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.  Google Scholar

[2]

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

[3]

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

[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.  Google Scholar

[5]

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

[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.  Google Scholar

[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.  Google Scholar

[8]

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

[9]

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

[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. Google Scholar

[11]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

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

[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.  Google Scholar

show all references

References:
[1]

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

[2]

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

[3]

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

[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.  Google Scholar

[5]

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

[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.  Google Scholar

[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.  Google Scholar

[8]

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

[9]

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

[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. Google Scholar

[11]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

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

[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.  Google Scholar

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