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