Morgan-Voyce polynomial approach for ordinary linear delay integro-differential equations with variable delays and variable bounds

Year 2021, Volume: 50 Issue: 5, 1434 - 1447, 15.10.2021

Mustafa Özel , Mehmet Tarakçı , Mehmet Sezer

https://doi.org/10.15672/hujms.569245

Abstract

An effective matrix method to solve the ordinary linear integro-differential equations with variable coefficients and variable delays under initial conditions is offered in this article. Our method consists of determining the approximate solution of the matrix form of Morgan-Voyce and Taylor polynomials and their derivatives in the collocation points. Then, we reconstruct the problem as a system of equations and solve this linear system. Also, some examples are given to show the validity and the residual error analysis is investigated.

Keywords

Morgan-Voyce polynomials , integro differential equations with variable delays , matrix-collocation method , residual error analysis

References

• [1] H.A. Ali, Expansion Method For Solving Linear Delay Integro-Differential Equation Using B-Spline Functions, Engineering and Technology Journal, 27 (10), 1651–1661, 2009.
• [2] C. Angelamaria, I.D. Prete and C. Nitsch, Gaussian direct quadrature methods for double delay Volterra integral equations, Electron. Trans. Numer. Anal. 35, 201–216, 2009.
• [3] A. Ardjouni and D. Ahcene, Fixed points and stability in linear neutral differential equations with variable delays, Nonlinear Anal. Theory, Methods Appl. 74 (6), 2062– 2070, 2011.
• [4] Z. Ayati and J. Biazar, On the convergence of Homotopy perturbation method, J. Egyptian Math. Soc. 23 (2), 424-428, 2015.
• [5] M.A. Balci and M. Sezer, Hybrid Euler-Taylor matrix method for solving of generalized linear Fredholm integro-differential difference equations, Appl. Math. Comput. 273, 33–41, 2016.
• [6] A. Bellour and M. Bousselsal, Numerical solution of delay integrodifferential equations by using Taylor collocation method, Math. Methods Appl. Sci. 37 (10), 1491–1506, 2014.
• [7] A.H. Bhrawy and S.S. Ezz-Eldien, A new Legendre operational technique for delay fractional optimal control problems, Calcolo, 53 (4), 521–543, 2016.
• [8] A.H. Bhrawy et al., A Legendre-Gauss collocation method for neutral functional- differential equations with proportional delays, Adv. Differ. Equ. 2013 (1), 63, 2013.
• [9] J. Biazar et al., Numerical solution of functional integral equations by the variational iteration method, J. Comput. Appl. Math. 235 (8), 2581–2585, 2011.
• [10] A.M. Bica and M. Sorin, Smooth dependence by LAG of the solution of a delay integro- differential equation from Biomathematics, Commun. Math. Anal. 1 (1), 64–74, 2006.
• [11] H. Brunner and H. Qiya, Optimal superconvergence results for delay integro- differential equations of pantograph type, SIAM J. Numer. Anal. 45 (3), 986–1004, 2007.
• [12] H. Brunner, Recent advances in the numerical analysis of Volterra functional dif- ferential equations with variable delays, J. Comput. Appl. Math. 228 (2), 524–537, 2009.
• [13] B. Cahlon and D. Schmidt, Stability criteria for certain delay integral equations of Volterra type, J. Comput. Appl. Math.84 2, 161–188, 1997.
• [14] A. Canada and A. Zertiti, Positive solutions of nonlinear delay integral equations modelling epidemics and population growth, Extracta Math. 8 (2-3), 153–157, 1993.
• [15] A. Canada and A. Zertiti, Systems of nonlinear delay integral equations modelling population growth in a periodic environment, Comment. Math. Univ. Carolinae, 35 (4), 633–644, 1994.
• [16] E.A. Dads and K. Ezzinbi, Existence of positive pseudo-almost-periodic solution for some nonlinear infinite delay integral equations arising in epidemic problems, Non- linear Anal. Theory, Methods Appl. 41 (1), 1–13, 2000.
• [17] H.S. Dink, T-J. Xiao and J. Liang, Existence of positive almost automorphic solutions to nonlinear delay integral equations, Nonlinear Anal. 70, 2216–2231, 2009.
• [18] E. Eleonora, E. Russo and A. Vecchio, Comparing analytical and numerical solution of a nonlinear two-delay integral equations, Math. Comput. Simul. 81 (5), 1017–1026, 2011.
• [19] S.S. Ezz-Eldien, On solving systems of multi-pantograph equations via spectral tau method, Appl. Math. Comput. 321, 63–73, 2018.
• [20] S.S. Ezz-Eldien and E.H. Doha, Fast and precise spectral method for solving panto- graph type Volterra integro-differential equations, Numer. Algorithms, 81 (1), 57–77, 2019.
• [21] A.M. Fink and J.A. Gatica, Positive almost periodic solutions of some delay integral equations, J. Differ. Equ. 83 (1), 166–178, 1990.
• [22] L. Fortuna and M. Frasca, Generating passive systems from recursively defined poly- nomials, Int. J. Circuits, Syst. Signal Process. 6, 179–188, 2012.
• [23] E. Gokmen, G. Yuksel and M. Sezer, A numerical approach for solving Volterra type functional integral equations with variable bounds and mixed delays, J. Comput. Appl. Math. 311, 354–363, 2017.
• [24] M. Gulsu and M. Sezer, Approximations to the solution of linear Fredholm integrod- ifferentialdifference equation of high order, J. Franklin Inst. 343 (7), 720–737, 2006.
• [25] H.G. He and L.P. Wen, Dissipativity of -methods and one-leg methods for nonlinear neutral delay integro-differential equations, WSEAS Trans. Math. textbf12, 405–415, 2013.
• [26] Ö.K. Kürkçü, E. Aslan and M. Sezer, A numerical approach with error estimation to solve general integro-differentialdifference equations using Dickson polynomials, Appl. Math. Comput. 276, 324–339, 2016.
• [27] Ö.K. Kürkçü, E. Aslan and M. Sezer, A novel collocation method based on residual er- ror analysis for solving integro-differential equations using hybrid Dickson and Taylor polynomials, Sains Malays. 46, 335–347, 2017.
• [28] Ö.K. Kürkçü, E. Aslan and M. Sezer, A numerical method for solving some model problems arising in science and convergence analysis based on residual function, Appl. Numer. Math. 121, 134–148, 2017.
• [29] T. Luzyanina, D. Roose and K. Engelborghs, Numerical stability analysis of steady state solutions of integral equations with distributed delays, Appl. Numer. Math.50 (1), 75–92, 2004.
• [30] M. Özel, Ö.K. Kürkçü and M. Sezer, Morgan-Voyce matrix method for generalized functional integro-differential equations of Volterra type, Journal of Science and Arts, 47 (2), 295–310, 2019.
• [31] M. Özel, M. Tarakçı and M. Sezer, A numerical approach for a nonhomogeneous differential equations with variable delays, Mathematical Sciences, 12, 145–155, 2018.
• [32] I. Özgül and N. Şahin, On Morgan-Voyce polynomials approximation for linear dif- ferential equations, Turk. J. Math. Comput. Sci. 2 (1), 1–10, 2014.
• [33] M.T. Rashed, Numerical solution of functional differential, integral and integro- differential equations, Appl. Math. Comp. 156 (2), 485–492, 2004.
• [34] S.Y. Reutskiy, The backward substitution method for multipoint problems with linear Volterra–Fredholm integro-differential equations of the neutral type, J. Comput. Appl. Math. 296, 724–738, 2016.
• [35] M. Sezer and A.A. Daşcıoğlu, Taylor polynomial solutions of general linear differential–difference equations with variable coefficients, Appl. Math. Comput. 174 (2), 1526–1538, 2006.
• [36] M. Sezer and A.A. Daşcıoğlu, A Taylor method for numerical solution of generalized pantograph equations with linear functional argument, J. Comput. Appl. Math. 200 (1), 217–225, 2007.
• [37] T. Stoll and R.F. Tichy, Diophantine equations for Morgan-Voyce and other modified orthogonal polynomials, Math. Slovaca, 58 (1), 11–18, 2008.
• [38] M.N.S. Swamy, Further Properties of Morgan-Voyce Polynomials, Fibonacci Quart. 6.2 , 167–175, 1968.
• [39] N. Şahin, Ş. Yüzbaşı and M. Sezer, A Bessel polynomial approach for solving general linear Fredholm integro-differential–difference equations, Int. J. Comput. Math. 88 (14), 3093–3111, 2011.
• [40] K. Toshiyuki, Stability of RungeKutta methods for delay integro-differential equations, J. Comput. Appl. Math. 145 (2), 483–492, 2002.
• [41] K. Toshiyuki, Stability of -methods for delay integro-differential equations, J. Com- put. Appl. Math. 161 (2), 393–404, 2003.