Şuayip Yüzbaşi And Nurbol İsmailov YÜZBAŞİ, Nurbol ISMAİLOV

Differential transform method to solve two-dimensional Volterra integral equations with proportional delays

In this paper, the differential transform method is extended by providing a new theorem to two-dimensional Volterra integral equations with proportional delays. The method is useful for both linear and nonlinear equations. If solutions of governing equations can be expanded for Taylor series, then the method gives opportunity determine coefficients Taylor series, i.e. the exact solutions are obtained in series form. In illustrate examples the method applying to a few type equations.

Keywords:

Two-dimensional Volterra integral equations with proportional delays, partial differential equations,

PDF

___

V. Volterra, Sopra alcune questioni di inversione di integrali definite, Ann.Mat. Pura Appl., (2) 25, 139-178, 1897.
K. L. Cooke, J. A. Yorke, Some equations modelling growth processes and epidemics, Math. Biosci., 16, 75-101, 1973.
P. Waltham, Deterministic Threshold models in the Theory of Epidemics, Lecture Notes in Biomath., Vol. 1, Springer-Verlag (Berlin-Heidelberg), 1974.
H. L. Smith, On periodic solutions of a delay integral equation modelling epidemics, J. Math. Biol., 4, 69-80, 1977.
S. Busenberg, K. L. Cooke, The effect of integral conditions in certain equations modelling epidemics and population growth, J. Math. Biol., 10, 13-32, 1980.
J. A. J. Metz, O. Diekmann, The Dynamics of Physiologically Structured Populations, Lecture Notes in Biomath., Vol. 68, Springer-Verlag (Berlin- Heidelberg), 1986.
H. W. Hethcote, P. van den Driessche, Two SIS epidemiologic models with delays, J. Math. Biol., 40, 3-26, 2000.
F. Brauer, P. van den Driessche, Some directions for mathematical epidemiology, in Dynamical Systems and Their Applications in Biology, Fields Institute Communications, Vol. 36, American Mathematical Society (Providence), 95-112, 2003.
H. Brunner, On the discretization of differential and Volterra integral equations with variable delay, BIT 37, 1-12, 1997.
N. Takama, Y. Muroyaand, E. Ishiwata, On the discretization of differential and Volterra integral equations with variable delay, BIT37,1-12, 2000.
C. J. Zhangand, S. Vandewalle, On the attainable order of collocation methods for the delay differential equations with proportional delay, BIT40, 374-394, 2008.
Bellen, Stability criteria for exatand discrete solutions of neutral multidelay-integro-differential equations, Adv.Comput.Math., 28, 383-399, 2002.
Ş. Yüzbaşı, Laguerre approach for solving pantograph-type Volterra integro-differential equations, Applied Mathematics and Computation, 232, 1183-1199, 2014.
N. Şahin, Ş. Yüzbaşı, M. Gülsu, A collocation approach for solving systems of linear Volterra integral equations with variable coefficients, Computers and Mathematics with Applications, 62, 755-769, 2011.
E. Yusufoğlu, A homotopy perturbation algorithm to solve a system of Fredholm-Volterra type integral equations, Mathematical and Computer Modelling 47 (2008) 1099-1107.
J. Saberi-Nadjafi, M. Tamamgar, The variational iteration method: A highly promising method for solving the system of integro-differential equations, Computers and Mathematics with Applications 56 (2008) 346-351.
K. Maleknejad, M. Tavassoli Kajani, Solving linear integro-differential equation system by Galerkin methods with hybrid functions, Applied Mathematics and Computation 159 (2004) 603-612.
J. Biazar, E. Babolian, R. Islam, Solution of a system of Volterra integral equations of the first kind by Adomian method, Applied Mathematics and Computation 139, 249-258 (2003).
G. E. Pukhov, Differential transforms and circuit theory, Int. J. Circ. Theor. App. 10, 265, 1982.
J. K. Zhou, Differential transformation and its applications for electrical circuits, in Chinese, Huarjung University Press, Wuuhahn, China, 1986.
A. Tari, M. Y. Rahimi, S. Shahmorad, F. Talati, Solving a class of two-dimensional linear and nonlinear Volterra integral equations by the differential transform method, Journal of Computational and Applied Mathematics, 228, 70-76, 2009.
A. Tari, S. Shahmorad, Differential transform method for the system of two-dimensional nonlinear Volterra integro-differential equations, Computers and Mathematics with Applications, 61, 2621-2629, 2011.
Bongsoo Jang, Comments on ”Solving a class of two-dimensional linear and nonlinear Volterra integral equations by the differential transform method”, Journal of Computational and Applied Mathematics, 233, 224-230, 2009.
R. Abazari, M. Ganji, Extended two-dimensional DTM and its application on nonlinear PDEs with proportional delay, International Journal of Computer Mathematics, Vol. 88, No. 8, 1749-1762, 2011.