*American Journal of Applied Mathematics and Statistics*.

**2014**, 2(3), 129-142

DOI: 10.12691/ajams-2-3-7

Open AccessReview Article

### Various Numerical Methods for Singularly Perturbed Boundary Value Problems

**Hradyesh Kumar Mishra**^{1, } and **Sonali saini**^{1}

^{1}Department of Mathematics, Jaypee University of Engineering &Technology, Madhya Pradesh, India

Pub. Date: May 09, 2014

**Cite this paper:**

Hradyesh Kumar Mishra and Sonali saini. Various Numerical Methods for Singularly Perturbed Boundary Value Problems. *American Journal of Applied Mathematics and Statistics*. 2014; 2(3):129-142. doi: 10.12691/ajams-2-3-7

### Abstract

The numerical treatment of singular perturbation problems is currently a field in which active research is going on these days. Singular perturbation problems in which the term containing the highest order derivative is multiplied by a small parameter ε, occur in a number of areas of applied mathematics, science and engineering among them fluid mechanics (boundary layer problems) elasticity (edge effort in shells) and quantum mechanics. In this paper, we consider few numerical methods for singularly perturbed boundary value problems developed by numerous researchers between 2006 to 2013. A Summary of the result of some recent methods is presented and this leads to conclusion and recommendations regarding methods to use on singular perturbation problem.**Keywords:**singular perturbation ordinary differential equation boundary layer two-point boundary value problem delay differential equations integral equations

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### References:

[1] | M. K. Kadalbajoo, Y.N. Reddy, Asymptotic and numerical analysis of singular perturbation problems (1989),Applied mathematics and computation 1989; 30: 223-259. | ||

[2] | M.K. Kadalbajoo and K.C. Patidar, A survey of numerical techniques for solving singularly perturbed ordinary differential equations, Applied Mathematics and Computation 130(2-3) (2002) 457-510. | ||

[3] | M. Kumar, Hradyesh Kumar Mishra, P. Singh, A recent survey on computational techniques for solving singularly perturbed boundary value problems, 2007,International Journal of Computer Mathematics 2007: 84: 1439-1463. | ||

[4] | S.A. Khuri, A. Sayfy, Self-adjoint singularly perturbed second-order two-point boundary value problems: A patching approach, Applied Mathematical Modelling, (Article in Press) December 2013. | ||

[5] | F.Z. Geng, S.P. Qian., Reproducing kernel method for singularly perturbed turning point problems having twin boundary layers, Applied Mathematics Letters, 2013; 26: 998-1004. | ||

[6] | Essam R. El-Zahar, Approximate analytical solutions of singularly perturbed fourth order boundary value problems using differential transform method, Journal of King Saud University - Science, 2013; 25: 257-265. | ||

[7] | Hradyesh Kumar Mishra, Sonali Saini, Numerical Solution of Singularly Perturbed Two-Point Boundary Value Problem via Liouville-Green Transform, American Journal of Computational Mathematics, 2013; 3: 1-5. | ||

[8] | M.Mokarram Shahraki, S. Mohammad Hosseini, Comparison of a higher order method and the simple upwind and non-monotone methods for singularly perturbed boundary value problems, Applied Mathematics and Computation 2006;182: 460-473. | ||

[9] | Kailash C. Patidar, High order parameter uniform numerical method for singular perturbation problems, Applied Mathematics and Computation 2007; 188: 720-733. | ||

[10] | S.Hemavathi, T. Bhuvaneswari, S. Valarmathi, J.J.H. Miller, A parameter uniform numerical method for a system of singularly perturbed ordinary differential equations, Applied Mathematics and Computation 2007;191:1-11. | ||

[11] | G.I. Shishkin, Mesh approximation of singularly perturbed boundary value problems for systems of elliptic and parabolic equations, Comput. Math. Math. Phys. 1995; 35 (4): 429-446. | ||

[12] | J. Rashidinia, R. Mohammadi, M. Ghasemi Cubic spline solution of singularly perturbed boundary value problems with significant first derivatives, Applied Mathematics and Computation 2007; 190: 1762-1766. | ||

[13] | L. Abrahamson, S. Osher, Monotone difference schemes for singular perturbation problems, SIAM J. Numer. Anal. 1982;19: 979-992. | ||

[14] | P.W. Hemker, J.J.H. Miller, Numerical Analysis of Singular Perturbation Problems, Academic Press, New York, 1979. | ||

[15] | B. Kreiss, H.O. Kreiss, Numerical methods for singular perturbation problems, SIAM J. Numer. Anal.1982; 46: 138-165. | ||

[16] | J.J.H. Miller, E.O. Riordan, G.I. Shishkin, On piecewise uniform meshes for upwind and central difference operators for solving singularly perturbed problems, IMA J. Numer. Anal. 1995; 15: 89-99. | ||

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

[18] | M.K. Kadalbajoo, K.C. Patidar, Numerical solution of singularly perturbed two point boundary value problems by spline in compression, Int. J. Comput. Math. 2001; 77: 263-283. | ||

[19] | R.B. Kellog, A. Tsan, Analysis of some difference approximations for a singular perturbation problem without turning points, Math. Comput. 1978; 32: 1025-1039. | ||

[20] | J. Kevorkian, J.D. Cole, Multiple Scale and Singular Perturbation Methods, Springer-Verlag, New York, 1996. | ||

[21] | R.K. Bawa, Spline based computational technique for linear singularly perturbed boundary value problems, Appl. Math. Comput. 2005; 167: 225-236. | ||

[22] | T. Aziz, A. Khan, A spline method for second order singularly perturbed boundary-value problems, J. Comput. Appl. Math. 2002; 147: 445-452. | ||

[23] | J. Rashidinia, Applications of spline to numerical solution of differential equations, M.Phil dissertation, A.M.U. India, 1990. | ||

[24] | J. Rashidinia, R. Mohammadi, R. Jalilian, The numerical solution of non-linear singular boundary value problems arising in physiology, Appl. Math. Comput. 2007; 185: 360-367. | ||

[25] | R.K. Mohanty, D.J. Evans, U. Arora, Convergent spline in tension methods for singularly perturbed two point singular boundary value problems, Int. J. Comput. Math. 2005; 82: 55-66. | ||

[26] | R.K. Mohanty, N. Jha, A class of variable mesh spline in compression method for singularly perturbed two point singular boundary value problems, Appl. Math. Comput. 2005; 168: 704-716. | ||

[27] | R.K. Mohanty, N. Jha, D.J. Evans, Spline in compression method for the numerical solution of singularly perturbed two point singular boundary value problems, Int. J. Comput. Math. 2004; 81: 615-627. | ||

[28] | R.K. Mohanty, Urvashi Arora, A family of non-uniform mesh tension spline methods for singularly perturbed two-point singular boundary value problems with significant first derivatives, Appl. Math. Comput. 2006; 172 531-544. | ||

[29] | M. Stojanovic, A first order accuracy scheme on non-uniform mesh, Publ. De L’Institut. Math. 1987; 42: 155-165. | ||

[30] | M.K. Kadalbajoo, V.K. Aggarwal, Fitted mesh B-spline method for solving a class of singular singularly perturbed boundary value problems, Int. J. Comput. Math. 2005; 82: 67-76. | ||

[31] | Jean M.-S. Lubuma a, Kailash C. Patidar bNon-standard methods for singularly perturbed problems possessing oscillatory/layer solutions, Applied Mathematics and Computation 2007; 187: 1147-1160. | ||

[32] | Dragoslav Herceg, Djordje Herceg, On a fourth-order finite-difference method for singularly perturbed boundary value problems Applied Mathematics and Computation 2008; 203: 828-837. | ||

[33] | R. Vulanovic´, On a numerical solution of a type of singularly perturbed boundary value problem by using a special discretization mesh, Univ. Novom Sadu Zb. Rad. Prirod. Mat. Fak. Ser. Mat.1983; 13: 187-201. | ||

[34] | D. Herceg, R. Vulanovic´, Some finite difference schemes for a singular perturbation problem on a nonuniform mesh, Univ. Novom Sadu Zb. Rad. Prirod. Mat. Fak. Ser. Mat. 1981; 11: 117-134. | ||

[35] | R. Vulanovic´, Mesh Construction for Discretization of Singularly Perturbed Boundary Value Problems, Doctoral Dissertation, Faculty of Science, Univesity of Novi Sad, 1986. | ||

[36] | N.S. Bakhvalov, Towards optimization of methods for solving boundary value problems in the presence of a boundary layer, Zh. vychisl. mat. i mat. fiz. 1969; 9: 841-859 (in Russian). | ||

[37] | D. Herceg, Uniform fourth order difference scheme for a singularly perturbation problem, Numer. Math. 1090; 56: 675-693. | ||

[38] | E.A. Bogucz, J.D.A. Walker, Fourth-order finite-difference methods for two-point boundary-value problems, IMA J. Numer. Anal. 1984; 4: 69-82. | ||

[39] | M. Cui, F. Geng, A computational method for solving third-order singularly perturbed boundary-value problems, Applied Mathematics and Computation 2008;198: 896-903. | ||

[40] | Vivek Kumar, Mani Mehra, Wavelet optimized finite difference method using interpolating wavelets for self-adjoint singularly perturbed problems, Journal of Computational and Applied Mathematics 2009; 230: 803-812. | ||

[41] | M.H. Protter, H.F. Weinberger, Maximum Principles in Differential Equations, Prantice-Hall, Inc., Englewood Cliffs, NJ, 1967. | ||

[42] | H.G. Roos, M. Stynes, L. Tobiska, Numerical Methods for Singularly Perturbed Differential Equations, Springer, 1996. | ||

[43] | J.J.H. Miller, E. O'Riordan, I.G. Shishkin, Fitted Numerical Methods for Singular Perturbation Problems, World Scientific, 1996. | ||

[44] | P.A. Farrell, A.F. Hegarty, J.J.H. Miller, E. O'Riordan, I.G. Shishkin, Robust Computational Techniques for Boundary Layers, Chapman and Hall, CRC, 2000. | ||

[45] | M.K. Kadalbajoo, V.K. Aggarwal, Fitted mesh B-spline collocation method for solving self adjoint singularly perturbed boundary value problems, Appl. Math. Comput.2005; 161 (3): 973-987. | ||

[46] | J.M.S. Lubuma, K.C. Patidar, Uniformly convergent non-standard finite difference methods for self-adjoint singular perturbation problems, J. Comput. Appl. Math. 2006; 191: 228-238. | ||

[47] | L. Jameson, A wavelet-optimized, very high order adaptive grid and numerical method, SIAM J. Sci. Comput. 1998; 19: 1980-2013. | ||

[48] | V. Kumar, M. Mehra, Cubic spline adaptive wavelet scheme to solve singularly perturbed reaction diffusion problems, Int. J. Wavelets Multiresoult. Inf. Process. 2007; 5 (2): 317-331. | ||

[49] | Manoj Kumar, Hradyesh Kumar Mishra, Peetam Singh, A boundary value approach for a class of linear singularly perturbed boundary value problems, Advances in Engineering Software 2009; 40: 298-304. | ||

[50] | Fazhan Geng, A novel method for solving a class of singularly perturbed boundary value problems based on reproducing kernel method. Applied Mathematics and Computation 2011; 218: 4211-4215. | ||

[51] | Djordje Herceg, Fourth-order finite-difference method for boundary value problems with two small parameters. Applied Mathematics and Computation 2011; 218: 616-627. | ||

[52] | N. Levinson, A boundary value problem for a singularly perturbed differential equation, Duke Math. J. 1958; 25: 331-343. | ||

[53] | D.M. Goecke, Third-order differential inequalities and singular perturbations, Doctoral Dissertation, Univ. Oklahoma, Norman, 1979. | ||

[54] | F.A. Howes, Differential inequalities of higher order and the asymptotic solution of nonlinear boundary value problems, SIAM J. Appl. Math. 1982; 13: 61-80. | ||

[55] | F.A. Howes, The asymptotic solution of a class of third-order boundary value problem arising in the theory of thin film flow, SIAM J. Appl. Math.1983; 43: 993-1004. | ||

[56] | Jihuan He, A simple perturbation approach to Blasius equation, Appl. Math. Comput. 2003; 140: 217-222. | ||

[57] | Weili Zhao, Singular perturbations of boundary value problems for a class of third-order nonlinear ordinary differential equations, J. Differen. Equat. 1990; 88: 265-278. | ||

[58] | M. Feckon, Singularly perturbed high order boundary value problems, J. Differen. Equat. 1994; 11: 79-102. | ||

[59] | Zengji Du, Weigao Ge, Mingru Zhou, Singular perturbations for third-order nonlinear multi-point boundary value problem, J. Differen. Equat. 2005; 218 (1): 69-90. | ||

[60] | S. Valarmathi, N. Ramanujam, An asymptotic numerical method for singular perturbed third-order differential equations of convection–diffusion type, Comput. Math. Appl. 2002; 44: 693-710. | ||

[61] | Xiaojie Lin, Singular perturbations of third-order nonlinear differential equations with full nonlinear boundary conditions, Applied Mathematics and Computation, 2013; 224:88-95. | ||

[62] | Zengji Du, Singularly perturbed third-order boundary value problem for nonlinear systems,applied Mathematics and Computation 2007;189 : 869-877. | ||

[63] | Tzu-Chu Lin, David H. Schultz, Weiqun Zhang, Numerical solutions of linear and nonlinear singular perturbation problems, Computers and Mathematics with Applications 2008; 55: 2574-2592. | ||

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

[65] | Mohan K. Kadalbajoo, Devendra Kumar, 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 2008; 202: 738-746. | ||

[66] | F.D. Van Niekerk, Non-linear one-step methods for initial value problems, Comput. Math. Appl. 1987; 13: 367-371. | ||

[67] | Zhiming Wang, Wuzhong Lin, Gexia Wang, Differentiability and its asymptotic analysis for nonlinear singularly perturbed boundary value problem Nonlinear Analysis 2008; 69: 2236-2250. | ||

[68] | Mohan K. Kadalbajoo, Devendra Kumar, Initial value technique for singularly perturbed two point boundary value problems using an exponentially fitted finite difference scheme, Computers and Mathematics with Applications 2009; 57: 1147-1156. | ||

[69] | H.G. Roos, A second order monotone upwind scheme, Computing 1986; 36: 57-67. | ||

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

[71] | Chein-Shan Liu, The Lie-group shooting method for solving nonlinear singularly perturbed boundary value problems, Commun Nonlinear Sci Numer Simulat 2012;17:1506-1521. | ||

[72] | R.E. O'Mally, Singular Perturbation Methods for Ordinary Differential Equations, Springer, New York, 1991. | ||

[73] | Basem S. Attili, Numerical treatment of singularly perturbed two point boundary value problems exhibiting boundary layers Commun Nonlinear Sci Numer Simulat 2011; 16: 3504-3511. | ||

[74] | G.M. Amiraliyev, Mustafa Kudu, Hakki Duru, Uniform difference method for a parameterized singular perturbation problem. Applied Mathematics and Computation2006; 175: 89-100. | ||

[75] | K. Zawischa, Uber die Differentialgleichung deren Losungskurve durch zwei gegebene Punkte hindurchgehen soll, Monatsh. Math. Phys. 1930; 37: 103-124. | ||

[76] | T. Pomentale, A constructive theorem of existence and uniqueness for problem y0 = f(x,y,k), y(a) = a, y(b) = b, Z. Angew. Math. Mech. 1976; 56: 387-388. | ||

[77] | M. Feckan, Parametrized singularly perturbed boundary value problems, J. Math. Anal. Appl.1994; 188:426-435. | ||

[78] | T. Jankowski, Monotone iterations for differential problems, Math. Notes, Miscolc 2001; 2: 31-38. | ||

[79] | T. Jankowski, V. Lakshmikantham, Monotone iterations for differential equations with a parameter, J. Appl. Math. Stoch. Anal. 1997; 10: 273-278. | ||

[80] | M. Ronto, T. Csikos-Marinets, On the investigation of some non-linear boundary value problems with parameters, Math. Notes, Miscolc 2000; 1: 157-166. | ||

[81] | S. Stanek, Nonlinear boundary value problem for second order differential equations depending on a parameter, Math. Slovaca 1997; 47: 439-449. | ||

[82] | A. Gulle, H. Duru, Convergence of the iterative process to the solution of the boundary problem with the parameter, Trans. Acad. Sci. Azerb., Ser. Phys. Tech. Math. Sci. 1998; 18: 34-40. | ||

[83] | A.H. Nayfeh, Introduction to Perturbation Techniques, Wiley, New York, 1993. | ||

[84] | E.R. Doolan, J.J.H. Miller, W.H.A. Schilders, Uniform Numerical Methods for Problems with Initial and Boundary Layers, Boole, Press, Dublin, 1980. | ||

[85] | G.M. Amiraliyev, Difference method for the solution one problem of the theory dispersive waves, USSR Diff. Equat. 1990; 26: 2146-2154. | ||

[86] | G.M. Amiraliyev, H. Duru, A uniformly convergent finite difference method for a singularly perturbed initial value problem, Appl. Math. Mech. (English Edition) 1999; 20: 379-387. | ||

[87] | G.M. Amiraliyev, H. Duru, A uniformly convergent difference method for the periodical boundary value problem, Comput. Math. Appl.2003; 46:695-703. | ||

[88] | Feng Xie, JianWang, Weijiang Zhang, Ming He, A novel method for a class of parameterized singularly perturbed boundary value problems Journal of Computational and Applied Mathematics 2008; 213: 258-267. | ||

[89] | M.Turkyilmazoglu, Analytic approximate solutions of parameterized unperturbed and singularly perturbed boundary value problems. Applied Mathematical Modelling 2011, 35: 3879-3886. | ||

[90] | G.M. Amiraliyev, H. Duru, A note on a parameterized singular perturbation problem, J. Comput. Appl. Math. 2005; 182: 233-242. | ||

[91] | W.K. Zahra, Ashraf M. El Mhlawy, Numerical solution of two-parameter singularly perturbed boundary value problems via exponential spline, Journal of King Saud University - Science, 2013; 25: 201-208. | ||

[92] | C.G. Lange, R.M. Miura, Singular perturbation analysis of boundary-value problems for differential-difference equations. v. small shifts with layer behavior, SIAM J. Appl. Math. 1994; 54: 249-272. | ||

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

[94] | M. Wazewska-Czyzewska, A. Lasota, Mathematical models of the red cell system, Mat. Stos. 1976; 6: 25-40. | ||

[95] | M.C. Mackey, G.L. Oscillations and chaos in physiological control systems, Science, 1977; 197: 287-289. | ||

[96] | A. Longtin, J. Milton, Complex oscillations in the human pupil light reflex with mixed and delayed feedback, Math. Biosci. 1988; 90: 183-199. | ||

[97] | V.Y. Glizer, Asymptotic analysis and solution of a finite-horizon H1 control problem for singularly-perturbed linear systems with small state delay, J. Optim. Theory Appl. 2003; 117: 295-325. | ||

[98] | H. Tian, Numerical treatment of singularly perturbed delay differential equations, PhD thesis, University of Manchester, 2000. | ||

[99] | 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.1994; 54: 273-283. | ||

[100] | M.K. Kadalbajoo, K.K. Sharma, Numerical analysis of singularly perturbed delay differential equations with layer behavior, Appl. Math. Comput. 2004; 157: 11-28. | ||

[101] | Mohan K. Kadalbajoo, Kapil K. Sharma,, A numerical method based on finite difference for boundary value problems for singularly perturbed delay differential equations, Applied Mathematics and Computation 2008;197: 692-707. | ||

[102] | Mohan K. Kadalbajoo, Devendra Kumar, A computational method for singularly perturbed nonlinear differential-difference equations with small shift. Applied Mathematical Modelling 2010; 34: 2584-2596. | ||

[103] | Gabil M. Amiraliyev, Erkan Cimen, Numerical method for a singularly perturbed convection–diffusion problem with delay Applied Mathematics and Computation 2010; 216: 2351-2359. | ||

[104] | I.G. Amiraliyeva, F. Erdogan, G.M. Amiraliyev, A uniform numerical method for dealing with a singularly perturbed delay initial value problem Applied Mathematics Letters 2010; 23: 1221-1225. | ||

[105] | G.M. Amiraliyev, Y.D. Mamedov, Difference schemes on the uniform mesh for singularly perturbed pseudo-parabolic equations, Tr. J. Math. 1995; 19: 207-222. | ||

[106] | G.M. Amiraliyev, H. Duru, A uniformly convergent finite difference method for a initial value problem, Appl. Math. Mech.1999; 20 (4):363-370. | ||

[107] | Pratima Rai, Kapil K. Sharma, Numerical analysis of singularly perturbed delay differential turning point problem. Applied Mathematics and Computation 2011; 218: 3483-3498. | ||

[108] | Pratima Rai, Kapil K. Sharma, Numerical study of singularly perturbed differential–difference equation arising in the modeling of neuronal variability, International Journal of computer mathematics with applications, 2012; 63 (1): 118-132. | ||

[109] | R. Nageshwar Rao, P. Pramod Chakravarthy, A finite difference method for singularly perturbed differential-difference equations arising from a model of neuronal variability, Journal of Taibah University for Science,2013; 73: 128-136. | ||

[110] | Carol Shubin, Singularly perturbed integral equations, J. Math. Anal. Appl.2006; 313: 234-250. | ||

[111] | C. Lange, D. Smith, Singular perturbation analysis of integral equations, Stud. Appl. Math. 1988; 79: 1-63. | ||

[112] | C. Lange, D. Smith, Singular perturbation analysis of integral equations, Part II, Stud. Appl. Math. 1993; 90: 1-74. | ||

[113] | G. Eskin, Asymptotics of solutions of elliptic pseudo-differential equations with a small parameter, Dokl. Akad. Nauk USSR 211 (1973) 547-550. English translation: Soviet Math. Dokl. 1973; 14: 1080-1083. | ||

[114] | G.M. Amiraliyev, I.G. Amiraliyeva, Mustafa Kudu, A numerical treatment for singularly perturbed differential equations with integral boundary condition Applied Mathematics and Computation 2007; 185: 574-582. | ||