American Journal of Numerical Analysis

ISSN (Print): 2372-2118

ISSN (Online): 2372-2126

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

Article

Optimal Quadrature Formulas with Polynomial Weight in Sobolev Space

1Institute 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

In this paper we construct the optimal quadrature formula with polynomial weight in the Sobolev space L2(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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[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).
 
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[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 K2(P3) 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 L2(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 L2(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.
 
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[26]  Kh.M.Shadimetov, A.R.Hayotov, D.M.Akhmedov. Optimal quadrature formulas for the Cauchy type singular integral in the Sobolev space L2(2)(-1,-1). American Journal of Numerical Analysis, Science and Education Publishing, Vol.1, no1, 22-31, (2013).
 
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Article

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

1Department 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

In this paper we consider numerical methods for solving elliptic as well as time dependent advection- diffusion-reaction (ADR) equations in one spatial dimension. We consider the case in which the difference diffusion coefficients as well as advection coefficients and reaction coefficients are discontinuous across a fixed interface. Using the immersed interface method (IIM) for finite difference approximations, we demonstrate how to modify numerical methods constructed for the constant coefficient case around interfaces of discontinuity of the diffusion, advection, and reaction coefficient.

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

1Department 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

In this paper, by introducing the fractional derivative in the sense of Caputo, of the Adomian decomposition method and the variational iteration method are directly extended to Fokker – Planck equation with time-fractional derivatives, as result the realistic numerical solutions are obtained in a form of rapidly convergent series with easily computable components. The figures show the effectiveness and good accuracy of the proposed methods.

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).
 
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[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).
 
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[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).
 
[9]  A.Wazwaz, A new Algorithm for Calculation Adomian Polynomials For Nonlinear Operators, Appl. Math. Comput., 111, pp 53-69, (2000).
 
[10]  A.Wazwaz, SEl-Sayed, A new Modification of the Adomian Decomposition Method for Linear and Nonlinear Operators, Appl. Math. Comput., 122, pp 393-405, (2001).
 
[11]  Shahe Momani, An Explicit and Numerical Solution of the Fractional KDV Equation, Mathematics and Computers in Simulation, 70, pp 110-118, (2005).
 
[12]  Yong Chen, Hong-Li An, Numerical Solutions of Coupled Burgers Equations with Time-and Space-Fractional Derivatives, Appl. Math. Comput., 200, pp 87-95, (2008).
 
[13]  Zaid Odiba, Shaher Momani, Application of Variationalal Iteration Method to Nonlinear Differential Equation of Fractional Orders, Int. J. Nonlinear Sciences and Numerical Simulations, 1(7), pp 15-27, (2006).
 
[14]  J.H.He, Variationalal Iteration Method for Autonomous Ordinary Differential Systems, Journal Applied Mathematics and Computation Volume 114 Issue 2-3, pp 115-123, (2000).
 
[15]  J.H.He, Variationalal Principle for Nano Thin Film Lubrication, Int. J. Nonlinear Sci. Numer Simul., 4, (3), pp 313-314, (2003).
 
[16]  J.H.He, Variationalal Principle For Some Nonlinear Partial Differential Equations with Variable Coefficients, Chaos, Solitons Fractals 19 (4), pp 847-851, (2004).
 
[17]  M. Inokuti, H. Sekine,T. Mura, General use of the Lagrange multiplier in nonlinear mathematical physics, in: S. Nemat- Nasser (Ed.),Variationalal Method in the Mechanics of Solids, Pergamon Press, Oxford, pp. 156–162, (1978).
 
[18]  H. Risken, The Fokker–Planck Equation Springer, Berlin (1988).
 
[19]  Seakweng Vong, Zhibo Wang, A high order compact finite difference scheme for time fractional Fokker–Planck equations, Applied Mathematics Letters, 43, PP 38-43, (2015).
 
[20]  Yuxin Zhang, [3, 3] Padé approximation method for solving space fractional Fokker–Planck equations, Applied Mathematics Letters, 35, PP 109-114, (2014).
 
[21]  Chunhong Wu, Linzhang Lu, Implicit numerical approximation scheme for the fractional Fokker–Planck equation, Applied Mathematics and Computation, 216 ( 7), PP 1945-1955, 2010.
 
[22]  F. Liu, V. Anh, I. Turner, Numerical solution of space fractional Fokker–Planck equation, J. Comput. Appl. Math. Volume 166, PP 209-219, (2004).
 
[23]  F. Liu, V. Anh, I. Turner, Numerical simulation for solute transport in fractal porous media, ANZIAM J. (E) 45, PP 461-473. (2004).
 
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Article

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

1Department of Mathematics, National Institute of Technology, Jamshedpur, INDIA

2Department 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

This paper presents the application of Differential Quadrature Method (DQM) for finding the numerical solution of singularly perturbed two point singular boundary value problems. The DQM is an efficient discretization technique in solving initial and/or boundary value problems accurately using a considerably small number of grid points. This method is based on the approximation of the derivatives of the unknown functions involved in the differential equations at the mess point of the solution domain. To demonstrate the applicability of the method, we have solved model example problems and presented the computational results. The computed results have been compared with the exact solution to show the accuracy and efficiency of the method.

Keywords

References

[1]  R. E. Bellman and J. Casti, Differential Quadrature and Long-Term Integration, Journal of Mathematical Analysis and Application, 34 (1971), 235-238.
 
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[6]  P. A. Farrell, A. F. Hegarty, J.J.H. Miller, E. O’Riordan, and G. I. Shishkin, Robust Computational techniques for Boundary Layers, Chapman & Hall/CRC Press, 2000.
 
[7]  7. M. K. Kadalbajoo, and V. K. Aggarwal, Fitted mesh B-spline method for solving a class of singular singularly perturbed boundary value problems, Int. J. Computer Mathematics, vol. (82), 1, (2005), 67-76.
 
[8]  J. Li, A Computational Method for Solving Singularly Perturbed Two-Point Singular Boundary Value Problem, Int. Journal of Math. Analysis, Vol. 2, (2008), no. 22, 1089-1096.
 
[9]  J. J. H. Miller, E. O. Riordan and G. I. Shishkin, Fitted numerical Methods for Singular Perturbation Problems, World Scientific, Singapore, 1996.
 
[10]  R. K. Mohanty and U. Arora, A family of non-uniform mesh tension spline methods for singularly perturbed two-point singular boundary value problems with significant first derivatives, App. Math. And Comput. Vol. (172), 1, (2006), 531-544.
 
[11]  R. K. Mohanty, D. J. Evans and U. Arora, Convergent spline in tension methods for singularly perturbed two-point singular boundary value problems, Int. J. Comput. Math., 82 (1), (2005), 55-66.
 
[12]  R. K. Mohanty and N. Jha, A class of variable mesh spline in compression methods for singularly perturbed two point singular boundary value problems.App. Math. And Comput. Vol. (168), 1, (2004), 704-716
 
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[14]  R. E. O’Malley, Introduction to Singular Perturbations, Academic Press, New York, 1974.
 
[15]  H. S. Prasad, Y. N. Reddy, Numerical Solution of Singularly Perturbed Differential-Difference Equations with Small Shifts of Mixed Type by Differential Quadrature Method, American Journal of Comput. and Appl. Mathematics, 2 (1), (2012), pp. 46-52.
 
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[17]  J. R. Quan, and C. T. Chang, New insights in solving distributed system equations by the quadrature methods-II. Application, Comput. Chem. Engrg, 13, (1989), 1017-1024.
 
[18]  J. Rashidinia, R. Mohammadi, M. Ghasemij, Cubic Spline Solution of Singularly Perturbed boundary value problems with significant first derivatives, App. Math. And Comput. Vol. 190, 2, (2007), 1762-1766.
 
[19]  H. G. Roos, M. Stynes, and L. Tobiska, Numerical Methods for Singularly Perturbed differential equations, Springer, Berlin, 1996.
 
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Article

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

1Department of Mathematics, National Institute of Technology, Jamshedpur, INDIA

2Department 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

In this paper a fitted second-order finite difference method is presented for solving singularly perturbed two-point boundary value problems with the boundary layer at both end (left and right) points. We have introduced a fitting factor in second-order tri-diagonal finite difference scheme and it is obtained from the theory of singular perturbations. The efficient Thomas algorithm is used to solve the tri-diagonal system. Maximum absolute errors are presented in tables to show the efficiency of the method.

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.
 
[4]  Jain, M.K.: Numerical solution of differential equations, 2nd Ed., Wiley Eastern Ltd., New Delhi 1984.
 
[5]  Kevorkian, J., Cole, J.D.: Perturbation Methods in Applied Mathematics, Springer-Verlag, New York, 1981.
 
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[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.
 
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Article

Initial Value Approach for a Class of Singular Perturbation Problems

1Department 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

In this paper, we present an initial value approach for a class of singularly perturbed two point boundary value problems with a boundary layer at one end point. The idea is to replace the original two point boundary value problem by set of suitable initial value problems. This replacement is significant from the computational point of view. This method does not depend on asymptotic expansions. Several linear and non-linear singular perturbation problems have been solved and the numerical results are presented to support the theory.

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.
 
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[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.
 
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Article

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

1Department 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

In this paper, we discuss the solution of singularly perturbed differential-difference equations exhibiting dual layer using the higher order finite differences. First, the second order singularly perturbed differential-difference equations is replaced by an asymptotically equivalent second order singular perturbed ordinary differential equation. Then, fourth order stable finite difference scheme is applied to get a three term recurrence relation which is easily solved by Thomas algorithm. Some numerical examples have been solved to validate the computational efficiency of the proposed numerical scheme. To analyze the effect of the parameters on the solution, the numerical solution has also been plotted using graphs. The error bound and convergence of the method have also been established.

Keywords

References

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

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

1Department 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

In this paper, we study the numerical solution of singular singularly perturbed third-order boundary value problems (BVPs) by using Quartic B-spline method. An efficient algorithm is presented here to solve the approximate solution of the given problem. To understand our method, we introduce the Quartic B-spline basis function in the form of at the different knots. After that we derive our method by using numerical difference formulas to construct the approximate values. Then we use the linear sequence of Quartic B-spline to get the numerical solution of the system of equations. These systems of equations are solved by using MATLAB. Three examples are illustrated to understand the present method.

Keywords

References

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Article

Initial Value Method Extended for General Singular Perturbation Problems

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

2Department 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

In this paper the initial value method is extended for solving singularly perturbed two point boundary value problems with internal and terminal layers. It is distinguished by the following fact: The given singularly perturbed boundary value problem is replaced by two first order initial value problems. These first order problems are solved using Runge Kutta method. Model example for each is solved to demonstrate the applicability of the method.

Keywords

References

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Article

Analysis of Fractional Splines Interpolation and Optimal Error Bounds

1Faculty 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

This paper presents a formulation and a study of three interpolatory fractional splines these are in the class of mα, m = 2, 4, 6, α = 0:5. We extend fractional splines function with uniform knots to approximate the solution of fractional equations. The developed of spline method is to analysis convergence fractional order derivatives and estimating error bounds. We propose spline fractional method to solve fractional differentiation equations. Numerical example is given to illustrate the applicability and accuracy of the methods.

Keywords

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