American Journal of Numerical Analysis

ISSN (Print): 2372-2118

ISSN (Online): 2372-2126

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

Article

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

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

2Tashkent Institute of Railway Engineers, Tashkent, Uzbekistan


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

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

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

Abstract

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

Keywords

References

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[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.
 
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[20]  Mastroianni, G., Milovanovic, G.V.: Interpolation Processes – Basic Theory and Applications, Springer Monographs in Mathematics, Springer – Verlag, Berlin – Heidelberg (2008).
 
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[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).
 
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[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

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

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


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

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

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

Abstract

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

Keywords

References

[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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[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 Lp, 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 WL2(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.
 
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[19]  A. Sard, Best approximate integration formulas; best approximation formulas, Amer. J. Math. 71 (1949) 80-91.
 
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[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.
 
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[25]  Kh.M. Shadimetov, Optimal quadrature formulas in L2m(Ω) and L2m(R1), Dokl. Akad. Nauk UzSSR. no.3 (1983) 5-8. (in Russian).
 
[26]  Kh.M. Shadimetov. The discrete analogue of the differential operator d2m/dx2m and its construction, Questions of Computations and Applied Mathematics. Tashkent, (1985) 22-35. ArXiv: 1001.0556.v1 [math.NA] Jan. 2010.
 
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[28]  Kh.M. Shadimetov, Construction of weight optimal quadrature formulas in the space L2(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 L2(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 W2(m,m-1) space. Calcolo.
 
[31]  Shadimetov, Kh.M., Hayotov, A.R., Azamov, S.S.: Optimal quadrature formula in K2(P2). 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 L2(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.
 
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[38]  A.A. Zhensikbaev, Monosplines of minimal norm and the best quadrature formulas (Russian), Uspekhi Matem. Nauk, 36 (1981) 107-159. (in Russian).
 
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Article

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

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

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


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

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

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

Abstract

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

Keywords

References

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[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.
 
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Article

Fitted Second Order Scheme for Singularly Perturbed Differential-difference Equations

1Department 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

In this paper, we present a fitted second order stable central finite difference scheme for solving singularly perturbed differential-difference equations (with delay and advanced parameter). First, the given second order differential difference equation is replaced by an asymptotically equivalent second order singularly perturbation problem. Then, a fitting factor is introduced into the second order stable central difference scheme and determined its value from the theory of singular perturbations. Discrete Invariant Imbedding Algorithm is used to solve the resulting tri-diagonal system. The error analysis and convergence of the scheme are also discussed. To validate the applicability of the method, several model examples have been solved by taking different values for the delay parameter δ, advanced parameter ηand the perturbation parameter ε.

Keywords

References

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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

[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).
 
Show More References
[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 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.
 
[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 WL1(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].
 
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[20]  Shadimetov, Kh.M.: The discrete analogue of the differential operator d2m/dx2m and its construction, Questions of Computations and Applied Mathematics. Tashkent, (1985) 22-35. ArXiv:1001.0556.v1 [math.NA] Jan. 2010.
 
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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

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[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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[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.
 
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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

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