Morgan-Voyce polynomial approach for ordinary linear delay integro-differential equations with variable delays and variable bounds. (English) Zbl 1513.65254
Summary: 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.
MSC:
| 65L60 | Finite element, Rayleigh-Ritz, Galerkin and collocation methods for ordinary differential equations |
| 34K06 | Linear functional-differential equations |
| 34K07 | Theoretical approximation of solutions to functional-differential equations |
| 45A05 | Linear integral equations |
| 65R20 | Numerical methods for integral equations |
Keywords:
Morgan-Voyce polynomials; integro-differential equations with variable delays; matrix-collocation method; residual error analysisReferences:
| [1] | [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] | [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. · Zbl 1196.65198 |
| [3] | [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. · Zbl 1216.34069 |
| [4] | [4] Z. Ayati and J. Biazar, On the convergence of Homotopy perturbation method, J. Egyptian Math. Soc. 23 (2), 424-428, 2015. · Zbl 1328.65112 |
| [5] | [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. · Zbl 1410.65237 |
| [6] | [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. · Zbl 1291.45009 |
| [7] | [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. · Zbl 1377.49032 |
| [8] | [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. · Zbl 1380.65116 |
| [9] | [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. · Zbl 1209.65143 |
| [10] | [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. · Zbl 1126.45004 |
| [11] | [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. · Zbl 1144.65083 |
| [12] | [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. · Zbl 1170.65103 |
| [13] | [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. · Zbl 0886.45002 |
| [14] | [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. · Zbl 1032.45500 |
| [15] | [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. · Zbl 0816.45002 |
| [16] | [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. · Zbl 0964.45003 |
| [17] | [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. · Zbl 1161.45004 |
| [18] | [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. · Zbl 1215.65201 |
| [19] | [19] S.S. Ezz-Eldien, On solving systems of multi-pantograph equations via spectral tau method, Appl. Math. Comput. 321, 63-73, 2018. · Zbl 1426.65114 |
| [20] | [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. · Zbl 1447.65014 |
| [21] | [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. · Zbl 0693.45006 |
| [22] | [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] | [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. · Zbl 1416.65534 |
| [24] | [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. · Zbl 1113.65122 |
| [25] | [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] | [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. · Zbl 1410.65240 |
| [27] | [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. · Zbl 1431.65244 |
| [28] | [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. · Zbl 1372.65222 |
| [29] | [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. · Zbl 1052.65119 |
| [30] | [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] | [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. · Zbl 1417.65145 |
| [32] | [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] | [33] M.T. Rashed, Numerical solution of functional differential, integral and integro- differential equations, Appl. Math. Comp. 156 (2), 485-492, 2004. · Zbl 1061.65146 |
| [34] | [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. · Zbl 1342.65163 |
| [35] | [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. · Zbl 1090.65087 |
| [36] | [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. · Zbl 1112.34063 |
| [37] | [37] T. Stoll and R.F. Tichy, Diophantine equations for Morgan-Voyce and other modified orthogonal polynomials, Math. Slovaca, 58 (1), 11-18, 2008. · Zbl 1164.11018 |
| [38] | [38] M.N.S. Swamy, Further Properties of Morgan-Voyce Polynomials, Fibonacci Quart. 6.2 , 167-175, 1968. · Zbl 0155.08204 |
| [39] | [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. · Zbl 1242.65288 |
| [40] | [40] K. Toshiyuki, Stability of RungeKutta methods for delay integro-differential equations, J. Comput. Appl. Math. 145 (2), 483-492, 2002. · Zbl 1002.65148 |
| [41] | [41] K. Toshiyuki, Stability of -methods for delay integro-differential equations, J. Com- put. Appl. Math. 161 (2), 393-404, 2003. · Zbl 1042.65108 |
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