Numerical Solution of Generalized Delay Integro-Differential Equations via Galerkin-Vieta-Lucas Polynomials
DOI :
https://doi.org/10.62054/ijdm/0102.02Mots-clés :
Delay Integro-Differential equations; Galerkin method; Shifted Vieta-Lucas polynomialsRésumé
In this article, the Galerkin-Vieta-Lucas scheme is presented to find an approximatesolution to the generalised delay integro-differential equation using the Vieta-Lucas
polynomial as an approximation. The Galerkin approach transforms the delay integro-differential
equation into a set of n × (n + m) algebraic equations, which, together with the attached
conditions, give (n + m) × (n + m) equations. The effectiveness and accuracy of the proposed
technique were tested on some existing examples in the literature, and obviously,
the results obtained justify the accuracy of the proposed scheme.
Références
Agarwal, P. and El-Sayed, A. A. (2020). Vieta-Luca polynomial for solving a fractional order mathematical physics model. Adv. in difference Eqns, 2020, 626. DOI: 10.22034/cmde.2020.42106.1818
Baharum, N. A., Abdulmajid, Z, Senu N. and Rosali, H. (2022). Numerical Approach for Delay Volterra Integro-Differential Equation. Sains Malaysiana, 51, 4125–4144. http://doi.org/10.17576/jsm-2022-5112-20
Biazar, J. and Salehi, F. (2016). Chebyshev Galerkin method for integro-differential equations of the second kind. Iranian J. of Numer. Analy. and Opt., 6, 31–42. DOI:10. 22067/IJNAO.V6I1.37480
Chen, C. and Shih, T. (1997). Finite Element Methods for Integro-differential Equations, Singapore: World Scientific Publishing Co. Ltd. https://doi.org/10.1142/3594
Daşcioġlu, A. and Bayram, D. V. (2019). Solving Fractional Fredholm Integro-Differential Equations by Laguerre Polynomials. Sains Malaysiana, 48(1), 251–257. http://dx.doi.org/10.17576/jsm-2019-4801-29
Dehghan, M. and Shakeri, F. (2008). Solution of an integro-differential equation arising in oscillating magnetic fields using He’s homotopy perturbation method, Progr. In Electromagnetics Res, PIER 78, 361-376. DOI:10.2528/PIER07090403
Dzhumabaev, D. S. (2018). New general solutions to linear Fredholm integro-differential equations and their applications on solving the boundary value problems. Journal of Computational and Applied Mathematics, 327, 79–108. DOI:10.1016/j.cam.2017. 06.010
Fathy, M., El-Gamel, M. and El-Azab, M. S. (2014). Legendre-Galerkin method for linear Fredholm integro-differential equations, Applied Math. and Comput., 243, 789-800. https://doi.org/10.1016/j.amc.2014.06.057
Golbabai, A. and Seifollahi, S. (2007). Radial basis function networks in the numerical solution of linear integro-differential equations, Appl. Math. Comput. 188, 427-432. https://doi.org/10.1016/j.amc.2006.10.004
Hetmaniok, E., Pleszczyski, M. and Khan, Y. (2022). Solving the Integral Differential Equations with Delayed Argument by Using the DTM Method. Sensors, 22, 4124. https://doi: 10.3390/s22114124.
Hou, J. and Yang, C. (2013). Numerical Method in Solving Fredholm Integro-differential Equations by Using Hybrid Function Operational Matrix of Derivative, J. Inf and Computational Sci. 10, 2757-2764. DOI:10.12733/jics20101830
Hosseini, S. M. and Shahmorad, S. (2003a). Tau numerical solution of Fredholm integro-differential equations with arbitrary polynomial bases. Applied Mathematical Modelling, 27, 145-154. https://doi.org/10.1016/S0307-904X(02)00099-9
Hosseini S. M. and Shahmorad, S. (2003b). Numerical solution of a class of Integro-Differential equations by the Tau Method with an error estimation”, Applied Mathematics and Computation, 136, 559-570. https://doi.org/10.1016/S0096-3003(02) 00081-4
Idrees, S. and Saeed, U. (2022). Vieta-Lucas wavelets method for fractional linear and nonlinear delay differential equations. Engineering Computations, 9, 3211-3231. https: //doi.org/10.1108/ec-02-2022-0094
Issa, K. and Salehi, F. (2017). Approximate solution of perturbed Volterra-Fredholm integro-differential equations by Chebyshev-Galerkin method, Journal of Mathematics, 8213932. https://doi.org/10.1155/2017/8213932
Issa, K., Biazar, J., Agboola, T. O. and Aliu, T. (2022). Perturbed Galerkin method for solving integro-differential equations, Journal of Applied Mathematics, (2022) 9748558. https://doi.org/10.1155/2022/9748558
Issa, K., Biazar, J. and Yisa, B. M. (2019). Shifted Chebyshev Approach for the Solution of Delay Fredholm and Volterra Integro-Differential Equations via Perturbed Galerkin Method. Iranian Journal of Optimization, 11, 149–159. DOI:20.1001.1.25885723.2019. 11.2.8.9
Issa, K. Bello R. A. and Abubakar, U. J. (2024). Approximate analytical solution of fractional order generalized integro-differential equations via fractional derivative of shifted Vieta Lucas polynomial. J. Nig. Phys. Sci. 6, 1821. DOI:10.46481/jnsps.2024.1821
Jangveladze, T., Kiguradze, Z. and Neta, B. (2011). Galerkin finite element method for one nonlinear integro-differential model. Appl. Math. Comp. 217(16), 6883-6892. https://doi.org/10.1016/j.amc.2011.01.053
Oyedepo, T., Ayinde, A. M. and Didigwu, E. N. (2024). Vieta-Lucas polynomial computational technique for Volterra integro-differential equations, Electronic Journal of Mathematical Analysis and Applications, 12. DOI:10.21608/EJMAA.2023.232998.1064
Peykrayegan, N., Ghovatmand, M. and Noori Skandari, M. H. (2021). An efficient method for linear fractional delay integro-differential equations. Computational and Applied Mathematics, 40, 249. https://doi.org/10.1007/s40314-021-01640-1
Saberi-Nadjafi, J. and Ghorbani, A. (2009). He’s homotopy perturbation method: an effective tool for solving nonlinear integral and integro-differential equations, Comput. Math. Appl. 58, 2379-2390. https://doi.org/10.1016/j.camwa.2009.03.032
Shahmorad, S. and Ostadzad, M. H. (2016). An Operational Matrix Method for Solving Delay Fredholm and Volterra Integro-differential Equations”, International Journal of Computational Methods, 13(6), 1650040. DOI:10.1142/S0219876216500407
Shahsavara, A. (2010). Numerical solution of linear Volterra and Fredholm integro differential equations using Haar wavelets. Math. Sci. J. 6, 85-96.
Shahmorad, S. (2005). Numerical solution of the linear Fredholm-Volterra Integro-Differential equations by the Tau method with an error estimation, Appl. Math. and Computation, 167(2), 1418-1429. https://doi.org/10.1016/j.amc.2004.08.045
Saadatmandi, A. and Dehghan, M. (2010). Numerical solution of the higher-order linear Fredholm integro-differential-difference equation with variable coefficients, Computers and Mathematics with Applications, 59, 2996–3004. https://doi.org/10.1016/j. camwa.2010.02.018
Tunç ¸C. (2016). New stability and boundedness results to Volterra integro-differential equations with delay. Journal of the Egyptian Mathematical Society, 24, 210–213. https://doi.org/10.1016/j.joems.2015.08.001.
Youssef, M. Z. Khader, M. M., Al-Dayel, I. and Ahmed, W. E. (2022). Solving fractional generalized Fisher-Kolmogorov-Petrovsky-Piskunov’s equation using compact-finite methods together with spectral collocation algorithms. Journal in Math. 2022, 1901131. https://doi.org/10.1155/2022/1901131
Yũzbaşi, Ş. (2017). Shifted Legendre method with residual error estimation for delay linear Fredholm integro-differential equations. Journal of Taibah University for Science, 11, 344-352. https://doi.org/10.1016/j.jtusci.2016.04.001
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