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Singh, Deepak, Verma, Sag. (1405). A study of a fractal multi-pantograph delay model with varying coefficients using fractional order wavelets. سامانه مدیریت نشریات علمی, 14(2), 828-851. doi: 10.22034/cmde.2025.62311.2735
Deepak Singh; Sag Ram Verma. "A study of a fractal multi-pantograph delay model with varying coefficients using fractional order wavelets". سامانه مدیریت نشریات علمی, 14, 2, 1405, 828-851. doi: 10.22034/cmde.2025.62311.2735
Singh, Deepak, Verma, Sag. (1405). 'A study of a fractal multi-pantograph delay model with varying coefficients using fractional order wavelets', سامانه مدیریت نشریات علمی, 14(2), pp. 828-851. doi: 10.22034/cmde.2025.62311.2735
Singh, Deepak, Verma, Sag. A study of a fractal multi-pantograph delay model with varying coefficients using fractional order wavelets. سامانه مدیریت نشریات علمی, 1405; 14(2): 828-851. doi: 10.22034/cmde.2025.62311.2735

A study of a fractal multi-pantograph delay model with varying coefficients using fractional order wavelets

Computational Methods for Differential Equations
مقاله 21، دوره 14، شماره 2، تیر 2026، صفحه 828-851    اصل مقاله (3.21 M)
نوع مقاله: Research Paper
شناسه دیجیتال (DOI): 10.22034/cmde.2025.62311.2735
نویسندگان
Deepak Singh؛ Sag Ram Verma*
Department of Mathematics and Statistics, Gurukula Kangari (Deemed to be University), Haridwar, 249404, Uttarakhand, India.
چکیده
Apart from using fractal dimensions to describe statistical self-similarity, exploring a direct measurement approach involves considering mathematical models capable of constructing a real-world fractal entity, as classical differential and integral operators cannot efficiently handle such problems. In this study, the fractal derivative is applied to develop a fractal model for multi-pantograph delay differential equations with variable coefficients. The wavelet approach, employing Jacobi fractional order wavelets, has been developed to attain a numerical solution. The proposed methodology relies on the utilization of the fractal integral operational matrix of Jacobi fractional-order wavelets combined with the collocation method. We have outlined pseudo-code and conducted a stability analysis for the methods proposed within the specified model. Furthermore, the convergence analysis of the approximate solution is presented through some lemmas and theorems. The numerical results and error analysis of some illustrative examples are also shown in the tables and graphs. In the proposed methods, numerical results are derived across various values of the fractal $(\mu)$ and fractional $(\phi)$ parameters. It is important to highlight that the classical scenario is retrieved when $\mu=1$.
کلیدواژه‌ها
Fractal operator؛ Multi-pantograph delay differential equation؛ Fractional order Jacobi wavelets؛ Function approximation؛ Integral operational matrix
مراجع
  • [1] A. Allwright and A. Atangana, Fractal advection-dispersion equation for groundwater transport in fractured aquifers with self-similarities, The European Physical Journal Plus, 133 (2018), 1-20.
  • [2] A. A. Arbuzov and R. R. Nigmatullin, Three-dimensional fractal models of electrochemical processes, Russian Journal of Electrochemistry, 45 (2009), 1276-1286.
  • [3] A. Atangana and S. Jain, A new numerical approximation of the fractal ordinary differential equation, The European Physical Journal Plus, 133 (2018), 1-15.
  • [4] M. M. Bahşi and M. Çevik, Numerical solution of pantograph-type delay differential equations using perturbationiteration algorithms, Journal of Applied Mathematics, 2015(1) (2015), 139821.
  • [5] C. T. Baker, C. A. Paul, and D. R. Willé, Issues in the numerical solution of evolutionary delay differential equations, Advances in Computational Mathematics, 3 (1995), 171-196.
  • [6] C. C. Barton, P. A. Hsieh, J. Angelier, F. Bergerat, C. Bouroz, and M. Dettinger, Physical and Hydrologicflow Properties of Fractures: Las Vegas, Nevada-Zion Canyon, Utah-Grand Canyon, Arizona-Yucca Mountain, Nevada, July 20-24, 1989, Washington, DC: American Geophysical Union, (1989).
  • [7] M. Basim, A. Ahmadian, N. Senu, and Z. B. Ibrahim, Numerical simulation of variable-order fractal-fractional delay differential equations with nonsingular derivative, Engineering Science and Technology, an International Journal, 42 (2023), 101412.
  • [8] A. H. Bhrawy and M. A. Zaky, Shifted fractional-order Jacobi orthogonal functions: application to a system of fractional differential equations, Appl. Math. Model, 40(2) (2016), 832-845.
  • [9] W. Chen, H. Sun, X. Zhang, and D. Korôsak, Anomalous diffusion modeling by fractal and fractional derivatives, Computers & Mathematics with Applications, 59(5) (2010), 1754-1758.
  • [10] Y. Chen, Fractal Modeling and fractal dimension description of urban morphology, Entropy, 22(9) (2020), 961.
  • [11] G. Chi, and G. Li, Numerical identification of the fractal orders in the generalized nonlocal elastic model, Journal of Engineering Mathematics 142(1) (2023), 4.
  • [12] C. K. Chui, Wavelets: a mathematical tool for signal analysis, Society for Industrial and Applied Mathematics, 1997.
  • [13] W. Dahmen Wavelet and multiscale methods for operator equations, Acta numerica, 6 (1997), 55-228.
  • [14] I. Daubechies, Ten lectures on wavelets, Society for industrial and applied mathematics, 1992.
  • [15] H. Dehestani, Y. Ordokhani, and M. Razzaghi, Fractional-order Bessel functions with various applications, Applications of Mathematics, 64 (2019), 637-662.
  • [16] A. El-Ajou, N. O. Moa’ath, Z. Al-Zhour, and S. Momani, Analytical numerical solutions of the fractional multipantograph system: Two attractive methods and comparisons, Results in Physics, 14 (2019), 102500.
  • [17] N. A. Elkot, E. H. Doha, I. G. Ameen, A. S. Hendy, and M. A. Zaky, A re-scaling spectral collocation method for the nonlinear fractional pantograph delay differential equations with non-smooth solutions, Communications in Nonlinear Science and Numerical Simulation, 118 (2023), 107017.
  • [18] J. Hăskovec, Asymptotic and exponential decay in mean square for delay geometric Brownian motion, Applications of Mathematics, (2022), 1-13.
  • [19] J. H. He, A new fractal derivation, Thermal science, 15(suppl.1) (2011), 145-147.
  • [20] O. R. Isik, and T. Turkoglu, A rational approximate solution for generalized pantograph-delay differential equations, Mathematical Methods in the Applied Sciences, 39(8) (2016), 2011-2024.
  • [21] M. T. Kajani, Numerical solution of fractional pantograph equations via Müntz–Legendre polynomials, Mathematical Sciences, 18(3) (2024), 387-395.
  • [22] A. Karoui, Wavelets: Properties and approximate solution of a second kind integral equation, Computers & Mathematics with Applications, 46(2-3) (2003), 263-277.
  • [23] M. A. Khan, and A. Atangana, Numerical methods for fractal-fractional differential equations and engineering: simulations and modeling, CRC Press, (2023).
  • [24] C. King, Fractal geography of the Riemann zeta function, arXiv preprint arXiv:1103.5274, (2011).
  • [25] M. Li, M. Fěckan, and J. Wang, Finite time stability and relative controllability of second order linear differential systems with pure delay, Applications of Mathematics, 68(3) (2023), 305-327.
  • [26] Y. Li, and Y. Shao, Dynamic analysis of an impulsive differential equation with time-varying delays, Applications of Mathematics, 59(1) (2014), 85-98.
  • [27] D. Lu, Y. Chen, R. Mehdi, S. Jabeen, and A. Rashid, Approximate solution of multi-pantograph equations with variable coefficients via collocation method based on hermite polynomials, Communications in Mathematics and Applications, 9(4) (2018), 601.
  • [28] J. R. Ockendon, and A. B. Tayler, The dynamics of a current collection system for an electric locomotive, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences, 322(1551) (1971), 447468.
  • [29] N. Pashmakian, A. Farajzadeh, N. Parandin, and N. Karamikabir,A numerical approach for solving the Fractal ordinary differential equations, Computational Methods for Differential Equations, 12(4) (2024), 780-790.
  • [30] Y. Rahrovi, Y. Mahmoudi, A. S. Shamloo, and M. J. Rad, Jacobi wavelets method for numerical solution of fractional population growth model, Computational Methods for Differential Equations, 11(2) (2023), 387-398.
  • [31] S. A. Rakhshan, and S. Effati, A generalized Legendre–Gauss collocation method for solving nonlinear fractional differential equations with time varying delays, Applied Numerical Mathematics, 146 (2019), 342-360.
  • [32] A. Rayal, and S. R. Verma, An approximate wavelets solution to the class of variational problems with fractional order, Journal of Applied Mathematics and Computing, 651 (2021), 735-769.
  • [33] A. Rayal, and S. R. Verma, Numerical analysis of pantograph differential equation of the stretched type associated with fractal-fractional derivatives via fractional order Legendre wavelets, Chaos, Solitons & Fractals, 139 (2020), 110076.
  • [34] J. A. Rodrigues, and V. C. Pandolfelli, Insights on the fractal-fracture behaviour relationship, Materials Research, 1 (1998), 47-52.
  • [35] A. Roy, E. Perfect, W. M. Dunne, and L. D. McKay, Fractal characterization of fracture networks: An improved box-counting technique, Journal of Geophysical Research: Solid Earth, 112(B12) (2007).
  • [36] S. Sabermahani, Y. Ordokhani, and M. Razzaghi, Ritz-generalized Pell wavelet method: Application for two classes of fractional pantograph problems, Communications in Nonlinear Science and Numerical Simulation, 119 (2023), 107138.
  • [37] S. Sabermahani, Y. Ordokhani, and P. Rahimkhani, Application of generalized Lucas wavelet method for solving nonlinear fractal-fractional optimal control problems, Chaos, Solitons & Fractals, 170 (2023), 113348.
  • [38] S. Sedaghat, Y. Ordokhani, and M. Dehghan, Numerical solution of the delay differential equations of pantograph type via Chebyshev polynomials, Communications in Nonlinear Science and Numerical Simulation, 17(12) (2012), 4815-4830.
  • [39] M. Sezer, and N. S¸ahin, Approximate solution of multi-pantograph equation with variable coefficients, Journal of Computational and Applied Mathematics, 214(2) (2008), 406-416.
  • [40] T. Shojaeizadeh, M. Mahmoudi, and M. Darehmiraki, Optimal control problem of advection-diffusion-reaction equation of kind fractal-fractional applying shifted Jacobi polynomials, Chaos, Solitons & Fractals, 143 (2021), 110568.
  • [41] A. B. Tayler, Mathematical models in applied mechanics, Oxford University Press, 4 (2001).
  • [42] K. L. Wang, A novel computational approach to the local fractional Lonngren wave equation in fractal media, Mathematical Sciences, 18(3) (2024), 413-418.
  • [43] X. J. Yang, D. Baleanu, and H. M. Srivastava, Local fractional integral transforms and their applications, Academic Press, (2015).
  • [44] F. Zhao, J. Hu, T. Liu, T. Zhou, and Q. Ren, Study of the macro and micro characteristics of and their relationships in cemented backfill based on SEM, Materials, 16(13) (2023), 4772.
  • [45] H. Zhou, F. Duan, Z. Liu, L. Chen, Y. Song, and Y. Zhang, Study on electric spark discharge between pantograph and catenary in electrified railway, IET Electrical Systems in Transportation, 12(2) (2022), 128-142.
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تعداد مشاهده مقاله: 339
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