On Approximate Solution of First-Order Weakly-Singular Volterra Integro-Dynamic Equation on Time Scales

Year 2015, Volume: 28 Issue: 4, 651 - 658, 16.12.2015

Adil Mısır , Süleyman Öğrekçi

Abstract

Many mathematical formulations of physical phenomena contain integro-dynamic equations. In this paper, we present a new and simple approach to resolve linear and nonlinear weakly-singular Volterra integro-dynamic equations of first and second order on any time scales. These equations occur in many applications shuch as in heat transfer, nuclear reactor dynamics, dynamics of linear viscoelastic materyal with long memory etc. In order to eliminate the singularity of the equation, nabla derivative is used and then transforming the given first-order integro-dynamic equations onto an firstorder dynamic equations on time scales. The validity of the method is illustrated with some examples. It has been observed that the numerical results efficiently approximate the exact solutions

Keywords

Time scales, İntegro-dynamic equations, Approximation of solutions

References

• R. P. Agarwal, Boundary value problems for higher order integro-differential equations, Nonlinear anal. 7 (1983) 259-270.
• B. Aulbach, S. Hilger, A unified approach to continuous and discrete dynamics, Qualitative Theory of Differential Equations, Szeged, 1988, Colloq. Math. Soc. János Bolyai, vol. 53, North-Holland, Amsterdam, (1990), 37-56.
• M. Bohner, A. Peterson, Dynamic equations on time scales, Birkhauser Boston, 2001.A.M.
• M. Bohner, A. Peterson, Advances in dynamic equations on time scales, Birkhauser Boston, 2003.
• L. Bougoffa, R. C. Rach and A. Mennouni, An approximate method for solving a class of weakly-singular equations, Applied Mathematics and Comput. Vol. 217, Issue.22 (2011), 8907-8913.
• integro-differential
• H. Brunner, On the numerical solution of nonlinear Voltera integro-differential equations, BIT Numerical Mathematics, 13 (1973) 381-390.
• H. Brunner, On the numerical solution of nonlinear Fredholm collocation methods, SIAM J. Numer. Anal. 27 (1990) 987-1000. equations by
• G. Ebadi, M. Rahimi-Ardabili, S.Shahmorad, Numerical integro-diferential equations by Tau method, Appl. Math. Comput. 188 (2007) 1580-1586. Voltera
• A. L. Jensen, Dynamics of populations with nonoverlapping generations, continuous mortality, and discrete reproductive periods. Edol. Modelling 74 (1994), 305--309.
• O. Lepik, Hear wavelet method for nonlinear integro-diferential equations, Appl. Math. comput. 176 (2006) 324-333.
• K. Maleknejad, Y. Mahmoudi, Taylor polynomial solution of higher-order nonlinear Volterra-Fredholm integro-differential equations, Appl. Math. Comput. 145 (2003) 641-653.
• Y. Mahmoudi, Waveled Galerkin method for numerical solution of integral equations, Appl. Math. Comput. 167(2005) 1119-1129.
• L. V. Nedorezova, B. N. Nedorezova, Correlations between models of population dynamics in continuous and discrete time. Ecol. Modelling 82 (1995), 93--97.
• D. B. Pachpatte, On approximate solutions of a Volterra type integrodifferential equation on time scales, Int. Journal of Math. Analysis, Vol. 4, No. 34 (2010) 1651-1659.
• J. Saberi-Nadja, A. Ghorbani, He's homotopy perturbation method :An effective tool for solving nonlinear integral and integro-differential equations, Comput. Math. Appl., 58 (2009) 2379-2390.M.
• A. M. Wazwaz, A reliable algorithm for solving boundary value problems for higher-order M. Wazwaz, The transforms-Adomain decomposition method for handling equations, Appl. Math. Comput. 216 (2010) 1304-1309. combined Laplace nonlinear Voltera integro-diferential
• M. Zarebnia, Z. Nikpour, Solution of linear Voltera integro-diferentials equations via Sinc functions, Int.J. Appl. Math. Comput. 2 (2010) 1-10.