Lagrange Operational Matrix Methods to Lane–Emden, Riccati’s and Bessel’s Equations

  • Vinita Devi
  • Rahul Kumar Maurya
  • Vijay Kumar Patel
  • Vineet Kumar SinghEmail author
Original paper


The current study is presented to develop two approaches and methodologies to find the numerical solution of linear and non-linear initial value problems such as Lane–Emden type equation, Riccati’s equation and Bessel’s equation of order zero based on approximation. The function approximations (scheme-I and scheme-II) are presented to find the numerical solutions of linear and non-linear initial value problems by using Gauss Legendre roots as node points and random node points in the domain [0, 1]. In the scheme-I, the roots of Legendre polynomial are used as node points for Lagrange polynomials and in scheme-II, we have taken random node points in the domain [0, 1] and orthogonalize the resulting Lagrange polynomials using Gram–Schmidt orthogonalization process. Firstly, we have introduced the function approximations by using generating interpolating scaling functions (ISF) and orthonormal Lagrangian basis functions (OLBF) over the space \(L^{2}[0,1]\) then we have constructed the operational matrices of integration and product operational matrices based on newly designed approximations namely ISF and OLBF. These operational matrices convert given linear and non-linear initial value problems into the associated system of algebraic equations. Finally, we have established error bounds (Lemmas 1, 2) of both scheme-I and scheme-II including the function approximations. The efficiency of the proposed schemes has been confirmed with several test examples including numerical stability. So, the schemes are simple, efficient and produces very accurate numerical results in considerably small number of basis functions and hence reduces computational effort.


Lagrange polynomials Lane–Emden type equation Riccati’s equation Bessel’s equation 



The first author acknowledges the financial support from Council of Scientific and Industrial Research (CSIR), India, under JRF scheme. The second author acknowledges the financial support from Ministry of Human Resource and Development (MHRD), India, under Junior Research Fellow (JRF) scheme. In addition, the corresponding author acknowledges the financial support from Science and Engineering Research Board, India, with Sanction Order No. YSS/2015/001017. The authors are thankful to the reviewers and editor of the journal for their valuable suggestions which have improved the quality of the paper.


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Copyright information

© Springer Nature India Private Limited 2019

Authors and Affiliations

  • Vinita Devi
    • 1
  • Rahul Kumar Maurya
    • 1
  • Vijay Kumar Patel
    • 2
  • Vineet Kumar Singh
    • 1
    Email author
  1. 1.Department of Mathematical SciencesIndian Institute of Technology (Banaras Hindu University)VaranasiIndia
  2. 2.Department of MathematicsNational Institute of TechnologyTiruchirappalliIndia

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