A Collocation Method Based on Morgan-Voyce Polynomials for Approximating Solutions to Nonlinear Pantograph Differential Equations

Öz

This study proposes a novel collocation technique based on Morgan-Voyce polynomials to obtain approximate solutions for a certain class of nonlinear pantograph differential equations that are subject to initial and boundary conditions. The method utilizes specially selected collocation points to transform the original nonlinear differential problem into an equivalent nonlinear algebraic system. The solutions of this algebraic system correspond to the unknown coefficients in the approximate solution expression. The main advantage of the proposed approach lies in its ability to reduce the complexity of solving nonlinear differential equations by converting them into manageable algebraic systems, while maintaining a high level of accuracy. To validate the accuracy and efficiency of the technique, several benchmark problems are considered. Numerical experiments are carried out, and the absolute error functions are used to analyze the performance of the method. All numerical computations and graphical illustrations presented in this paper have been conducted using a program developed in MATLAB R2022b.

Anahtar Kelimeler

• Nonlinear pantograph differential equation (NPDE)
• Pantograph equations
• Morgan-Voyce collocation method
• Collocation points
• Morgan-Voyce polynomials

Doğrusal Olmayan Pantograf Diferansiyel Denklemlerinin Yaklaşık Çözümleri için Morgan-Voyce Polinomlarına Bağlı bir Sıralama Yöntemi

Öz

Bu çalışma, başlangıç ve sınır koşullarına tabi belirli bir sınıftaki doğrusal olmayan pantograf diferansiyel denklemler için yaklaşık çözümler elde etmek amacıyla Morgan-Voyce polinomlarına dayalı yeni bir sıralama tekniği önermektedir. Yöntem, özel olarak seçilen sıralama noktalarını kullanarak orijinal doğrusal olmayan diferansiyel problemi eşdeğer bir doğrusal olmayan cebirsel sisteme dönüştürür. Bu cebirsel sistemin çözümleri, yaklaşık çözüm ifadesindeki bilinmeyen katsayılara karşılık gelir. Önerilen yaklaşımın temel avantajı, doğrusal olmayan diferansiyel denklemlerin çözümündeki karmaşıklığı azaltarak bunları daha yönetilebilir cebirsel sistemlere dönüştürmesi ve aynı zamanda yüksek doğruluk düzeyini korumasıdır. Yöntemin doğruluğunu ve etkinliğini doğrulamak amacıyla çeşitli karşılaştırmalı test problemleri ele alınmıştır. Sayısal deneyler gerçekleştirilmiş ve yöntemin performansını analiz etmek için mutlak hata fonksiyonları kullanılmıştır. Bu çalışmada sunulan tüm sayısal hesaplamalar ve grafiksel gösterimler, MATLAB R2022b ortamında geliştirilen bir program aracılığıyla gerçekleştirilmiştir.

Anahtar Kelimeler

• Doğrusal olmayan pantograf diferansiyel denklemler
• Pantograf Denklemler
• Morgan- Voyce Sıralama Yöntemi
• Sıralama noktaları
• Morgan-Voyce polinomları

Kaynakça

1. Aiello, W. G., Freedman, H. I., & Wu, J. (1992). Analysis of a model representing stage-structured population growth with state-dependent time delay. SIAM Journal on Applied Mathematics, 52(3), 855–869. https://doi.org/10.1137/0152048
2. Buhmann, M. D., & Iserles, A. (1993). Stability of the discretized pantograph differential equation. Mathematics of Computation, 60(201), 575–589. https://doi.org/10.1090/S0025-5718-1993-1176707-2
3. 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. https://doi.org/10.1098/rspa.1971.0078
4. Fox, L., Mayers, D. F., Ockendon, J. A., & Tayler, A. B. (1971). On a functional differential equation. Journal of the Institute of Mathematics and Its Applications, 8(3), 271–307. https://doi.org/10.1093/imamat/8.3.271
5. Kumar, P., Erturk, V. S., Yusuf, A., & Kumar, S. (2021). Fractional time-delay mathematical modeling of Oncolytic Virotherapy. Chaos, Solitons and Fractals, 150, Article ID 111123. https://doi.org/10.1016/j.chaos.2021.111123
6. Kumar, P., Baleanu, D., Erturk, V. S., Inc, M., & Govindaraj, V. (2022). A delayed plant disease model with Caputo fractional derivatives. Advances in Continuous and Discrete Models, 2022(1), 11–22. https://doi.org/10.1186/s13662-022-03684-x
7. Derfel, G. (1980). On compactly supported solutions of a class of functional-differential equations. In Modern Problems of Functions Theory and Functional Analysis (in Russian). Karaganda University Press.
8. Derfel, G., Dyn, N., & Levin, D. (1995). Generalized refinement equation and subdivision process. Journal of Approximation Theory, 80(2), 272–297, https://doi.org/10.1006/jath.1995.1019

9. Ahmed, I., Kumam, P., Abdeljawad, T., Jarad, F., Borisut, P., Demba M. A. & Kumam W. (2020). Existence and uniqueness results for ψ-Caputo implicit fractional pantograph differential equation with generalized anti-periodic boundary condition. Advances in Differential Equations, 2020(1), 1–19. https://doi.org/10.1186/s13662-020-03008-x
10. Balachandran, K., Kiruthika, S., & Trujillo, J. J. (2013). Existence of solutions of nonlinear fractional pantograph equations. Acta Mathematica Scientia, 33(3), 712–720. https://doi.org/10.1016/S0252-9602(13)60032-6
11. Fan, Z., Liu, M., & Cao, W. (2007). Existence and uniqueness of the solutions and convergence of semi-implicit Euler methods for stochastic pantograph equations. Journal of Mathematical Analysis and Applications, 325(2), 1142–1159. https://doi.org/10.1016/j.jmaa.2006.02.063
12. Tohidi, E., Bhrawy, A. H., & Erfani, K. (2013). A collocation method based on Bernoulli operational matrix for numerical solution of generalized pantograph equation. Applied Mathematical Modelling, 37(6), 4283–4294. https://doi.org/10.1016/j.apm.2012.09.032
13. Gümgüm, S., Sava¸saneril, 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. https://doi.org/10.15672/hujms.460975
14. Yılmaz, B., & Yaman, V. (2020). Numerical solutions of nonlinear boundary value pantograph type delay differential equations. International Journal of Advanced Engineering and Pure Sciences, 32(3), 333–339. https://doi.org/10.7240/jeps.696635
15. Sezer, M., Yalçınba¸s, S., & ¸ Sahin, N. (2008). Approximate solution of multi-pantograph equation with variable coefficients. Journal of Computational and Applied Mathematics, 214(2), 406–416. https://doi.org/10.1016/j.cam.2007.03.024
16. 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 Differential Equations, 2020(1), 1–25. https://doi.org/10.1186/s13662-020-03107-9
17. Ilhan, Ö. (2017). An improved Morgan-Voyce collocation method for numerical solution of generalized pantograph equations. Journal of Scientific and Engineering Research, 4(10), 320–332. https://doi.org/10.20852/ntmsci.2017.236
18. Ilhan, Ö. (2017). An improved Morgan-Voyce collocation method for numerical solution of multi-pantograph equations. New Trends in Mathematical Sciences, 4(5), 248–260. https://doi.org/10.20852/ntmsci.2017.236
19. Bonyah, E., Sagoe, A. K., Kumar, D., & Deniz, S. (2021). Fractional optimal control dynamics of coronavirus model with Mittag–Leffler law. Ecological Complexity, 45, 100880. https://doi.org/10.1016/j.ecocom.2020.100880
20. Jaaffar, N. T., Abdul Majid, Z., & Senu, N. (2020). Numerical approach for solving delay differential equations with boundary conditions. Mathematics, 8, 1073. https://doi.org/10.3390/math8071073
21. Wazwaz, A.-M., Raja, M. A. Z., & Syam, I. M. (2017). Reliable treatment for solving boundary value problems of pantograph delay differential equation. Romanian Reports in Physics, 69, 102.
22. He, J. H., & Latifizadeh, H. (2020). A general numerical algorithm for nonlinear differential equations by the variational iteration method. International Journal of Numerical Methods for Heat & Fluid Flow, 30(11), 4797–4810. https://doi.org/10.1108/HFF-01-2020-0029
23. Rani, D., & Mishra, V. (2020). Numerical inverse Laplace transform based on Bernoulli polynomials operational matrix for solving nonlinear differential equations. Results in Physics, 16, 102836. https://doi.org/10.1016/j.rinp.2019.102836
24. Yuttanan, B., Razzaghi, M., & Vo, T. N. (2021). A fractional-order generalized Taylor wavelet method for nonlinear fractional delay and nonlinear fractional pantograph differential equations. Mathematical Methods in the Applied Sciences, 44(5), 4156–4175. https://doi.org/10.1002/mma.7020
25. Bahgat, M. S. (2020). Approximate analytical solution of the linear and nonlinear multi-pantograph delay differential equations. Physica Scripta, 95(5), 055219. https://doi.org/10.1088/1402-4896/ab6ba2
26. Jafari, H., Tuan, N. A., & Ganji, R. M. (2021). A new numerical scheme for solving pantograph type nonlinear fractional integro-differential equations. Journal of King Saud University - Science, 33(1), 101185. https://doi.org/10.1016/j.jksus.2020.08.029
27. 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. https://doi.org/10.1016/j.cnsns.2012.05.009
28. Wang, L. P., Chen, Y. M., Liu, D. Y., & Boutat, D. (2019). Numerical algorithm to solve generalized fractional pantograph equations with variable coefficients based on shifted Chebyshev polynomials. International Journal of Computer Mathematics, 96(12), 2487–2510. https://doi.org/10.1080/00207160.2019.1573992
29. Saeed, U. (2014). Hermite wavelet method for fractional delay differential equations. Journal of Difference Equations, 2014, Article ID 359093, 1–8. https://doi.org/10.1155/2014/359093
30. Nemati, S., Lima, P., & Sedaghat, S. (2018). An effective numerical method for solving fractional pantograph differential equations using modification of hat functions. Applied Numerical Mathematics, 131, 174–189. https://doi.org/10.1016/j.apnum.2018.05.005
31. 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. https://doi.org/10.1016/j.cam.2016.06.005
32. Rabiei, K., & Ordokhani, Y. (2019). Solving fractional pantograph delay differential equations via fractional-order Boubaker polynomials. Engineering with Computers, 35, 1431–1441. https://doi.org/10.1007/s00366-018-0673-8
33. Anapalı, A., Öztürk, Y., & Gülsu, M. (2015). Numerical approach for solving fractional pantograph equation. International Journal of Computer Application, 113(9), 45–52. https://doi.org/10.5120/19857-1801
34. Vichitkunakorn, P., Vo, T. N., & Razzaghi, M. (2020). A numerical method for fractional pantograph differential equations based on Taylor wavelets. Transactions of the Institute of Measurement and Control, 42(7), 1334–1344. https://doi.org/10.1177/014233121989017
35. Isah, A., Phang, C., & Phang, P. (2017). Collocation method based on Genocchi operational matrix for solving generalized fractional pantograph equations. International Journal of Differential Equations, 2017, 1–10. https://doi.org/10.1155/2017/2097317
36. Işık, O. R., Güney, & Sezer, M. (2012). Bernstein series solutions of pantograph equations using polynomial interpolation. Journal of Difference Equations and Applications, 18(3), 357–374. https://doi.org/10.1080/10236198.2010.496456
37. Alsuyuti, M. M., Doha, E. H., Ezz-Eldien, S. S., & Youssef, I. K. (2021). Spectral Galerkin schemes for a class of multi-order fractional pantograph equations. Journal of Computational and Applied Mathematics, 384, Article ID 113157. https://doi.org/10.1016/j.cam.2020.113157
38. Çakmak, M., & Alkan, S. (2022). A numerical method for solving a class of systems of nonlinear pantograph differential equations. Alexandria Engineering Journal, 61(4), 2651–2661. https://doi.org/10.1016/j.aej.2021.07.028
39. Şahin, G. (2024). Epidemik sır modelinin ve lineer olmayan pantograf diferansiyel denklemlerinin Morgan-Voyce polinomları yardımıyla yakla¸sık çözümleri. (Tez no. 854202) [Yüksek Lisans Tezi, Muğla Sıtkı Koçman Üniversitesi].
40. Ezz-Eldien, S. S., Wang, Y., Abdelkawy, M. A., Zaky, M. A., Aldraiweesh, A. A., & Machado, J. T. (2020). Chebyshev spectral methods for multi-order fractional neutral pantograph equations. Nonlinear Dynamics, 100(4), 3785–3797. https://doi.org/10.1007/s11071-020-05728-x
41. Tarakçı, M., Özel, M., & Sezer, M. (2020). Solution of nonlinear ordinary differential equations with quadratic and cubic terms by Morgan-Voyce matrix-collocation method. Turkish Journal of Mathematics, 44 (3), 906–918. https://doi.org/10.3906/mat-1908-102
42. Swamy, M. N. S., & Bhattacharryya, B. B. (1967). A study of recurrent ladders using the polynomials defined by Morgan-Voyce. IEEE Transactions on Circuit Theory, CT-14(3), 260–264. https://doi.org/10.1109/TCT.1967.1082705
43. IIhan, Ö. (2012). Lineer diferansiyel, integral ve integro-diferansiyel denklemlerin Morgan-Voyce polinom çözümleri. (Tez no. 305210) [Yüksek Lisans Tezi, Muğla Sıtkı Koçman Üniversitesi].
44. Çakmak, M. (2022). Fibonacci collocation method for solving a class of nonlinear pantograph differential equations. Mathematical Methods in the Applied Sciences, 45(17), 11962–11976. https://doi.org/10.1002/mma.8636