American Institute of Mathematical Sciences

Advanced
2013, 2013(special): 355-363. doi: 10.3934/proc.2013.2013.355

Nonpolynomial spline finite difference scheme for nonlinear singuiar boundary value problems with singular perturbation and its mechanization

Navnit Jha 1,

1. 

Department of Mathematics, South Asian University, Akbar Bhawan, Chanakyapuri, New Delhi-110021, India

Received  September 2012 Published  November 2013

A general scheme for the numerical solution of nonlinear singular perturbation problems using nonpolynomial spline basis is proposed in the paper. The special non-equidistant formulation of mesh takes into account the boundary and interior layer structures. The proposed scheme is almost fourth order accurate and applicable to both singular and nonsingular cases. Convergence analysis of the scheme is briefly discussed. Maple program for the generation of difference scheme is presented. Computational illustrations characterized by boundary and interior layers show that the practical order of accuracy is close to the theoretical order of the method.
Keywords: reaction-convection-diffusion, finite difference, compact operator., nonpolynomial splines, Singular perurbation.
Mathematics Subject Classification: Primary: 65L10, 65L12; Secondary: 34B1.
Citation: Navnit Jha. Nonpolynomial spline finite difference scheme for nonlinear singuiar boundary value problems with singular perturbation and its mechanization. Conference Publications, 2013, 2013 (special) : 355-363. doi: 10.3934/proc.2013.2013.355
References:
[1]

A. H. Nayfeh, "Introduction to Perturbation Technique," A Wiley-Interscience Publication. Wiley-Interscience [John Wiley & Sons], New York, 1981.

[2]

K. W. Chang and F. A. Howes, "Nonlinear Singular Perturbation Phenomena: Theory and Applications," Applied Mathematical Sciences, 56. Springer-Verlag, New York , 1984.

[3]

J. Kevorkian and J. D. Cole, "Multi Scale and Singular Perturbation Methods," Applied Mathematical Sciences, 114. Springer-Verlag, New York , 1996.

[4]

E.O'Riordan and M. Stynes, A uniformly accurate finite element method for a singularly perturbed one-dimensional reaction diffusion problem, Math. Comput., 47 (1986), 555-570.

[5]

R. Vulanovic, Fourth order algorithms for semilinear singular perturbation problems, Numer. Algorithms, 16 (1997), 117-128.

[6]

R. K. Mohanty, N. Jha and D. J. Evans, capitalized., Spline in compression method for the numerical solution of singularly perturbed two point singular boundary value problems, Int. J. Comput. Math., 81 (2004), 615-627.

[7]

M. Kumar, P. Singh and H. K. Mishra, capitalized., An initial value technique for singularly perturbed boundary value problems via cubic spline, Int. J. Comput. Meth. Eng. Sc. Mech., 8 (2007), 419-427.

[8]

C. Y. Jung and R. Temam, Finite volume approximation of one dimensional stiff convection-diffusion equation, J. Sci. Comput., 41 (2009), 384-410.

[9]

R. Lin, A robust finite element method for singularly perturbed convection-diffusion problems, Discrete Contin. Dyn. Syst., 9 (2009), 496-505.

[10]

B. Lin, K. Li and Z. Cheng, B-spline solution of a singularly perturbed boundary value problem arising in biology, Chaos, Solitons Fractals, 42 (2009), 2934-2948.

[11]

F. Xie, On a class of singular boundary value problems with singular perturbation, J. Differential Equations, 252 (2012), 2370-2387.

[12]

I. A. Tirmizi, F. I. Haq and S. I. Islam, capitalized., Nonpolynomial spline solution of singularly perturbed boundary value problems, Appl. Math. Comput., 196 (2008), 6-16.

[13]

L. K. Bieniasz, Two new compact finite difference schemes for the solution of boundary value problems in second order nonlinear ordinary differential equations, using non-uniform grids, J. Comput. Methods Sci. Eng., 8 (2008), 3-18.

[14]

R. K. Mohanty, A class of non-uniform mesh three point arithmetic average discretizations for y"=f(x,y,y') and the estimates of y', Appl. Math. Comput., 183 (2006), 477-485.

[15]

A. Khan, I. Khan and T. Aziz, Sextic spline solution of a singularly perturbed boundary value problems, Appl. Math. Comput., 181 (2006), 432-439.

[16]

M. K. Kadalbajoo and R. K. Bawa, Variable mesh difference scheme for singularly perturbed boundary value problems using splines, J. Optim. Theory Appl., 90 (1996), 405-416.

[17]

M. C. Natividad and M. Stynes, Richardson extrapolation for a convection-diffusion problem using a Shishkin mesh, Appl. Numer. Math., 45 (2003), 315-329.

[18]

C. E. Pearson, On non-linear ordinary differential equations of boundary layer type, J. Math. Phy., 47 (1968), 351-358.

[19]

M. K. Kadalbajoo and K. C. Patidar, Numerical solution of singularly perturbed nonlinear two point boundary value problems by spline in compression, Int. J. Comput. Math., 79 (2002), 271-288.

show all references

References:
[1]

A. H. Nayfeh, "Introduction to Perturbation Technique," A Wiley-Interscience Publication. Wiley-Interscience [John Wiley & Sons], New York, 1981.

[2]

K. W. Chang and F. A. Howes, "Nonlinear Singular Perturbation Phenomena: Theory and Applications," Applied Mathematical Sciences, 56. Springer-Verlag, New York , 1984.

[3]

J. Kevorkian and J. D. Cole, "Multi Scale and Singular Perturbation Methods," Applied Mathematical Sciences, 114. Springer-Verlag, New York , 1996.

[4]

E.O'Riordan and M. Stynes, A uniformly accurate finite element method for a singularly perturbed one-dimensional reaction diffusion problem, Math. Comput., 47 (1986), 555-570.

[5]

R. Vulanovic, Fourth order algorithms for semilinear singular perturbation problems, Numer. Algorithms, 16 (1997), 117-128.

[6]

R. K. Mohanty, N. Jha and D. J. Evans, capitalized., Spline in compression method for the numerical solution of singularly perturbed two point singular boundary value problems, Int. J. Comput. Math., 81 (2004), 615-627.

[7]

M. Kumar, P. Singh and H. K. Mishra, capitalized., An initial value technique for singularly perturbed boundary value problems via cubic spline, Int. J. Comput. Meth. Eng. Sc. Mech., 8 (2007), 419-427.

[8]

C. Y. Jung and R. Temam, Finite volume approximation of one dimensional stiff convection-diffusion equation, J. Sci. Comput., 41 (2009), 384-410.

[9]

R. Lin, A robust finite element method for singularly perturbed convection-diffusion problems, Discrete Contin. Dyn. Syst., 9 (2009), 496-505.

[10]

B. Lin, K. Li and Z. Cheng, B-spline solution of a singularly perturbed boundary value problem arising in biology, Chaos, Solitons Fractals, 42 (2009), 2934-2948.

[11]

F. Xie, On a class of singular boundary value problems with singular perturbation, J. Differential Equations, 252 (2012), 2370-2387.

[12]

I. A. Tirmizi, F. I. Haq and S. I. Islam, capitalized., Nonpolynomial spline solution of singularly perturbed boundary value problems, Appl. Math. Comput., 196 (2008), 6-16.

[13]

L. K. Bieniasz, Two new compact finite difference schemes for the solution of boundary value problems in second order nonlinear ordinary differential equations, using non-uniform grids, J. Comput. Methods Sci. Eng., 8 (2008), 3-18.

[14]

R. K. Mohanty, A class of non-uniform mesh three point arithmetic average discretizations for y"=f(x,y,y') and the estimates of y', Appl. Math. Comput., 183 (2006), 477-485.

[15]

A. Khan, I. Khan and T. Aziz, Sextic spline solution of a singularly perturbed boundary value problems, Appl. Math. Comput., 181 (2006), 432-439.

[16]

M. K. Kadalbajoo and R. K. Bawa, Variable mesh difference scheme for singularly perturbed boundary value problems using splines, J. Optim. Theory Appl., 90 (1996), 405-416.

[17]

M. C. Natividad and M. Stynes, Richardson extrapolation for a convection-diffusion problem using a Shishkin mesh, Appl. Numer. Math., 45 (2003), 315-329.

[18]

C. E. Pearson, On non-linear ordinary differential equations of boundary layer type, J. Math. Phy., 47 (1968), 351-358.

[19]

M. K. Kadalbajoo and K. C. Patidar, Numerical solution of singularly perturbed nonlinear two point boundary value problems by spline in compression, Int. J. Comput. Math., 79 (2002), 271-288.

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