Authors:
Inderdeep Singh,Preeti,DOI NO:
https://doi.org/10.26782/jmcms.spl.11/2024.05.00009Keywords:
Wavelets,Chebyshev wavelets of the second kind,Haar wavelets,Operational metrics of integrations,Logistic differential equations,Numerical examples.,Abstract
This research paper focuses on the comparison study of wavelet solutions for solving logistic differential equations. For this purpose, we are utilizing Chebyshev wavelets of the second kind and Haar wavelets. Various numerical tests have been conducted to demonstrate the ease of use, precision, and effectiveness of the solutions provided by various wavelet techniques. The implications of these results are discussed within the broader context of mathematical and scientific research.Refference:
I. Alsaedi, D. Baleanu, S. Etemad, & S. Rezapour. : ‘On coupled systems of time-fractional differential problems by using a new fractional derivative’. Journal of Function Spaces, 2016, (2016). 10.1155/2016/4626940
II. A. Turan Dincel, & S. N. Tural Polat. : ‘Fourth kind Chebyshev Wavelet Method for the solution of multi-term variable order fractional differential equations’. Engineering Computations. Vol. 39(4), pp. 1274-1287, (2022). https://www.emerald.com/insight/content/doi/10.1108/EC-04-2021-0211/full/html
III. B. Sripathy, P. Vijayaraju, & G. Hariharan. : ‘Chebyshev wavelet-based approximation method to some non-linear differential equations arising in engineering’. Int. J. Math. Anal., Vol. 9(20), pp. 993-1010, (2015) 10.12988/ijma.2015.5393
IV. C. H. Hsiao, & S. P. Wu. : ‘Numerical solution of time-varying functional differential equations via Haar wavelets’. Applied Mathematics and Computation. Vol. 188(1), pp. 1049-1058, (2007) https://doi.org/10.1016/j.amc.2006.10.070
V. Debnath Lokenath. : ‘A brief historical introduction to fractional calculus’. International Journal of Mathematical Education in Science and Technology. Vol. 35, pp. 487-501, (2004) 10.1080/00207390410001686571
VI. E. N. Petropoulou. : ‘A discrete equivalent of the logistic equation’. Advances in Difference Equations, 2010, pp. 1-15, (2010) doi:10.1155/2010/457073
VII. F. A. Shah, R. Abass, & L. Debnath. : ‘Numerical solution of fractional differential equations using Haar wavelet operational matrix method’. International Journal of Applied and Computational Mathematics. Vol. 3, pp. 2423-2445, (2017) https://link.springer.com/article/10.1007/s40819-016-0246-8
VIII. G. Hariharan, & K. Kannan. : ‘Review of wavelet methods for the solution of reaction–diffusion problems in science and engineering’. Applied Mathematical Modelling. Vol. 38(3), pp. 799-813, (2014) https://doi.org/10.1016/j.apm.2013.08.003
IX. H. Jafari, S. A. Yousefi, M. A. Firoozjaee, S. Momani, & C. M Khalique. : ‘Application of Legendre wavelets for solving fractional differential equations’. Computers & Mathematics with Applications’. Vol. 62(3), pp. 1038-1045, (2011). https://doi.org/10.1016/j.camwa.2011.04.024
X. I. Aziz, & R. Amin. : ‘Numerical solution of a class of delay differential and delay partial differential equations via Haar wavelet’. Applied Mathematical Modelling. Vol. 40(23-24), pp. 10286-10299, (2016) 10.1016/j.apm.2016.07.018
XI. K. Srinivasa, & R. A. Mundewadi. : ‘Wavelets approach for the solution of nonlinear variable delay differential equations’. International Journal of Mathematics and Computer in Engineering. Vol. 1(2), pp. 139-148, (2023). 10.2478/ijmce-2023-0011
XII. M. E. Benattia, & K. Belghaba. : ‘Numerical solution for solving fractional differential equations using shifted Chebyshev wavelet’. Gen. Lett. Math., Vol. 3(2), pp. 101-110, (2017). https://www.refaad.com/Files/GLM/GLM-3-2-3.pdf
XIII. M. Faheem, A. Raza, & A. Khan. : ‘Wavelet collocation methods for solving neutral delay differential equations’. International Journal of Nonlinear Sciences and Numerical Simulation. Vol. 23(7-8), pp. 1129-1156, 10.1515/ijnsns-2020-0103
XIV. M. Ghergu, and V. Radulescu. : ‘Existence and nonexistence of entire solutions to the logistic differential equation’. In Abstract and Applied Analysis. pp. 995-1003, (2003). 10.1155/S1085337503305020
XV. M. H. Heydari, M. R. Hooshmandasl, F. M. Maalek Ghaini, & F. Mohammadi. : ‘Wavelet collocation method for solving multiorder fractional differential equations’. Journal of Applied mathematics. 2012. 10.1155/2012/542401
XVI. M. H. Heydari, M. R. Mahmoudi, A. Shakiba, & Z. Avazzadeh. : ‘Chebyshev cardinal wavelets and their application in solving nonlinear stochastic differential equations with fractional Brownian motion’. Communications in Nonlinear Science and Numerical Simulation. Vol. 64, pp. 98-121, (2018). https://doi.org/10.1016/j.cnsns.2018.04.018
XVII. M. M. Khader and M. M. Babatin. : ‘Numerical study of fractional Logistic differential equation using implementation of Legendre wavelet approximation’. Journal of Computational and Theoretical Nanoscience. Vol. 13(1), pp. 1022-1026, (2016). 10.1166/jctn.2016.4510
XVIII. M. S. Hafshejani, S. K. Vanani, & J. S. Hafshejani. : ‘Numerical solution of delay differential equations using Legendre wavelet method’. World Applied Sciences Journal. Vol. 13, pp. 27-33, (2011). https://www.researchgate.net/profile/Mahmoud-Sadeghi-6/publication/351267182_Numerical_Solution_of_Delay_Differential_Equations_Using_Legendre_Wavelet_Method/links/608e63c1458515d315ee5639/Numerical-Solution-of-Delay-Differential-Equations-Using-Legendre-Wavelet-Method.pdf
XIX. N. A. Kudryashov. : ‘Logistic function as solution of many nonlinear differential equations’. Applied Mathematical Modelling. Vol. 39(18), pp. 5733-5742, (2015) 10.1016/j.apm.2015.01.048
XX. N. I. Mahmudov, M. Awadalla, and K. Abuassba. : ‘Nonlinear sequential fractional differential equations with nonlocal boundary conditions’. Advances in Difference Equations, 2017, pp. 1-15, (2017) https://link.springer.com/article/10.1186/s13662-017-1371-3
XXI. P. Manchanda, & M. Rani. : ‘Second kind Chebyshev wavelet method for solving system of linear differential equations’. Int. J. Pure Appl. Math., Vol. 114, pp. 91-104, (2017) 10.12732/ijpam.v114i1.8
XXII. S. C. Shiralashetti, B. S. Hoogar, & S. Kumbinarasaiah. : ‘Hermite wavelet-based method for the numerical solution of linear and nonlinear delay differential equations’. Int. J. Eng. Sci. Math., Vol. 6(8), pp. 71-79, (2017) https://www.researchgate.net/profile/Kumbinarasaiah-S/publication/322438497_Hermite_Wavelet_Based_Method_for_the_Numerical_Solution_of_Linear_and_Nonlinear_Delay_Differential_Equations/links/5cb854f292851c8d22f3511d/Hermite-Wavelet-Based-Method-for-the-Numerical-Solution-of-Linear-and-Nonlinear-Delay-Differential-Equations.pdf
XXIII. S. G. Hosseini, & F. Mohammadi. : ‘A new operational matrix of derivative for Chebyshev wavelets and its applications in solving ordinary differential equations with non-analytic solution’. Applied mathematical sciences. Vol. 5(51), pp. 2537-2548, (2011). https://m-hikari.com/ams/ams-2011/ams-49-52-2011/mohammadiAMS49-52-2011.pdf
XXIV. S. Kumbinarasaiah, & K. R. Raghunatha. : ‘The applications of Hermite wavelet method to nonlinear differential equations arising in heat transfer’. International Journal of Thermofluids. Vol. 9, 100066, (2021). 10.1016/j.ijft.2021.100066
XXV. S. S. Zeid. : ‘Approximation methods for solving fractional equations, Chaos’. Solitons & Fractals. Vol. 125, pp. 171-193, (2019) 10.1016/j.chaos.2019.05.008
XXVI. T. Abdeljawad, R. Amin, K. Shah, Q. Al-Mdallal, & F. Jarad. : ‘Efficient sustainable algorithm for numerical solutions of systems of fractional order differential equations by Haar wavelet collocation method’. Alexandria Engineering Journal. Vol. 59(4), pp. 2391-2400, (2020). 10.1016/j.aej.2020.02.035
XXVII. V. Bevia, J. Calatayud, J. C. Cortés, & M. Jornet. : ‘On the generalized logistic random differential equation: Theoretical analysis and numerical simulations with real-world data’. Communications in Nonlinear Science and Numerical Simulation. Vol. 116, 106832, (2023). 10.1016/j.cnsns.2022.106832
XXVIII. X. J. Yang, D. Baleanu, & H. M. Srivastava. : ‘Local fractional integral transforms and their applications’. Academic Press. (2015).
XXIX. X. Xu, & F. Zhou. : ‘Chebyshev wavelet-Picard technique for solving fractional nonlinear differential equations’. International Journal of Nonlinear Sciences and Numerical Simulation. (0), (2022). 10.1515/ijnsns-2021-0413.
XXX. Y. Y. Yameni Noupoue, Y. Tandoğdu, & M. Awadalla. : ‘On numerical techniques for solving the fractional logistic differential equation’. Advances in Difference Equations. 2019, pp. 1-13, (2019). https://link.springer.com/article/10.1186/s13662-019-2055-y