Quartic Non-polynomial Spline Method for Singularly Perturbed Differential-difference Equation with Two Parameters
Keywords:Quartic Non-polynomial, Differential-difference, Two-parameters, Accurate Solution
Quartic non-polynomial spline method is presented to solve the singularly perturbed differential-difference equation containing two parameters. The considered equation is transformed into an asymptotical equivalent differential equation, and the derivatives are replaced finite difference approximation using the quartic non-polynomial spline method. The convergence analysis of the method has been established. Numerical experimentation is carried out on model examples, and the results are presented both in tables and graphs. Furthermore, the present method gives a more accurate solution than some existing methods reported in the literature.
File, G. and Reddy, Y.N., 2013. Computational method for solving singularly perturbed delay differential equations with negative shift. International Journal of Applied Science and Engineering, 11(1), pp.101-113. DOI: https://doi.org/10.1155/2012/572723
File, G., Gadisa, G., Aga, T. and Reddy, Y.N., 2017. Numerical solution of singularly perturbed delay reaction-diffusion equations with layer or oscillatory behaviour. American Journal of Numerical Analysis, 5(1), pp.1-10.
Gadisa, G., File, G. and Aga, T., 2018. Fourth order numerical method for singularly perturbed delay differential equations. International Journal of Applied Science and Engineering, 15(1), pp.17-32.
Kadalbajoo, M.K. and Kumar, V., 2007. B-spline method for a class of singular two-point boundary value problems using optimal grid. Applied Mathematics and Computation, 188(2), pp.1856-1869. DOI: https://doi.org/10.1016/j.amc.2006.11.050
Kadalbajoo, M.K. and Sharma, K.K., 2004. Numerical analysis of singularly perturbed delay differential equations with layer behavior. Applied Mathematics and Computation, 157(1), pp.11-28. DOI: https://doi.org/10.1016/j.amc.2003.06.012
Kadalbajoo, M.K. and Sharma, K.K., 2008. A numerical method based on finite difference for boundary value problems for singularly perturbed delay differential equations. Applied Mathematics and Computation, 197(2), pp.692-707. DOI: https://doi.org/10.1016/j.amc.2007.08.089
Kadalbajoo, M.K. and Yadaw, A.S., 2008. B-Spline collocation method for a two-parameter singularly perturbed convection–diffusion boundary value problems. Applied Mathematics and Computation, 201(1-2), pp.504-513. DOI: https://doi.org/10.1016/j.amc.2007.12.038
Kadalbajoo, M.K., Gupta, V. and Awasthi, A., 2008. A uniformly convergent B-spline collocation method on a nonuniform mesh for singularly perturbed one-dimensional time-dependent linear convection–diffusion problem. Journal of Computational and Applied Mathematics, 220(1-2), pp.271-289. DOI: https://doi.org/10.1016/j.cam.2007.08.016
Sahu, S.R. and Mohapatra, J., 2019. Parameter uniform numerical methods for singularly perturbed delay differential equation involving two small parameters. International Journal of Applied and Computational Mathematics, 5(5), pp.1-19. DOI: https://doi.org/10.1007/s40819-019-0713-0
Akram, G. and Talib, I., 2014. Quartic non-polynomial spline solution of a third order singularly perturbed boundary value problem. Research Journal of Applied Sciences, Engineering and Technology, 7(23), pp.4859-4863. DOI: https://doi.org/10.19026/rjaset.7.875
Ala'yed, O.H., Ying, T.Y. and Saaban, A., 2015. New fourth order quartic spline method for solving second order boundary value problems. MATEMATIKA: Malaysian Journal of Industrial and Applied Mathematics, 31(2), pp.149-157.
Chakravarthy, P.P., Kumar, S.D., Rao, R.N. and Ghate, D.P., 2015. A fitted numerical scheme for second order singularly perturbed delay differential equations via cubic spline in compression. Advances in Difference Equations, 2015(1), pp.1-14. DOI: https://doi.org/10.1186/s13662-015-0637-x
Chakravarthy, P.P., Kumar, S.D. and Rao, R.N., 2017. An exponentially fitted finite difference scheme for a class of singularly perturbed delay differential equations with large delays. Ain Shams Engineering Journal, 8(4), pp.663-671. DOI: https://doi.org/10.1016/j.asej.2015.09.004
Dugassa, T., File, G. and Aga, T., 2019. Stable Numerical Method for Singularly Perturbed Boundary Value Problems with Two Small Parameters. Ethiopian Journal of Education and Sciences, 14(2), pp.9-27.
Erdogan, F., 2009. An exponentially fitted method for singularly perturbed delay differential equations. Advances in Difference Equations, 2009, pp.1-9. DOI: https://doi.org/10.1155/2009/781579
Siraj, M.K., Duressa, G.F. and Bullo, T.A., 2019. Fourth-order stable central difference with Richardson extrapolation method for second-order self-adjoint singularly perturbed boundary value problems. Journal of the Egyptian Mathematical Society, 27(1), pp.1-14. DOI: https://doi.org/10.1186/s42787-019-0047-4
Zahra, W.K. and El Mhlawy, A.M., 2013. Numerical solution of two-parameter singularly perturbed boundary value problems via exponential spline. Journal of King Saud University-Science, 25(3), pp.201-208. DOI: https://doi.org/10.1016/j.jksus.2013.01.003
Smith, G.D., Smith, G.D. and Smith, G.D.S., 1985. Numerical solution of partial differential equations: finite difference methods. Oxford University Press.
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