Approximate Solution of Multi-Pantograph Equations With Variable Coefficients via Collocation Method Based on Hermite Polynomials

Authors

  • Dianchen Lu School of Sciences, Jiangsu University, Zhenjiang, Jiangsu
  • Chen Yuan School of Sciences, Jiangsu University, Zhenjiang, Jiangsu
  • Rabia Mehdi Department of Mathematics, Gomal University, Dera Ismail Khan
  • Shamoona Jabeen School of Mathematics and System Sciences, Beihang University, Beijing
  • Abdur Rashid 5School of Sciences, Jiangsu University, Zhenjiang, Jiangsu

DOI:

https://doi.org/10.26713/cma.v9i4.844

Keywords:

Multi-Pantograph equation, Collocation method, Hermite polynomials, Matrix equation, Approximate results, Accuracy

Abstract

This research article presents an approximate solution of the non-homogenous Multi-Pantograph equation comprising of variable coefficients by utilizing a collocation method based on Hermite polynomials. These orthogonal polynomials along with its collocation points transform the equation and the initial conditions into matrix equation comprising of a system of linear algebraic equations. Subsequently, by solving this system, the unknown Hermite coefficients are calculated. To reveal the accuracy and efficiency of the method applied, the approximate results obtained by this technique have been compared with exact solutions. Moreover, some numerical illustrations in the form of examples are given to exhibit the applicability of the proposed technique.

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References

M. Arnold and B. Simeon, Pantograph and catenary dynamics: a benchmark problem and its numerical solution, Applied Numerical Mathematics 34(4) (2000), 345 – 362, DOI: 10.1016/S0168-9274(99)00038-0.

F. Awawdeh, A. Adawi and S. Al-Shara, Analytic solution of multipantograph equation, Advances in Decision Sciences 2008 (2008), Article ID 760191, 12 pages.

G. Derfel and A. Iserles, The pantograph equation in the complex plane, Journal of Mathematical Analysis and Applications 213 (1997), 117 – 132, DOI: 10.1006/jmaa.1997.5483.

G.A. Derfel and F. Vogl, On the asymptotics of solutions of a class of linear functional-differential equations, European Journal of Applied Mathematics 7 (1996), 511 – 518.

E.H. Doha, A.H. Bhrawy, D. Baleanu and R.M. Hafez, A new Jacobi rational-Gauss collocation method for numerical solution of generalized pantograph equations, Applied Numerical Mathematics 77 (2014), 43 – 54, DOI: 10.1016/j.apnum.2013.11.003.

D.J. Evens and K.R. Raslan, The adomian decomposition method for solving delay differential equation, International Journal of Computational Mathematics 82 (2005), 49 – 54, DOI: 10.1080/00207160412331286815.

X. Feng, An analytic study on the multi-pantograph delay equations with variable coefficients, Bulletin mathématique de la Société des Sciences Mathématiques de Roumanie 56 (2013), 205 – 215.

D. Funaro, Polynomial Approximations of Differential Equations, Springer-Verlag (1992).

B. Ibis and M. Bayram, Numerical solution of the neutral functional-differential equations with proportional delays via collocation method based on Hermite polynomials, Communication in Mathematical Modeling and Applications 3 (2016), 22 – 30.

A. Iserles, On the generalized pantograph functional-differential equation, European Journal of Applied Mathematics 4 (1993), 1 – 38.

M.A. Jafari and A. Aminataei, Method of successive approximations for solving the multipantograph delay equations, Gen. Math. Notes 8 (2012), 23 – 28.

Y. Keskin, A. Kurnaz, M.E. Kiris and G. Oturanc, Approximate solutions of generalized pantograph equations by the differential transform method, International Journal of Nonlinear Sciences and Numerical Simulation 8 (2007), 159 – 167.

D. Li and M. Liu, Runge-Kutta methods for the multi-pantograph delay equation, Journal of Applied Mathematics and Computation 163 (2005), 383 – 395, DOI: 10.1016/j.amc.2004.02.013.

M. Liu and D. Li, Properties of analytic solution and numerical solution of multipantograph equation, Journal of Applied Mathematics and Computation 155 (2004), 853 – 871, DOI: 10.1016/j.amc.2003.07.017.

Y. Muroya, E. Ishiwata and H. Brunner, On the attainable order of collocation methods for pantograph integro-differential equations, Journal of Computational and Applied Mathematics 152 (2003), 347 – 366.

J.R. Ockendon and A.B. Tayler, The dynamics of a current collection system for an electric locomotive, Proc. Roy. Soc. London Ser. A 322 (1971), 447 – 468.

M. Sezer and A. Aky¸segül-Da¸scioˇglu, A Taylor method for numerical solution of generalized pantograph equations with linear functional argument, Journal of Computational and Applied Mathematics 200 (2007), 217 – 225, DOI: 10.1016/j.cam.2005.12.015.

M. Sezer, S. Yalcinbas and N. Sahina, Approximate solution of multi-pantograph equation with variable coefficients, Journal of Computational and Applied Mathematics 214 (2008), 406 – 416, DOI: 10.1016/j.cam.2007.03.024.

J. Shen, T. Tang and L.L. Wang, Spectral Methods Algorithms, Analysis and Applications, Springer (2011).

E. Tohidi, A.H. Bhrawy and K. Erfani, A collocation method based on Bernoulli operational matrix for numerical solution of generalized pantograph equation, Applied Mathematical Modelling 37 (2013), 4283 – 4294, DOI: 10.1016/j.apm.2012.09.032.

G.C.Wake, S. Cooper, H.-K. Kim and B. Van-Brunt, Functional differential equations for cell-growth models with dispersion, Communications in Applied Analysis 4 (2000), 561 – 573.

S. Yüzba¸si, N. S and M. Sezer, A Bessel collocation method for numerical solution of generalized pantograph equations, Numerical Methods for Partial Differential Equations 28 (2012), 1105 – 1123, DOI: https://doi.org/10.1002/num.20660.

S. Yalsinbas, M. Aynigül and M. Sezer, A collocation method using Hermite polynomials for approximate solution of pantograph equations, Journal of the Franklin Institute 348 (2011), 1128 – 1139, DOI: http://dx.doi.org/10.1016/j.jfranklin.2011.05.003.

Z. Yu, Variational iteration method for solving the multi-pantograph delay equation, Physics Letter A 372 (2008), 6475 – 6479, DOI: 10.1016/j.physleta.2008.09.013.

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Published

25-12-2018
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How to Cite

Lu, D., Yuan, C., Mehdi, R., Jabeen, S., & Rashid, A. (2018). Approximate Solution of Multi-Pantograph Equations With Variable Coefficients via Collocation Method Based on Hermite Polynomials. Communications in Mathematics and Applications, 9(4), 601–614. https://doi.org/10.26713/cma.v9i4.844

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Research Article