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American Journal of Numerical Analysis

## 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

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## 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

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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

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[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. | ||

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## 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

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[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. | ||

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[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. | ||

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[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. | ||

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## Article

# Quartic B – Spline Method for Solving a Singular Singularly Perturbed Third-Order Boundary Value Problems

^{1}Department of Mathematics Jaypee University of Engineering & Technology Guna-473226(M.P) India

*American Journal of Numerical Analysis*.

**2015**, 3(1), 18-24

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

**Copyright © 2015 Science and Education Publishing**

**Cite this paper:**

Hradyesh Kumar Mishra, Sonali saini. Quartic B – Spline Method for Solving a Singular Singularly Perturbed Third-Order Boundary Value Problems.

*American Journal of Numerical Analysis*. 2015; 3(1):18-24. doi: 10.12691/ajna-3-1-3.

Correspondence to: Hradyesh Kumar Mishra, Department of Mathematics Jaypee University of Engineering & Technology Guna-473226(M.P) India. Email: hk.mishra@juet.ac.in

## Abstract

## Keywords

## References

[1] | Ghazala Akram, Quartic Spline Solution Of A Third Order Singularly Perturbed Boundary Value Problem, ANZIAM J, 53, (2012), 44-58. | ||

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## Article

# Initial Value Method Extended for General Singular Perturbation Problems

^{1}Department of Mathematics, Osmania University College For Women, Hyderabad, Telangana, India

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

*American Journal of Numerical Analysis*.

**2015**, 3(1), 25-29

**DOI:**10.12691/ajna-3-1-4

**Copyright © 2015 Science and Education Publishing**

**Cite this paper:**

Loka Pavani, Y. N. Reddy. Initial Value Method Extended for General Singular Perturbation Problems.

*American Journal of Numerical Analysis*. 2015; 3(1):25-29. doi: 10.12691/ajna-3-1-4.

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

## Abstract

## Keywords

## References

[1] | Awoke A.T. Numerical Treatment of a class of singular perturbation problems – Ph. D Thesis, NITW, India, April 2008. | ||

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

[3] | Kadalbajoo M.K. and Reddy Y.N. An Initial Value Technique for a class of Non-linear Singular Perturbation Problems, Journal of Optimization Theory and Applications. - 1987. - Vol. 53. 395-406. | ||

[4] | Kadalbajoo M.K. and Reddy Y.N. Numerical solution of singular perturbation problems by Terminal Boundary Value Technique, Journal of Optimization Theory and Applications. - 1987. - Vol. 52. - pp. 243-254. | ||

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

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

[10] | M. Van Dyke, Perturbation Methods in Fluid Mechanics, Academic Press, New York, 1964. | ||

## Article

# Analysis of Fractional Splines Interpolation and Optimal Error Bounds

^{1}Faculty of Science and Science Education, School of Science Education, Sulaimani Univ., Sulaimani, Iraq

*American Journal of Numerical Analysis*.

**2015**, 3(1), 30-35

**DOI:**10.12691/ajna-3-1-5

**Copyright © 2015 Science and Education Publishing**

**Cite this paper:**

Faraidun K. Hamasalh, Pshtiwan O. Muhammad. Analysis of Fractional Splines Interpolation and Optimal Error Bounds.

*American Journal of Numerical Analysis*. 2015; 3(1):30-35. doi: 10.12691/ajna-3-1-5.

Correspondence to: Faraidun K. Hamasalh, Faculty of Science and Science Education, School of Science Education, Sulaimani Univ., Sulaimani, Iraq. Email: faraidunsalh@gmail.com

## Abstract

## Keywords

## References

[1] | S. G. Samko, A. A. Kilbas, O. I. Marichev, Fractional Integrals and Derivatives-Theory and Applications, Gordon and Breach Science, Amsterdam, 1993. | ||

[2] | I. Podlubny, Fractional Differentional Equations, Academic Press, San Diego, 1999. | ||

[3] | J. E. Phythian and R. Williams, Direct cubic spline approximation to integrals, Int. j. numer. methodseng., 23 (1986), 305-315. | ||

[4] | P. G. Clarleft, M. H. Schultz and R. S. Varga, Numerical methods of high order accuracy, Numer. Math., 9 (1967), 394-430. | ||

[5] | Zahra W. K. and Elkholy S. M., Quadratic spline solution for boundary value problem of fractional order, Numer Algor, 59 (2012), 373-391. | ||

[6] | G. Micula, T. Fawzy, and Z. Ramadan, A polynomial spline approximation method for solving system of ordinary differential equations, Babes-Bolyai Cluj-Napoca. Mathematica, vol. 32, no. 4 (1987) 55-60. | ||

[7] | M. A. Ramadan, Spline solutions of first order delay differential equations, Journal of the Egyptian Mathematical Society, vol. 13, no. 1 (2005) 7-18. | ||

[8] | G. Birkhoff and A. Priver, Hermite interpolation errors for derivatives, J. Math. Phys., 46 (1967) 440-447. | ||

[9] | A. K. Varma and G. Howell, Best error bounds for derivatives in two point Birkhoff interpolation problems, J. Approx. Theory, 38 (1983) 258-268. | ||

[10] | Richard Herrmann, Fractional calculus: an introduction for physicists, Giga Hedron, Germany, 2nd edition, 2014. | ||

[11] | M. Ishteva, Properties and applications of the Caputo fractional operator, Msc. Thesis, Dept. of Math., Universität Karlsruhe (TH), Sofia, Bulgaria, 2005. | ||

[12] | A. K. Varma and G. Howell, Cantor-type cylindrical coordinate method for differential equations with local fractional derivatives, Physics Letters A, 377 (28) (2013) 1696-1700. | ||

[13] | A. K. Varma and G. Howell, Thermal impedance estimations by semi-derivatives and semi-integrals: 1-D semiinfinite cases, Thermal Science, 17 (2) (2013) 581-589. | ||

[14] | Faraidun K. Hamasalh and Pshtiwan O. Muhammad, Generalized Quartic Fractional Spline Interpolation with Applications, Int. J. Open Problems Compt. Math, Vol. 8, No. 1 (2015)., 67-80. | ||

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