Gaussian Quadrature for Two-Point Singularly Perturbed Boundary Value Problems with Exponential Fitting

K. Phaneendra, M. Lalu


In this paper, the Gaussian quadrature method with exponential fitting is proposed for the solution of two-point singularly perturbed boundary value problems with layer at one endpoint, dual boundary layers and internal boundary layers. The given boundary value problem is reduced into an equivalent first order differential equation with the perturbation parameter as deviating argument. Then, Gaussian two-point quadrature technique with exponential fitting is implemented to solve the first order equation with deviating parameter. The analysis of the convergence of the method is discussed. Several numerical examples are illustrated with a layer at one end, a layer at two ends and internal layers. Comparison of maximum errors in the solution of the examples with other methods is shown to justify the method.


Singular perturbation problem; Boundary layer; Gaussian quadrature; Dual layer; Internal layer

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B. S. Attili, Numerical treatment of singularly perturbed two point boundary value problems exhibiting boundary layers, Communications in Nonlinear Science and Numerical Simulation 16 (2011), 3504 – 3511, DOI: 10.1016/j.cnsns.2011.01.001.

A. Awoke and Y. N. Reddy, An exponentially fitted special second-order finite difference method for solving singular perturbation problems, Applied Mathematics and Computation 190 (2007), 1767 – 1782, DOI: 10.1016/j.amc.2007.02.051.

T. Aziz and A. Khan, A spline method for second-order singularly perturbed boundary value problems, Journal of Computational and Applied Mathematics 147 (2002), 445 – 452, DOI: 10.1016/S0377-0427(02)00479-X.

R. K. Bawa, Spline based computational technique for linear singularly perturbed boundary value problems, Applied Mathematics and Computation 167 (2005), 225 – 236 DOI: 10.1016/j.amc.2004.06.112.

C. M. Bender and S. A. Orszag, Advanced Mathematical Methods for Scientists and Engineers, McGraw-Hill, New York (1978).

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

P. Gupta and M. Kumar, Multiple-scales method and numerical simulation of singularly perturbed boundary layer problems, Appl. Math. Inf. Sci. 10 (2016), 1119 – 1127, DOI: 10.18576/amis/100330.

M. K. Jain, Numerical Solution of Differential Equations, Wiley Eastern, New Delhi (1984).

M. K. Kadalbajoo and K. C. Patidar, Uniformly convergent fitted mesh finite difference methods for general singular perturbation problems, Applied Mathematics and Computation 179 (2006), 248 – 266, DOI: 10.1016/j.amc.2005.11.096.

J. J. H. Miller, E. O’Riordan and G. I. Shishkin, 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), DOI: 10.1142/8410.

R. Mohammadi, Numerical solution of general singular perturbation boundary value problems based on adaptive cubic spline, TWMS Jour. Pure Appl. Math. 3 (2012), 11 – 21.

S. Natesan, J. Jayakumar and J. Vigo-Aguiar, Parameter uniform numerical method for singularly perturbed turning point problems exhibiting boundary layers, J. Comput. Appl. Math. 158 (2003), 121 – 134, DOI: 10.1016/S0377-0427(03)00476-X.

R. E. O’Malley, Introduction to Singular Perturbations, Academic Press, New York (1974).

K. Phaneendra, S. Rakmaiah and M. Chenna Krishna Reddy, Numerical treatment of singular perturbation problems exhibiting dual boundary layers, Ain Shams Engineering Journal 6 (2015), 1121 – 1127, DOI: 10.1016/j.asej.2015.02.012.

Y. N. Reddy, A numerical integration method for solving singular perturbation problems, Applied Mathematics and Computation 37 (1990), 83 – 95, DOI: 10.1016/0096-3003(90)90037-4.

G. Roos, M. Stynes and L. Tobiska, Numerical Methods for Singularly Perturbed Differential Equations, Springer Verlag, Berlin (1996), DOI: 10.1007/978-3-662-03206-0.

M. M. Shahraki and S. M. 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 182 (2006), 460 – 473, DOI: 10.1016/j.amc.2006.04.007.

G. B. S. L. Soujanya, Y. N. Reddy and K. Phaneendra, A fitted Galerkin method for singularly perturbed differential equations with layer behaviour, International Journal of Applied Science and Engineering 9 (2011), 195 – 206.



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