Abstract
This article introduces a proficient method for solving both linear and nonlinear second-order singular value differential equations within the framework of Fibonacci wavelets and the collocation technique. Two key theorems are presented to facilitate a discussion on the convergence analysis of the method. The efficacy, ease of application, and computational speed of this approach are demonstrated through its application to diverse problem scenarios. The resulting solutions are compared with existing numerical solutions, further affirming the correctness and effectiveness of the proposed method. Notably, the method consistently yields solutions that align with the exact answers for a multitude of issues. Graphs and figures are employed to visually demonstrate the higher accuracy achieved by the Fibonacci wavelet approach for specific problems. All calculations and data processing are conducted using MATLAB software.
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References
Kumar M, Singh N (2009) A collection of computational techniques for solving singular boundary-value problems. Adv Eng Softw 40(4):288–297
Shiralashetti SC, Deshi AB (2018) The numerical solution of singular initial value problems using Chebyshev wavelet collocation method. Ain Shams Eng J 9(4):1451–1456
Shiralashetti SC, Deshi AB (2018) Legendre wavelet collocation method for the numerical solution of singular initial value problems. Int J Appl Math Stat 3(4):121–129
Yüzbaşı Ş (2011) A numerical approach for solving a class of the nonlinear Lane-Emden type equations arising in astrophysics. Math Methods Appl Sci 34(18):2218–2230
Yüzbaşı Ş (2011) A numerical approach for solving the high-order linear singular differential-difference equations. Comput Math Appl 62(5):2289–2303
Yüzbaşı Ş (2017) A numerical scheme for solutions of a class of nonlinear differential equations. J Taibah Univ Sci 11(6):1165–1181
Shiralashetti LL (2020) Taylor wavelets operational matrix method for the numerical solution of stochastic Volterra-Fredholm integral equations. Stoch Anal Appl 24(2):121–240
Tunç C, Tunç O (2015) A note on certain qualitative properties of a second order linear differential system. Appl Math Inf Sci 9(2):953
Tunç C, Tunç O (2016) On the boundedness and integration of non-oscillatory solutions of certain linear differential equations of second order. J Adv Res 7(1):165–168
Tunç C, Tunç O (2017) A note on the stability and boundedness of solutions to non-linear differential systems of second order. Arab J Basic Appl Sci 24:169–175
Hasan YQ, Zhu LM (2007) Solving singular initial value problems in the second-order ordinary differential equations. J Appl Sci 7(17):2505–2508
Mohammadi F, Hosseini MM (2011) A new Legendre wavelet operational matrix of derivative and its applications in solving the singular ordinary differential equations. J Frank Inst 348(8):1787–1796
Iqbal J, Abass R, Kumar P (2018) Solution of linear and nonlinear singular boundary value problems using Legendre wavelet method. Ital J Pure Appl Math 40:311–328
Shiralashetti SC, Deshi AB, Desai PM (2016) Haar wavelet collocation method for the numerical solution of singular initial value problems. Ain Shams Eng J 7(2):663–670
Zhou F, Xu X (2016) Numerical solutions for the linear and nonlinear singular boundary value problems using Laguerre wavelets. Adv Differ Equ 2016(1):1–5
Babolian E, Fattahzadeh F (2007) Numerical solution of differential equations by using Chebyshev wavelet operational matrix of integration. Appl Math Comput 188(1):417–426
Shiralashetti SC, Kumbinarasaiah S (2018) Hermite wavelets operational matrix of integration for the numerical solution of nonlinear singular initial value problems. Alex Eng J 57(4):2591–2600
Vivek, Kumar M, Mishra SN (2023) A fast Fibonacci wavelet-based numerical algorithm for the solution of HIV-infected CD4+ T cells model. Eur Phys J Plus 138(5):458
Vivek, Kumar M, Mishra SN (2023) Solution of linear and nonlinear singular value problems using operational matrix of integration of Taylor wavelets. J Taibah Univ Sci 17(1):2241716
Vivek, Kumar M (2023) Spreading behavior of biological SIR system of a COVID-19 disease through a fast Taylor wavelet based numerical algorithm. Results Control Optim 13:100316
Tunç C, Tunç O (2022) On the fundamental analyses of solutions to nonlinear integro-differential equations of the second order. Mathematics 10(22):4235
Sohaib M, Haq S, Mukhtar S, Khan I (2018) Numerical solution of sixth-order boundary-value problems using Legendre wavelet collocation method. Results Phys 8:1204–1208
Sohaib M, Haq S (2020) An efficient wavelet-based method for numerical solution of nonlinear integral and integro-differential equations. Math Methods Appl Sci
Haq S, Sohaib M (2021) An enhanced wavelet based method for numerical solution of high order boundary value problems. J Mt Area Res 6:63–76
Ali I, Haq S, Ullah R, Arifeen SU (2023) Approximate solution of second order singular perturbed and obstacle boundary value problems using meshless method based on radial basis functions. J Nonlinear Math Phys 30(1):215–234
Haq S, Ali I (2022) Approximate solution of two-dimensional Sobolev equation using a mixed Lucas and Fibonacci polynomials. Eng Comput 38(Suppl 3):2059–2068
Vivek, Mishra SN, Kumar M (2023) Taylor wavelet approach for the solution of the Fredholm integro-differential equation of the second kind. Jñānābha 53(2):273–286
Bobisud LE (1990) Existence of solutions for nonlinear singular boundary value problems. Appl Anal 35(1–4):43–57
Abukhaled M, Khuri SA, Sayfy A (2011) A numerical approach for solving a class of singular boundary value problems arising in physiology. Int J Numer Anal Model 8(2):353–363
Singh R, Guleria V, Singh M (2020) Haar wavelet quasilinearization method for numerical solution of Emden-Fowler type equations. Math Comput Simul 174:123–133
Dunninger DR, Kurtz JC (1986) Existence of solutions for some nonlinear singular boundary value problems. J Math Anal 115(2):396–405
Pandey RK, Verma AK (2009) A note on existence-uniqueness results for a class of doubly singular boundary value problems. Nonlinear Anal Theory Methods Appl 71(7–8):3477–3487
Chawla MM, Shivakumar PN (1987) On the existence of solutions of a class of singular nonlinear two-point boundary value problems. J Comput Appl 19(3):379–388
Pandey RK (1996) On a class of weakly regular singular two-point boundary value problems II. Differ Equ 127(1):110–123
Biles DC, Robinson MP, Spraker JS (2002) A generalization of the Lane-Emden equation. J Math Anal 273(2):654–666
Aslanov A (2014) A singular initial-value problem for second order differential equations. Abstr. Appl. Anal. 2014:1085–3375
Sun J, Zhang G (2007) Nontrivial solutions of singular sublinear Sturm-Liouville problems. J Math Anal 326(1):242–251
Liu Y, Yu H (2005) Existence and uniqueness of positive solution for singular boundary value problem. Comput Math Appl 50(1–2):133–143
Pandey RK (1997) On a class of regular singular two point boundary value problems. J Math Anal 208(2):388–403
Pandey RK (1996) On a class of weakly regular singular two point boundary value problems-I. Nonlinear Anal Theory Methods Appl 27(1):1–2
Kumbinarasaiah S, Mulimani M (2022) A novel scheme for the hyperbolic partial differential equation through Fibonacci wavelets. J Taibah Univ 16(1):1112–1132
Manohara G, Kumbinarasaiah S (2023) Fibonacci wavelets operational matrix approach for solving chemistry problems. J Umm Al-Qura Univ Appl Sci 9:393–410
Yadav P, Jahan S, Nisar KS (2023) Fibonacci wavelet collocation method for Fredholm integral equations of second kind. Qual Theory Dyn Syst 22(2):82
Manohara G, Kumbinarasaiah S (2023) Fibonacci wavelets operational matrix approach for solving chemistry problems. JUQU 9(4):1–18
Yousefi SA (2007) Legendre scaling function for solving generalized Emden-Fowler equations. Int J Inf Syst Sci 3(2):243–250
Nasab AK, Kılıçman A, Babolian E, Atabakan ZP (2013) Wavelet analysis method for solving linear and nonlinear singular boundary value problems. Appl Math Model 37(8):5876–5886
Doha EH, Abd-Elhameed WM, Youssri YH (2013) Second kind Chebyshev operational matrix algorithm for solving differential equations of Lane-Emden type. New Astron 23:113–117
Acknowledgements
Both the authors developed the fundamental concept of the article, wrote the text, and completed all phases of the research’s proofs. Vivek, one of the authors, performed the computational work related to the method using the MATLAB software.
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Vivek developed the fundamental concept of the article, wrote the text, and completed all phases of the research’s proofs.
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Vivek, Kumar, M. Solution of linear and nonlinear singular value problems using operational matrix of integration of Fibonacci wavelets. J Eng Math 145, 19 (2024). https://doi.org/10.1007/s10665-024-10350-6
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DOI: https://doi.org/10.1007/s10665-024-10350-6