Research Article
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Lineer Olmayan Pantograf Diferansiyel Denklemleri İçin Pell Sıralama Yaklaşımı

Year 2024, Volume: 9 Issue: 1, 167 - 183, 29.06.2024
https://doi.org/10.33484/sinopfbd.1401042

Abstract

Elektrodinamik, kontrol sistemleri ve kuantum mekaniği gibi teorik ve uygulamalı matematiğin dallarında karşılaştığımız Pantograf denklemleri, orantısal gecikmeli fonksiyonel diferansiyel denklemlerin özel bir türüdür. Bu çalışmada, Pantograf diferansiyel denklemin yaklaşık çözümleri üzerine çalışılmıştır. Bu denklem sınıfı için analitik çözüm olmadığından sadece yaklaşık çözümleri bulunabilir. Bu amaçla sayısal çözüm yöntemlerinden biri olan Pell sıralama yöntemi seçilmiştir. Yöntemin denkleme uygulanması sonucunda bir cebirsel denklem sistemi elde edilmiş ve MATHEMATICA programı kullanılarak verilen başlangıç koşulları ile yaklaşık çözüm bulunmuştur. Bu yöntem bazı test örneklerine uygulanmış ve sonuçlar hem grafiksel olarak hem de tablo olarak ifade edilmiştir. Hata analizleri bu yöntemin doğru ve etkili çalıştığını göstermiştir.

References

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  • Alkan, S., Aydin, M. N., & Coban, R. (2019). A numerical approach to solve the model of an electromechanical system. Mathematical Methods in the Applied Sciences, 42(16), 5266-5273.
  • Alkan, S., & Secer, A. (2018). A collocation method for solving boundary value problems of fractional order. Sakarya University Journal of Science, 22(6), 1601-1608.
  • Hesameddini, E., & Asadollahifard, E. (2015). Numerical solution of multi-order fractional differential equations via the sinc-collocation method. Iranian Joıurnal of Numerical Analysis and Optimization, 5(1), 37-48.
  • Nagy, A. M. (2017). Numerical solution of time fractional nonlinear Klein–Gordon equation using Sinc–Chebyshev collocation method. Applied Mathematics and Computation, 310, 139-148.
  • Zhi, M., Aiguo, X., Zuguo, Y., & Long, S. (2014). Finite difference and Sinc-collocation approximations to a class of fractional diffusion-wave equations. Journal of Applied Mathematics, 536030.
  • Moshtaghi, N., & Saadatmandi, A. (2021). Numerical solution of time fractional cable equation via the Sinc-Bernoulli collocation method. Journal of Applied and Computational Mechanics, 7(4), 1916-1924.
  • Jalili, P., Jalili, B., Ahmad, I., Hendy, A., Ali, M., & Ganji, D. D. (2024). Python approach for Using homotopy perturbation method to investigate heat transfer problems, Case Studies in Thermal Engineering, 54, 104049.
  • Hatipoglu, V. F. (2021). A novel model for the contamination of a system of three artificial lakes. Discrete and Continuous Dynamical Systems-S, 14(7), 2261-2272.
  • Hatipoglu, V. F. (2019). A numerical algorithm for the solution of nonlinear fractional differential equations via beta-derivatives. Mathematical Methods in the Applied Sciences, 42(16), 5258-5265.
  • Bayram, M., Hatipoglu, V. F., Alkan, S., & Das, S. E. (2018). A solution method for integro-differential equations of conformable fractional derivative. Thermal Science, 22(1), S7-S14.
  • Sedaghat, S., Ordokhani, Y., & Dehghan, M. (2012). Numerical solution of the delay differential equations of pantograph type via Chebyshev polynomials. Communications in Nonlinear Science and Numerical Simulation, 17(12), 4815-4830.
  • Jafari, H., Mahmoudi, M., & Skandari, M. H. N. (2021). A new numerical method to solve pantograph delay differential equations with convergence analysis. Advances in Difference Equations, 2021(1), 129.
  • Bahşi, M. M., & Çevik, M. (2015). Numerical solution of pantograph-type delay differential equations using perturbation-iteration algorithms. Journal of Applied Mathematics, 139821.
  • Alrebdi, R., & Al-Jeaid, H. K. (2023). Accurate solution for the pantograph delay differential equation via Laplace transform. Mathematics, 11(9), 2031.
  • Izadi, M., & Srivastava, H. M. (2021). A novel matrix technique for multi-order pantograph differential equations of fractional order. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 477(2253), 20210321.
  • Abdo, M. S., Abdeljawad, T., Kucche, K. D., Ali, S. M., & Jeelani, M.B. (2021). On nonlinear pantograph fractional differential equations with Atangana–Baleanu–Caputo derivative. Advances in Difference Equations, 2021(1), 65.
  • Rahimkhani, P., Ordokhani, Y., & Babolian, E. (2017). Numerical solution of fractional pantograph differential equations by using generalized fractional-order Bernoulli wavelet. Journal of Computational and Applied Mathematics, 309, 493–510.
  • Ali, I., Brunner, H., & Tang, T. (2009). Spectral methods for pantograph-type differential and integral equations with multiple delays. Frontiers of Mathematics in China, 4(1), 49–61.
  • Rabiei, K., & Ordokhani, Y. (2019). Solving fractional pantograph delay differential equations via fractional-order Boubaker polynomials. Engineering with Computers, 35(4), 1431–1441.
  • Xu, Y., Zhang, Y., & Zhao, J. (2019). Error analysis of the Legendre-Gauss collocation methods for the nonlinear distributed-order fractional differential equation. Applied Numerical Mathematics, 142, 122-138.
  • Gebril, E., El-Azab, M. S., & Sameeh, M. (2024). Chebyshev collocation method for fractional Newell-Whitehead-Segel equation. Alexandria Engineering Journal, 87, 39-46.
  • Manohara, G., & Kumbinarasaiah, S. (2024). An innovative Fibonacci wavelet collocation method for the numerical approximation of Emden-Fowler equations. Applied Numerical Mathematics, 201, 347-369.
  • Kumari, A., & Kukreja, V. K. (2023). Study of 4th order Kuramoto-Sivashinsky equation by septic Hermite collocation method. Applied Numerical Mathematics, 188, 88-105.
  • Saad, K. M. (2020). New fractional derivative with non-singular kernel for deriving Legendre spectral collocation method. Alexandria Engineering Journal, 59(4), 1909-1917.
  • Wang, K., &Wang, Q. (2013). Lagrange collocation method for solving Volterra–Fredholm integral equations. Applied Mathematics and Computation, 219(21), 10434-10440.
  • Horadam, A. F., & Mahon, J. M. (1985). Pell and pell-lucas polynomials. Fibonacci Quart, 23(1), 7-20.
  • Taghipour, M., & Aminikhah, H. (2022). A fast collocation method for solving the weakly singular fractional integro-differential equation. Computational and Applied Mathematics, 41(4), 142. 182
  • Sabermahani, S., Ordokhani, Y., & Razzaghi, M. (2023). Ritz-generalized Pell wavelet method: Application for two classes of fractional pantograph problems.Communications in Nonlinear Science and Numerical Simulation , 119, 107138.
  • Sahin, M., & Sezer, M. (2018). Pell-Lucas collocation method for solving high-order functional differential equations with hybrid delays. Celal Bayar University Journal of Science, 14(2), 141-149.
  • Cayan, S., & Sezer, M. (2019). Pell polynomial approach for Dirichlet problem related to partial differential equations, Journal of Science and Arts, 19(3), 613-628.
  • Yüzbasi, ¸ S., &Yildirim, G. (2020). Pell-Lucas collocation method to solve high-order linear Fredholm-Volterra integro-differential equations and residual correction, Turkish Journal of Mathematics, 44(4), 1065-1091.
  • Noori, S. R. M., & Taghizadeh, N. (2020). Modified differential transform method for solving linear and nonlinear pantograph type of differential and Volterra integro-differential equations with proportional delays. Advances in Difference Equations, 2020(1), 1-25.
  • Gümgüm, S., Savaşaneril, N. B., Kürkçü, Ö. K.,& Sezer, M. (2020). Lucas polynomial solution of nonlinear differential equations with variable delays. Hacettepe Journal of Mathematics and Statistics, 49(2), 553-564.

Pell Collocation Approach for the Nonlinear Pantograph Differential Equations

Year 2024, Volume: 9 Issue: 1, 167 - 183, 29.06.2024
https://doi.org/10.33484/sinopfbd.1401042

Abstract

Pantograph equations, which we encounter in the branches of pure and applied mathematics such as electrodynamics, control systems and quantum mechanics, are essentially a particular form of the functional differential equations characterized with proportional delays. This study focuses on exploring the approximate solution to the Pantograph differential equation. Since there is no analytic solutions for this equation class, only the approximate solutions can be obtain. For this purpose, Pell Collocation Method which is one of the numerical solution methods is chosen. As the result of applying the method to the equation, an algebraic equation system has been gained and then the approximate solution has been found by using MATHEMATICA via the given initial conditions. The method is applied to the some test examples and then the results are presented by both graphically and by table. The error estimations show that the method works accurately and efficiently.

References

  • Ockendon, J. R., & Tayler, A. B. (1971). The dynamics of a current collection system for an electric locomotive. Proceedings of the Royal Society of London, Series A, Mathematical and Physical Sciences, 322 (1551), 447-468.
  • Alkan, S., Aydin, M. N., & Coban, R. (2019). A numerical approach to solve the model of an electromechanical system. Mathematical Methods in the Applied Sciences, 42(16), 5266-5273.
  • Alkan, S., & Secer, A. (2018). A collocation method for solving boundary value problems of fractional order. Sakarya University Journal of Science, 22(6), 1601-1608.
  • Hesameddini, E., & Asadollahifard, E. (2015). Numerical solution of multi-order fractional differential equations via the sinc-collocation method. Iranian Joıurnal of Numerical Analysis and Optimization, 5(1), 37-48.
  • Nagy, A. M. (2017). Numerical solution of time fractional nonlinear Klein–Gordon equation using Sinc–Chebyshev collocation method. Applied Mathematics and Computation, 310, 139-148.
  • Zhi, M., Aiguo, X., Zuguo, Y., & Long, S. (2014). Finite difference and Sinc-collocation approximations to a class of fractional diffusion-wave equations. Journal of Applied Mathematics, 536030.
  • Moshtaghi, N., & Saadatmandi, A. (2021). Numerical solution of time fractional cable equation via the Sinc-Bernoulli collocation method. Journal of Applied and Computational Mechanics, 7(4), 1916-1924.
  • Jalili, P., Jalili, B., Ahmad, I., Hendy, A., Ali, M., & Ganji, D. D. (2024). Python approach for Using homotopy perturbation method to investigate heat transfer problems, Case Studies in Thermal Engineering, 54, 104049.
  • Hatipoglu, V. F. (2021). A novel model for the contamination of a system of three artificial lakes. Discrete and Continuous Dynamical Systems-S, 14(7), 2261-2272.
  • Hatipoglu, V. F. (2019). A numerical algorithm for the solution of nonlinear fractional differential equations via beta-derivatives. Mathematical Methods in the Applied Sciences, 42(16), 5258-5265.
  • Bayram, M., Hatipoglu, V. F., Alkan, S., & Das, S. E. (2018). A solution method for integro-differential equations of conformable fractional derivative. Thermal Science, 22(1), S7-S14.
  • Sedaghat, S., Ordokhani, Y., & Dehghan, M. (2012). Numerical solution of the delay differential equations of pantograph type via Chebyshev polynomials. Communications in Nonlinear Science and Numerical Simulation, 17(12), 4815-4830.
  • Jafari, H., Mahmoudi, M., & Skandari, M. H. N. (2021). A new numerical method to solve pantograph delay differential equations with convergence analysis. Advances in Difference Equations, 2021(1), 129.
  • Bahşi, M. M., & Çevik, M. (2015). Numerical solution of pantograph-type delay differential equations using perturbation-iteration algorithms. Journal of Applied Mathematics, 139821.
  • Alrebdi, R., & Al-Jeaid, H. K. (2023). Accurate solution for the pantograph delay differential equation via Laplace transform. Mathematics, 11(9), 2031.
  • Izadi, M., & Srivastava, H. M. (2021). A novel matrix technique for multi-order pantograph differential equations of fractional order. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 477(2253), 20210321.
  • Abdo, M. S., Abdeljawad, T., Kucche, K. D., Ali, S. M., & Jeelani, M.B. (2021). On nonlinear pantograph fractional differential equations with Atangana–Baleanu–Caputo derivative. Advances in Difference Equations, 2021(1), 65.
  • Rahimkhani, P., Ordokhani, Y., & Babolian, E. (2017). Numerical solution of fractional pantograph differential equations by using generalized fractional-order Bernoulli wavelet. Journal of Computational and Applied Mathematics, 309, 493–510.
  • Ali, I., Brunner, H., & Tang, T. (2009). Spectral methods for pantograph-type differential and integral equations with multiple delays. Frontiers of Mathematics in China, 4(1), 49–61.
  • Rabiei, K., & Ordokhani, Y. (2019). Solving fractional pantograph delay differential equations via fractional-order Boubaker polynomials. Engineering with Computers, 35(4), 1431–1441.
  • Xu, Y., Zhang, Y., & Zhao, J. (2019). Error analysis of the Legendre-Gauss collocation methods for the nonlinear distributed-order fractional differential equation. Applied Numerical Mathematics, 142, 122-138.
  • Gebril, E., El-Azab, M. S., & Sameeh, M. (2024). Chebyshev collocation method for fractional Newell-Whitehead-Segel equation. Alexandria Engineering Journal, 87, 39-46.
  • Manohara, G., & Kumbinarasaiah, S. (2024). An innovative Fibonacci wavelet collocation method for the numerical approximation of Emden-Fowler equations. Applied Numerical Mathematics, 201, 347-369.
  • Kumari, A., & Kukreja, V. K. (2023). Study of 4th order Kuramoto-Sivashinsky equation by septic Hermite collocation method. Applied Numerical Mathematics, 188, 88-105.
  • Saad, K. M. (2020). New fractional derivative with non-singular kernel for deriving Legendre spectral collocation method. Alexandria Engineering Journal, 59(4), 1909-1917.
  • Wang, K., &Wang, Q. (2013). Lagrange collocation method for solving Volterra–Fredholm integral equations. Applied Mathematics and Computation, 219(21), 10434-10440.
  • Horadam, A. F., & Mahon, J. M. (1985). Pell and pell-lucas polynomials. Fibonacci Quart, 23(1), 7-20.
  • Taghipour, M., & Aminikhah, H. (2022). A fast collocation method for solving the weakly singular fractional integro-differential equation. Computational and Applied Mathematics, 41(4), 142. 182
  • Sabermahani, S., Ordokhani, Y., & Razzaghi, M. (2023). Ritz-generalized Pell wavelet method: Application for two classes of fractional pantograph problems.Communications in Nonlinear Science and Numerical Simulation , 119, 107138.
  • Sahin, M., & Sezer, M. (2018). Pell-Lucas collocation method for solving high-order functional differential equations with hybrid delays. Celal Bayar University Journal of Science, 14(2), 141-149.
  • Cayan, S., & Sezer, M. (2019). Pell polynomial approach for Dirichlet problem related to partial differential equations, Journal of Science and Arts, 19(3), 613-628.
  • Yüzbasi, ¸ S., &Yildirim, G. (2020). Pell-Lucas collocation method to solve high-order linear Fredholm-Volterra integro-differential equations and residual correction, Turkish Journal of Mathematics, 44(4), 1065-1091.
  • Noori, S. R. M., & Taghizadeh, N. (2020). Modified differential transform method for solving linear and nonlinear pantograph type of differential and Volterra integro-differential equations with proportional delays. Advances in Difference Equations, 2020(1), 1-25.
  • Gümgüm, S., Savaşaneril, N. B., Kürkçü, Ö. K.,& Sezer, M. (2020). Lucas polynomial solution of nonlinear differential equations with variable delays. Hacettepe Journal of Mathematics and Statistics, 49(2), 553-564.
There are 34 citations in total.

Details

Primary Language English
Subjects Ordinary Differential Equations, Difference Equations and Dynamical Systems
Journal Section Research Articles
Authors

Pınar Albayrak 0000-0002-7973-3500

Publication Date June 29, 2024
Submission Date December 7, 2023
Acceptance Date May 27, 2024
Published in Issue Year 2024 Volume: 9 Issue: 1

Cite

APA Albayrak, P. (2024). Pell Collocation Approach for the Nonlinear Pantograph Differential Equations. Sinop Üniversitesi Fen Bilimleri Dergisi, 9(1), 167-183. https://doi.org/10.33484/sinopfbd.1401042