A numerical framework for solving high-order pantograph-delay Volterra integro-differential equations

Authors

  • Farshid Mirzaee Malayer University
  • Saeed Bimesl Malayer University
  • Emran Tohidi Department of Mathematics, Islamic Azad University, Zahedan Branch, Zahedan

Keywords:

Accuracy, euler polynomials, high-order, integro-differential equations, pantograph-delay.

Abstract

In this paper, we present an efficient numerical method to solve Volterra integrodifferentialequations of pantograph-delay type. We use the Euler polynomials toapproximate the solutions. The proposed method is discussed in detail and comparedby solving some numerical examples. Moreover, the error estimates of the proposedmethod is given. Special cases of the main results are also mentioned.

References

Abubakar, A. & Taiwo, O. (2014) Integral collocation approximation methods for the numerical solution

of high-orders linear Fredholm-Volterra integro-differential equations. American Journal of

Computational and Applied Mathematics, 4(4):111-117.

Ali, I., Brunner, H. & Tang, T. (2009) A spectral method for pantograph-type delay differential equations

and its convergence analysis. Journal of Computational Mathematics, 27:254-265.

Bhrawy, A.H., Tohidi, E. & Soleymani, F. (2012) A new Bernoulli matrix method for solving high-order

linear and nonlinear Fredholm integro-differential equations with piecewise intervals. Applied

Mathematics and Computation, 219:482-497.

Bellour, A. & Bousselsal, M. (2014) Numerical solution of delay integro-differential equations by using

Taylor collocation method. Mathematical Methods in Applied Science, 37:1491-1506.

Evans, D.J. & Raslan, K.R. (2005) The Adomian decomposition method for solving delay differential

equation. International Journal of Computer Mathematics, 82(1):49-54.

Fazeli, S. & Hojjati, G. (2015) Numerical solution of Volterra integro-differential equations by super

implicit multistep collocation methods. Numerical Algorithms, 68(4):741-768.

Ghany, H.A. & Hyder, A.A. (2014) Exact solutions for the wick-type stochastic time-fractional Kdv

equations. Kuwait Journal of Science, 41(1):75-84.

Gokdogan, A. & Merdan, M. (2013) Adaptive multi-step differential transformation method to solve

ODE systems. Kuwait Journal of Science, 40(1):33-35.

Gülsu, M. & Sezer, M. (2011) A collocation approach for the numerical solution of certain linear retarded

and advanced integro-differential equations with linear functional arguments. Numerical Methods

for Partial Differential Equations, 27(2):447-459.

Heydari, M., Loghmani, G.B. & Hosseini, S.M. (2013) Operational matrices of Chebyshevcardinal

functions and their application for solving delay differential equations arising in electrodynamics

with error estimation. Applied Mathematical Modelling, 37:7789-7809.

Horvat, V. (1999) On Polynomial spline collocation methods for neutral Volterr integro-differential

equations with delay arguments. Proceedings of the 1. Conference on Applied Mathematics and

Computation, 13:113-128.

Jerri, A. (1999) Introduction to integral equations with applications. Wiley, New York.

Jiang, Y. & Ma, J. (2013) Spectral collocation methods for Volterra-integro differential equations with

noncompact kernels. Journal of Computational and Applied Mathematics, 244:115-124.

Jiang, W. & Tian, T. (2015) Numerical solution of nonlinear Volterra integro-differential equations of

fractional order by the reproducing kernel method. Applied Mathematical Modelling, 39(16):4871-

Kanwal, R.P. & Liu, K.C. (1989) A Taylor expansion approach for solving integral equations.

International Journal of Mathematical Education in Science and Technology, 20(3):411-414.

Keskin, Y., Kurnaz, A., Kiris, M.E. & Oturance, G. (2007) Approximate solutions of generalized

pantograph equations by the differential transform method. International Journal of Nonlinear

Sciences and Numerical Simulation, 8:159-164.

Liu, J. & Jiang, Y.L. (2013) Convergence analysis of an Arnoldi order reduced Runge-Kutta method

for integro-differential equations of pantograph type. Applied Mathematics and Computation,

:11460-11470.

Lü, X. & Cui, M. (2008) Analytic solutions to a class of nonlinear infinite-delay-differential equations.

Journal of Mathematical Analysis and Applications, 343:724-732.

Mirzaee, F. & Bimesl, S. (2014) Application of Euler matrix method for solving linear and aclass of

nonlinear Fredholm integro-differentia equations. Mediterranean Journal of Mathematics, 11:999-

Mirzaee, F. & Bimesl, S. (2015) Solving systems of high-order linear differential-difference equations via

Euler matrix method. Journal of the Egyptian Mathematical Society, 23:286-291.

Muroya, Y., Ishiwata, E. & Brunner, H. (2003) On the attainable order of collocation methods for

pantograph integro-differential equations. Journal of Computational and Applied Mathematics,

:347-366.

Nas, S., Yalçinbas, S. & Sezer, M. (2000) A Taylor polynomial approach for solving high-order linear

Fredholm integro-differential equations. International Journal of Mathematical Education in Science

and Technology, 31(2):213-225.

Ockendon, J.R., Tayler, A.B. (1971) The dynamics of a current collection system for an electric

locomotive. Proceedings of the Royal Society, A 322:447-468.

Spiridonov, V. (1995) Universal superpositions of coherent states and self-similar potentials. Physical

Review Letters, 52:1909-1935.

Tang, T., Xu, X. & Cheng, J. (2008) On spectral methods for Volterra integral equations and the

convergence analysis. Journal of Computational Mathematics, 26:825-837.

Tohidi, E. & Kiliçman, A. (2014) An efficient spectral approximation for solving several types of

parabolic PDEs with nonlocal boundary conditions. Mathematical Problems in Enginering, 2014.

doi: 10.1155/20142014//369029.

Yalçinbas, S. & Sezer, M. (2000) The approximate solution of high-order linear Volterra Fredholm integrodifferential

equations in terms of Taylor polynomials. Applied Mathematics and Computation,

:291-308.

Yi, L. & Wang, Z. (2014) Legendre-Gauss spectral collocation method for second order nonlinear delay

differential equations. Numerical Mathematics: Theory, Methods and Applications, 7(2):149-178.

Yi, M. & Huang, J. (2015) CAS wavelet method for solving the fractional integro-differential equation

with a weakly singular kernel. International Journal of Computer Mathematics, 92(8):1715-1728.

Yüzbasi, S. (2014) Laguerre approach for solving pantograph-type Volterraintegro-differential equations.

Applied Mathematics and Computation, 232:1183-1199.

Yüzbasi, S. & Sezer, M. (2013) An exponential approximation for solutions of generalized pantographdelay

differential equations. Applied Mathematical Modelling, 37:9160-9173.

Downloads

Published

08-02-2016