Skip to main content
Springer Nature Link
Log in

Hybrid model for the optimal numerical solution of nonlinear ordinary differential systems

  • Published:
Computational and Applied Mathematics Aims and scope Submit manuscript

Abstract

In this article, we solve a class of coupled systems of nonlinear differential equations with appropriate initial, boundary, and four-point boundary conditions. We use quasilinearization to linearise these systems of equations and then use the Haar wavelets collocation approach to get the numerical solutions. We propose three quasilinearization schemes and observe that among them two schemes converge faster than the third one. We have also compared our results with other existing results that demonstrate the accuracy and effectiveness of our approach. The convergence of the schemes is also presented. The theory can be used to design and analyse the algorithm for the solution of SBVPs with the aid of Haar Wavelets.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8

Similar content being viewed by others

References

  • Abdeljawad T, Amin R, Shah K, Al-Mdallal Q, Jarad F (2020) Efficient sustainable algorithm for numerical solutions of systems of fractional order differential equations by Haar wavelet collocation method. Alexandria Eng J 59(4):2391–2400

    Google Scholar 

  • Aghazadeh N, Ahmadnezhad G, Rezapour S (2020) Haar wavelet iteration method for solving time fractional Fisher’s equation. Comput Methods Differ Equ 8(3):505–22

    MathSciNet  MATH  Google Scholar 

  • Alaidarous ES, Ullah MZ, Ahmad F, Al-Fhaid AS (2013) An efficient higher-order quasilinearization method for solving nonlinear bvps. J Appl Math 2013:11

    MathSciNet  MATH  Google Scholar 

  • Amin R, Shah K, Asif M, Khan I, Ullah F (2021) An efficient algorithm for numerical solution of fractional integro-differential equations via Haar wavelet. J Comput Appl Math 381:113028

    MathSciNet  MATH  Google Scholar 

  • Babolian E, Shahsavaran A (2009) Numerical solution of nonlinear Fredholm integral equations of the second kind using Haar wavelets. J Comput Appl Math 225(1):87–95

    MathSciNet  MATH  Google Scholar 

  • Barnwal AK, Pathak P (2019) Successive iteration technique for singular nonlinear system with four-point boundary conditions. J Appl Math Comput 62(1–2):301–24

    MathSciNet  MATH  Google Scholar 

  • Bellman RE, Kalaba RE (1965) Quasilinearization and non-linear boundary value problems, vol 52. American Elsevier Publishing Company Inc., New York, pp 212–212

    MATH  Google Scholar 

  • Carlo C (2001) Haar wavelet splines. J Interdiscip Math 4(1):35–47

    MathSciNet  MATH  Google Scholar 

  • Chen CF, Hsiao CH (1997) Haar wavelet method for solving lumped and distributed-parameter systems. IEE Proc Control Theory Appl 144(1):87–94

    MathSciNet  MATH  Google Scholar 

  • Flockerzi D, Sundmacher K (2011) On coupled Lane-Emden equations arising in dusty fluid models. J Phys Conf Ser 268:012006

    Google Scholar 

  • Hao T, Cong F, Shang Y (2018) An efficient method for solving coupled Lane-Emden boundary value problems in catalytic diffusion reactions and error estimate. J Math Chem 56(9):2691–2706

    MathSciNet  MATH  Google Scholar 

  • He J, Ji F (2019) Taylor series solution for Lane-Emden equation. J Math Chem 57(8):1932–1934

    MathSciNet  MATH  Google Scholar 

  • Heydari M, Avazzadeh Z, Hosseinzadeh N (2022) Haar wavelet method for solving high-order differential equations with multi-point boundary conditions. J Appl Comput Mech 8:528–544

    Google Scholar 

  • Hsiao CH (2004) Haar wavelet approach to linear stiff systems. Math Comput Simul 64(5):561–567

    MathSciNet  MATH  Google Scholar 

  • Hsiao C-H, Wang W-J (2001) Haar wavelet approach to nonlinear stiff systems. Math Comput Simul 57(6):347–353

    MathSciNet  MATH  Google Scholar 

  • Islam S, Aziz I, Šarler B (2010) The numerical solution of second-order boundary-value problems by collocation method with the Haar wavelets. Math Comput Model 52(9):1577–1590

    MathSciNet  MATH  Google Scholar 

  • Kaur H, Mittal RC, Mishra V (2011) Haar wavelet quasilinearization approach for solving nonlinear boundary value problems. Am J Comput Math 1(3):176–182

    Google Scholar 

  • Kaur H, Mittal RC, Mishra V (2013a) Haar wavelet approximate solutions for the generalized Lane-Emden equations arising in astrophysics. Comput Phys Commun 184(9):2169–2177

    MathSciNet  MATH  Google Scholar 

  • Kaur H, Mittal RC, Mishra V (2013b) A collocation approach with uniform Haar wavelets to solve Volterra population model for population growth of a species. Math Sci Int Res J 2

  • Kaur H, Mishra V, Mittal RC (2013c) Numerical solution of a laminar viscous flow boundary layer equation using uniform Haar wavelet quasi-linearization method. In: World Academy of Science, Engineering and Technology, p 79

  • Kaur H, Mittal RC, Mishra V (2014) Haar wavelet solutions of nonlinear oscillator equations. Appl Math Model 38(21):4958–4971

    MathSciNet  MATH  Google Scholar 

  • Kumar A, Hashmi MS, Ansari AQ, Arzykulov S (2020) Haar wavelet based algorithm for solution of second order electromagnetic problems in time and space domains. J Electromagn Waves Appl 34(3):362–374

    Google Scholar 

  • Lepik U (2005) Numerical solution of differential equations using Haar wavelets. Math Comput Simul 68:127–143

    MathSciNet  MATH  Google Scholar 

  • Lepik U (2007) Numerical solution of evolution equations by the Haar wavelet method. Appl Math Comput 185(695–704):02

    MathSciNet  MATH  Google Scholar 

  • Lepik U, Hein H (2014) Haar wavelets with applications. Springer, Cham

    MATH  Google Scholar 

  • Maarjus Kirs M, Mikola A, Haavajoe E, Ounapuu BS, Majak J (2016) Haar wavelet method for vibration analysis of nanobeams. Waves Wavelets Fract 2:20–28

    Google Scholar 

  • Mahalakshmi M, Hariharan G (2019) An efficient wavelet-based method for the numerical solutions of nonlinear coupled reaction-diffusion equations in biochemical engineering. J Math Chem 57(4):1154–1168

    MathSciNet  MATH  Google Scholar 

  • Majak J, Shvartsman B, Karjust K, Mikola M, Haavaj A, Pohlak M (2015a) On the accuracy of the Haar wavelet discretization method. Compos B Eng 80:321–327

    Google Scholar 

  • Majak J, Shvartsman BS, Kirs M, Pohlak M, Herranen H (2015b) Convergence theorem for the Haar wavelet based discretization method. Compos Struct 126:227–232

    Google Scholar 

  • Majak J, Shvartsman B, Ratas M, Bassir D, Pohlak M, Karjust K, Eerme M (2020) Higher-order Haar wavelet method for vibration analysis of nanobeams. Mater Today Commun 25:101290

    Google Scholar 

  • Oruç Ö (2019) A non-uniform Haar wavelet method for numerically solving two-dimensional convection-dominated equations and two-dimensional near singular elliptic equations. Comput Math Appl 77(7):1799–1820

    MathSciNet  MATH  Google Scholar 

  • Oruç Ö, Fatih B, Alaattin E (2015) A Haar wavelet finite difference hybrid method for the numerical solution of the modified Burgers’ equation. J Math Chem 53:1592–1607

    MathSciNet  MATH  Google Scholar 

  • Oruç Ö, Fatih B, Alaattin E (2016) Numerical solutions of regularized long wave equation by Haar wavelet method. Mediterr J Math 13:3235–3253

    MathSciNet  MATH  Google Scholar 

  • Oruç Ö, Bulut F, Esen A (2017) A numerical treatment based on Haar wavelets for coupled KdV equation. Int J Optimiz Control Theor Appl (IJOCTA) 7:195–204

    MathSciNet  Google Scholar 

  • Oruç Ö, Esen A, Bulut F (2019) A Haar wavelet approximation for two-dimensional time fractional reaction-subdiffusion equation. Eng Comput 35:75–86

    Google Scholar 

  • Ozturk Y (2018) An efficient numerical algorithm for solving system of Lane-Emden type equations arising in engineering. Nonlinear Eng 8(1):429–437

    Google Scholar 

  • Pandey RK, Verma AK (2010a) Monotone method for singular bvp in the presence of upper and lower solutions. Appl Math Comput 215(11):3860–3867

    MathSciNet  MATH  Google Scholar 

  • Pandey RK, Verma AK (2010b) On solvability of derivative dependent doubly singular boundary value problems. J Appl Math Comput 33(1):489–511

    MathSciNet  MATH  Google Scholar 

  • Pervaiz N, Aziz I (2020) Haar wavelet approximation for the solution of cubic nonlinear Schrodinger equations. Phys A 545:123738

    MathSciNet  Google Scholar 

  • Rach R, Duan J, Wazwaz A (2014) Solving coupled Lane-Emden boundary value problems in catalytic diffusion reactions by the Adomian decomposition method. J Math Chem 52:255–267

    MathSciNet  MATH  Google Scholar 

  • Rehman M, Khan RA (2012) A numerical method for solving boundary value problems for fractional differential equations. Appl Math Model 36(3):894–907

    MathSciNet  MATH  Google Scholar 

  • Ruyun M (2000) Multiple nonnegative solutions of second-order systems of boundary value problems. Nonlinear Anal Theory Methods Appl 42(6):1003–1010

    MathSciNet  MATH  Google Scholar 

  • Singh R (2020) Solving coupled Lane-Emden equations by Green’s function and decomposition technique. Int J Appl Comput Math 6:80

    MathSciNet  MATH  Google Scholar 

  • Singh M, Verma AK (2017) Nonlinear three point singular bvps: a classification. Commun Appl Anal 21(4):513–532

    MathSciNet  Google Scholar 

  • Singh R, Wazwaz AM (2019) An efficient algorithm for solving coupled Lane-Emden boundary value problems in catalytic diffusion reactions: the homotopy analysis method. MATCH Commun Math Comput Chem 81:785–800

    MATH  Google Scholar 

  • Singh M, Verma AK, Agarwal RP (2015) Maximum and anti-maximum principles for three point sbvps and nonlinear three point sbvps. J Appl Math Comput 47(1):249–263

    MathSciNet  MATH  Google Scholar 

  • Singh R, Garg H, Guleria V (2019a) Haar wavelet collocation method for Lane-Emden equations with Dirichlet, Neumann and Neumann-Robin boundary conditions. J Comput Appl Math 346:150–161

    MathSciNet  MATH  Google Scholar 

  • Singh R, Shahni J, Garg H, Garg A (2019b) Haar wavelet collocation approach for Lane-Emden equations arising in mathematical physics and astrophysics. Eur Phys J Plus 134(11):548

    Google Scholar 

  • 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

    MathSciNet  MATH  Google Scholar 

  • Swati KS, Verma AK, Singh M (2020) Higher order Emden-Fowler type equations via uniform Haar wavelet resolution technique. J Comput Appl Math 376:112836

  • Talwalkar S, Mankar S, Katariya A, Aghalayam P, Ivanova M, Sundmacher K, Mahajani S (2007) Selectivity engineering with reactive distillation for dimerization of c4 olefins: experimental and theoretical studies. Ind Eng Chem Res 46:3024–3034

    Google Scholar 

  • Verma AK (2011) The monotone iterative method and zeros of Bessel functions for nonlinear singular derivative dependent bvp in the presence of upper and lower solutions. Nonlinear Anal Theory Methods Appl 74(14):4709–4717

    MathSciNet  MATH  Google Scholar 

  • Verma AK, Singh M (2014) Existence of solutions for three-point bvps arising in bridge design. Electron J Differ Equ 2014(173):1–11

    MathSciNet  MATH  Google Scholar 

  • Verma AK, Singh M (2015a) A note on existence results for a class of three-point nonlinear bvps. Math Model Anal 20(4):457–470

    MathSciNet  MATH  Google Scholar 

  • Verma AK, Singh M (2015b) Singular nonlinear three point bvps arising in thermal explosion in a cylindrical reactor. J Math Chem 53(2):670–684

    MathSciNet  MATH  Google Scholar 

  • Verma AK, Tiwari D (2019a) Higher resolution methods based on quasilinearization and Haar wavelets on Lane-Emden equations. Int J Wavelets Multiresol Inf Process 17(03):1950005

    MathSciNet  MATH  Google Scholar 

  • Verma AK, Tiwari D (2019b) A note on Legendre, Hermite, Chebyshev, Laguerre and Gegenbauer wavelets with an application on sbvps arising in real life. arXiv

  • Verma AK, Tiwari D (2019c) On some computational aspects of Hermite wavelets on a class of sbvps arising in exothermic reactions. arXiv

  • Verma AK, Pandit B, Verma L, Agarwal RP (2020a) A review on a class of second order nonlinear singular BVPs. Mathematics 8(7):1045

    Google Scholar 

  • Verma AK, Kumar N, Tiwari D (2020b) Haar wavelets collocation method for a system of nonlinear singular differential equations. Eng Comput 38(2):659–98

    Google Scholar 

  • Verma AK, Urus N, Singh M (2020c) Monotone iterative technique for a class of four point bvps with reversed ordered upper and lower solutions. Int J Comput Methods 17(9):1950066

    MathSciNet  MATH  Google Scholar 

  • Verma L, Pandit B, Verma AK (2020d) Taylor series solution of some real life problems: Odes and pdes. Preprint

  • Wazwaz AM, Rach R, Duan J (2014) A study on the systems of the Volterra integral forms of the Lane-Emden equations by the Adomian decomposition method. Math Methods Appl Sci 37(1):10–19

    MathSciNet  MATH  Google Scholar 

  • Xie L, Zhou C, Xu S (2019) Solving the systems of equations of Lane-Emden type by differential transform method coupled with Adomian polynomials. Mathematics 7(4):377

    Google Scholar 

  • Zhang H, Sun J (2009) Existence of positive solution to singular systems of second-order four-point bvps. J Appl Math Comput 29(1):325–339

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, IIT Jodhpur, Karwar, Jodhpur, 342037, India

    Narendra Kumar

  2. Faculty of Mathematics, University of Vienna, Vienna, Austria

    Diksha Tiwari

  3. Department of Mathematics, IIT Patna, Bihta, Patna, 801103, India

    Amit K. Verma

  4. Engineering School (DEIM), University of Tuscia, Largo dell’Università, 01100, Viterbo, Italy

    Carlo Cattani

Authors
  1. Narendra Kumar
    View author publications

    You can also search for this author inPubMed Google Scholar

  2. Diksha Tiwari
    View author publications

    You can also search for this author inPubMed Google Scholar

  3. Amit K. Verma
    View author publications

    You can also search for this author inPubMed Google Scholar

  4. Carlo Cattani
    View author publications

    You can also search for this author inPubMed Google Scholar

Corresponding author

Correspondence to Amit K. Verma.

Additional information

Communicated by Eduardo Souza de Cursi.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kumar, N., Tiwari, D., Verma, A.K. et al. Hybrid model for the optimal numerical solution of nonlinear ordinary differential systems. Comp. Appl. Math. 42, 322 (2023). https://doi.org/10.1007/s40314-023-02468-7

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s40314-023-02468-7

Keywords

Mathematics Subject Classification