Modified Non-Polynomial Fractional Spline for Solving a System of Differential Equations

Authors

  • Paywast Department of Mathematics, College of Education, University of Sulaimani, Sulaimani, Iraq
  • Faraidun K. Hamasalh Bakrajo Technical Institute, Sulaimani Polytechnic University, Sulaimani, Iraq

DOI:

https://doi.org/10.56714/bjrs.51.2.4

Keywords:

Gamma function, Caputo fractional derivative, convergence analysis, Non-polynomial spline interpolation

Abstract

This work presents a fractional non-polynomial spline approach for solving a system of differential equations based on physics, engineering, and optimal control. The approach that produces a system of differential equations and more clearly illustrates the variety of the non-polynomial interpolation method demonstrates the diversity of methods within the Caputo derivative frame. The proposed methodology shows the possibility of obtaining numerical results for a system of differential equations in a variety of scientific fields.

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References

[1] S. R. Jena and A. Senapati, “Explicit and implicit numerical investigations of one-dimensional heat equation based on spline collocation and Thomas algorithm,” Soft Computing, vol. 28, no. 20, pp. 12227–12248, 2024, DOI: 10.21203/rs.3.rs-3800780/v1. DOI: https://doi.org/10.1007/s00500-024-09925-3

[2] S. Khan and A. Khan, “Non-polynomial cubic spline method for solution of higher order boundary value problems,” Computational Methods for Differential Equations, vol. 11, no. 2, pp. 225–240, 2023, DOI: 10.22034/cmde.2022.49827.2072.

[3] F. Geng and M. Cui, “Solving a nonlinear system of second order boundary value problems,” Journal of Mathematical Analysis and Applications, vol. 327, no. 2, pp. 1167–1181, 2007, DOI: 10.1016/j.jmaa.2006.05.011. DOI: https://doi.org/10.1016/j.jmaa.2006.05.011

[4] A. Khalid, I. Haq, A. Rehan, H. M. Alshehri, and M. S. Osman, “Employing CNPS and CPS approaches to calculate numerical roots of ninth-order linear and nonlinear boundary value problems,” International Journal of Modern Physics C, vol. 35, no. 10, p. 2450121, 2024, DOI: 10.1142/S0129183124501213. DOI: https://doi.org/10.1142/S0129183124501213

[5] R. Choudhary, D. Kumar, and S. Singh, “Second-order convergent scheme for time-fractional partial differential equations with a delay in time,” Journal of Mathematical Chemistry, vol. 61, no. 1, pp. 21–46, 2023, DOI: 10.1007/s10910-022-01409-9. DOI: https://doi.org/10.1007/s10910-022-01409-9

[6] J. Lu, “Variational iteration method for solving a nonlinear system of second-order boundary value problems,” Computers & Mathematics with Applications, vol. 54, no. 7–8, pp. 1133–1138, 2007, DOI: 10.1016/j.camwa.2006.12.060. DOI: https://doi.org/10.1016/j.camwa.2006.12.060

[7] A. S. Bataineh, M. S. Noorani, and I. Hashim, “Modified homotopy analysis method for solving systems of second-order BVPs,” Communications in Nonlinear Science and Numerical Simulation, vol. 14, no. 2, pp. 430–442, 2009, DOI: 10.1016/j.cnsns.2007.09.012. DOI: https://doi.org/10.1016/j.cnsns.2007.09.012

[8] K. H. Jwamer, F. K. Hamasalh, and R. C. Karem, “The study focuses on lacunary interpolation and error estimations by spline function,” Basrah Researches Sciences, vol. 50, no. 2, pp. 146–156, 2024, DOI: 10.56714/bjrs.50.2.12. DOI: https://doi.org/10.56714/bjrs.50.2.12

[9] A. Chaurasia, P. C. Srivastava, and Y. Gupta, “Solution of higher order boundary value problems by spline methods,” in AIP Conference Proceedings, vol. 1897, no. 1, p. 020018, 2017, DOI: 10.1063/1.5008697. DOI: https://doi.org/10.1063/1.5008697

[10] W. T. Aniley and G. F. Duressa, “A novel fitted numerical scheme for time-fractional singularly perturbed convection-diffusion problems with a delay in time via cubic B-spline approach,” Journal of Mathematical Modeling, vol. 12, no. 2, pp. 215–231, 2024, DOI: 10.22124/jmm.2023.25969.2303.

[11] N. Caglar and H. Caglar, “B-spline method for solving linear system of second-order boundary value problems,” Computers & Mathematics with Applications, vol. 57, no. 5, pp. 757–762, 2009, DOI: 10.1016/j.camwa.2008.09.033. DOI: https://doi.org/10.1016/j.camwa.2008.09.033

[12] S. A. Khuri and A. Sayfy, “Spline collocation approach for the numerical solution of a generalized system of second-order boundary-value problems,” Applied Mathematical Sciences, vol. 3, no. 45, pp. 2227–2239, 2009.

[13] K. Ragula, G. B. Soujanya, and D. Swarnakar, “Computational approach for a singularly perturbed differential equations with mixed shifts using a non-polynomial spline,” International Journal of Analysis and Applications, vol. 21, pp. 5–5, 2023, DOI: 10.28924/2291-8639-21-2023-5. DOI: https://doi.org/10.28924/2291-8639-21-2023-5

[14] H. Emadifar and F. K. Hamasalh, “Nonpolynomial spline interpolation for solving fractional subdiffusion equations,” Mathematical Problems in Engineering, vol. 2022, no. 1, p. 7354121, 2022, DOI: 10.1155/2022/7354121. DOI: https://doi.org/10.1155/2022/7354121

[15] A. K. Abu-Nab, Z. F. AbuShaeer, and A. F. Abu-Bakr, “Rebound micro-cavitation dynamics with ultrasound fields during histotripsy: a numerical investigation,” The European Physical Journal Special Topics, vol. 233, no. 23, pp. 3483–3495, 2024, DOI: 10.1140/epjs/s11734-024-01232-0. DOI: https://doi.org/10.1140/epjs/s11734-024-01232-0

[16] C. Milici, G. Drăgănescu, and J. T. Machado, Introduction to Fractional Differential Equations, vol. 25. Springer, 2018, DOI: 10.1007/978-3-030-00895-6. DOI: https://doi.org/10.1007/978-3-030-00895-6

[17] A. Chaurasia, Y. Gupta, and P. C. Srivastava, “Numerical scheme based on non-polynomial spline functions for the system of second order boundary value problems arising in various engineering applications,” Journal of Applied and Computational Mechanics, vol. 8, no. 1, pp. 144–152, 2022, DOI: 10.22055/JACM.2020.32435.2012.

[18] A. Y. Bayati, R. K. Saeed, and F. K. Hama-Salh, “The existence, uniqueness and error bounds of approximation splines interpolation for solving second-order initial value problems,” Journal of Mathematics and Statistics, vol. 5, no. 2, p. 123, 2009. DOI: https://doi.org/10.3844/jmssp.2009.123.129

[19] F. Hamasalh, B. D. Yassin, and S. A. Tahir, “Quasi-linearized polynomial collocation method for solving MHD Casson nanofluid mixed convection,” Basrah Researches Sciences, vol. 51, no. 1, pp. 182–193, 2025, DOI: 10.56714/bjrs.51.1.16.

[20] D. C. Lay, Linear Algebra and Its Applications. Pearson Education India, 2003.

[21] F. K. Hamasalh and M. A. Headayat, “The applications of non-polynomial spline to the numerical solution for fractional differential equations,” in AIP Conference Proceedings, vol. 2334, no. 1, p. 060014, 2021, DOI: 10.1063/5.0042319. DOI: https://doi.org/10.1063/5.0042319

[22] S. K. Tesfaye, G. F. Duressa, T. G. Dinka, and M. M. Woldaregay, “Fitted tension spline scheme for a singularly perturbed parabolic problem with time delay,” Journal of Applied Mathematics, vol. 2024, no. 1, p. 9458277, 2024, DOI: 10.1155/2024/9458277. DOI: https://doi.org/10.1155/2024/9458277

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Published

31-12-2025

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How to Cite

Modified Non-Polynomial Fractional Spline for Solving a System of Differential Equations. (2025). Basrah Researches Sciences, 51(2), 13. https://doi.org/10.56714/bjrs.51.2.4

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