difference Method for Singular Perturbation Problems Exhibiting Dual Layers

^{1}Department of Mathematics, National Institute of Technology, Jamshedpur, INDIA

^{2}Department Mathematics, National Institute of Technology, Warangal, INDIA

*American Journal of Numerical Analysis*.

**2014**, 2(6), 184-189

**DOI:**10.12691/ajna-2-6-3

**Copyright © 2014 Science and Education Publishing**

**Cite this paper:**

H.S. Prasad, Y.N. Reddy. A Fitted Second Order Finite Difference Method for Singular Perturbation Problems Exhibiting Dual Layers.

*American Journal of Numerical Analysis*. 2014; 2(6):184-189. doi: 10.12691/ajna-2-6-3.

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

## Abstract

## Keywords

## References

[1] | Bender, C.M., Orszag, S.A.: Advanced Mathematical Methods for Scientists and Engineers, Mc. Graw-Hill, . 1978. | ||

[2] | Doolan, E.P., Miller, J.J.H., Schilders, W.H.A.: Uniform Numerical methods for problems with initial and boundary layers, Boole Press, Dublin. 1980. | ||

[3] | Hemker, P.W., Miller, J.J.H. (Editors).: Numerical Analysis of Singular Perturbation Problems, Academic Press, New York, 1978. | ||

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[5] | Kevorkian, J., Cole, J.D.: Perturbation Methods in Applied Mathematics, Springer-Verlag, New York, 1981. | ||

[6] | Kadalbajoo, M.K., Reddy, Y.N.: Asymptotic and Numerical Analysis of Singular Perturbation Problems: A Survey, Applied Mathematics and Computation, 30: 223-259, 1989. | ||

[7] | Miller, J.J.H., O’Riordan, E., Shishkin, G.I.: Fitted Numerical methods for singular perturbation problems, Error estimates in the maximum norm for linear problems in one and two dimensions, World Scientific Publishing Company Pvt. Ltd. 1996. | ||

[8] | Nayfeh, A.H.: Perturbation Methods, Wiley, New York. 1973. | ||

[9] | O’ Malley, R.E.: Introduction to Singular Perturbations, Academic Press, New York, 1974. | ||

[10] | Phaneendra, K., Pramod Chakravarthy, P., Reddy, Y. N.: A Fitted Numerov Method for Singular Perturbation Problems Exhibiting Twin Layers, Applied Mathematics & Information Sciences – An International Journal, Dixie W Publishing Corporation, U.S. A., 4 (3): 341-352, 2010. | ||

[11] | Reddy, Y.N. (1986). Numerical Treatment of Singularly Perturbed Two Point Boundary Value Problems, Ph.D. thesis, IIT, Kanpur, India. 1986. | ||

[12] | Reddy Y.N., Pramod Chakravarthy, P. (2004). An exponentially fitted finite difference method for singular perturbation problems, Applied Mathematics and Computation, 154: 83-101 2004. | ||

## Article

# Initial Value Approach for a Class of Singular Perturbation Problems

^{1}Department of Mathematics National Institute of Technology WARANGAL-506004, INDIA

*American Journal of Numerical Analysis*.

**2015**, 3(1), 1-7

**DOI:**10.12691/ajna-3-1-1

**Copyright © 2015 Science and Education Publishing**

**Cite this paper:**

P. Padmaja, Y.N. Reddy. Initial Value Approach for a Class of Singular Perturbation Problems.

*American Journal of Numerical Analysis*. 2015; 3(1):1-7. doi: 10.12691/ajna-3-1-1.

Correspondence to: Y.N. Reddy, Department of Mathematics National Institute of Technology WARANGAL-506004, INDIA. Email: ynreddy_nitw@yahoo.com

## Abstract

## Keywords

## References

[1] | Bellman, R.: Perturbation Techniques in Mathematics, Physics and Engineering, Holt, Rinehart, , 1964. | ||

[2] | Bender, C. M. and Orszag, S. A.: Advanced Mathematical Methods for Scientists and Engineers, McGraw-Hill, New York, 1978. | ||

[3] | Doolan, E. P., Miller, J. J. H. and Schilders, W. H. A.: Uniform Numerical Methods for problems with Initial and Boundary Layers, Boole Press, Dublin, 1980. | ||

[4] | El’sgol’ts L. E. and Norkin, S. B.: Introduction to the Theory and Applications of Differential Equations with Deviating Arguments, Academic Press, New York, 1973. | ||

[5] | Kevorkian, J. and Cole, J.D.: Perturbation Methods in Applied Mathematics, Springer- Verlag, New York, 1981. | ||

[6] | Miller, J.J.H., O’Riordan, E. and Shishkin, G.I.: Fitted Numerical Methods Singular Perturbation Problems, World Scientific, River Edge, NJ, 1996. | ||

[7] | Nayfeh, A. H.: Perturbation Methods, Wiley, New York, 1979. | ||

[8] | Nayfeh, A. H.: Introduction to Perturbation Techniques, Wiley, New York, 1981. | ||

[9] | Nayfeh, A. H: Problems in Perturbation, Wiley, New York, 1985. | ||

[10] | O’Malley, R. E.: Introduction to Singular Perturbations, Academic Press, New York, 1974. | ||

[11] | Pearson, C.E.: On a differential equation of boundary layer type, J. Math Phy, 47, 1968, pp 134-154. | ||

[12] | Smith, D. R.: Singular Perturbation Theory-An Introduction with applications, Cambridge University Press, Cambridge, 1985. | ||

## Article

# Solution of Singularly Perturbed Differential Difference Equations Using Higher Order Finite Differences

^{1}Department of Mathematics, National Institute of Technology, Warangal, India

*American Journal of Numerical Analysis*.

**2015**, 3(1), 8-17

**DOI:**10.12691/ajna-3-1-2

**Copyright © 2015 Science and Education Publishing**

**Cite this paper:**

Lakshmi Sirisha, Y. N. Reddy. Solution of Singularly Perturbed Differential Difference Equations Using Higher Order Finite Differences.

*American Journal of Numerical Analysis*. 2015; 3(1):8-17. doi: 10.12691/ajna-3-1-2.

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

## Abstract

## Keywords

## References

[1] | Bellman and R. K. L. Cooke, Differential-Difference Equations, Academic Press, , 1963. | ||

[2] | Derstine, M. W., Gibbs, H. M., Hopf, F. A. and Kalplan, D. L.: Bifurcation gap in hybrid optical system, Phys. Rev. A26(1982)3720-3722. | ||

[3] | Kolmanovskii, V. and Myshkis, A.:Applied Theory of Functional Differential Equations, Volume 85 of Mathematics and its Applications(Soviet Series), Kluwer Academic Publishers Group, Dordrecht, 1992. | ||

[4] | H. C. Tuckwell and W. Richter, Neuronal inter-spike time distributions and the estimation of neuro-physiological and neuro-anatomical parameters, J. Theor. Biol., 71 (1978) 167-183. | ||

[5] | Stein R.B., A theoretical analysis of neuronal variability, Biophys. J., 5 (1965) 173-194. | ||

[6] | C.G. Lange R.M. Miura, Singular perturbation analysis of boundary-value problems for differential-difference equations. II. Rapid oscillations and resonances, SIAM J. Appl. Math., 45 (1985) 687-707. | ||

[7] | C.G. Lange R.M. Miura, Singular perturbation analysis of boundary- value problems for differential-difference equations. III. Turning point problems, SIAM J. Appl. Math., 45 (1985) 708-734. | ||

[8] | C.G. Lange R.M. Miura, Singular perturbation analysis of boundary-value problems for differential-difference equations. V. Small shifts with layer behaviour, SIAM J. Appl. Math., 54 (1994) 249-272. | ||

[9] | C.G. Lange R.M. Miura, Singular perturbation analysis of boundary-value problems for differential-difference equations. VI. Small shifts with rapid oscillations, SIAM J. Appl. Math., 54 (1994) 273-283. | ||

[10] | C.G. Lange R.M. Miura, Singular perturbation analysis of boundary-value problems for differential difference equations, SIAM J. Appl. Math., 42 (1982) 502-531. | ||

[11] | M. K. Kadalbajoo and K. K. Sharma, Numerical analysis of boundary-value problems for singularly perturbed differential-difference equations with small shifts of mixed type, J. Optim. Theory Appl., 115 (1) (2002) 145-163. | ||

[12] | M. K. Kadalbajoo and K. K. Sharma, Numerical treatment of a mathematical model arising from a model of neuronal variability, J. Math. Anal. Appl., 307 (2005) 606-627. | ||

[13] | Sharma, K. K. and Kaushik, A.: A solution of the discrepancy occurs due to using the fitted mesh approach rather than to the fitted operator for solving singularly perturbed differential equations, Appl. Math. Comput., 181(2006) 756-766. | ||

[14] | Amiraliyeva, I. G., and Cimen, E: Numerical method for a singularly perturbed convection-diffusion problem with delay, Applied Mathematics and Computation, 216(2010) 2351-2359. | ||

[15] | Pratima, R. and Sharma, K. K.: Numerical method for singularly perturbed differential-difference equations with turning point, International Journal of Pure and Applied Mathematics, 73(4) (2011) 451-470. | ||

[16] | Chakarvarthy, P. P. and Rao R. N.: A modified Numerov method for solving singularly perturbed differential-difference equations arising in science and engineering, Results in Physics, 2 (2012) 100-103. | ||

[17] | L. E. El’sgolts and S. B. Norkin, Introduction to the Theory and Applications of Differential Equations with Deviating Arguments, Academic Press, New York, 1973. | ||

[18] | R. K. Mohanty and N. Jha, A class of variable mesh spline in compression methods for singularly perturbed two point singular boundary value problems, Applied Mathematics and Computation, 168 (2005) 704-716. | ||

[19] | J. Y. Choo and D. H.Schultz: Stable higher order methods for differential equations with small coefficients for the second order terms, Journal of Computers and Math. with Applications, 25 (1993), 105-123. | ||

[20] | Pratima, R. and Sharma, K. K.: Numerical study of singularly perturbed differential difference equation arising in the modelling of neuronal variability, Computers and Mathematics with Applications, 63 (2012) 118-132. | ||

[21] | Rao, R. N. and Chakravarthy, P. P.: An initial value technique for singularly perturbed differential-difference equations with a small negative shift, J. Appl. Math. & Informatics, 31 (2013) 131-145. | ||

[22] | Gemechis FD and Reddy Y.N.: A Non-Asymptotic Method for Singularly Perturbed Delay Differential Equations, Journal of Applied Mathematics and Informatics, 32:1(2014) 39-53. | ||

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