Legendre multi-wavelets collocation method for numerical solution of linear and nonlinear integral equations

Muhammad Asif, Imran Khan, Nadeem Haider, Qasem Al-Mdallal

Research output: Contribution to journalArticlepeer-review

15 Citations (Scopus)


In this article, a new collocation technique for numerical solution of Fredholm, Volterra and mixed Volterra-Fredholm integral equations of the second kind is introduced and also developed a numerical integration formula on the basis of linear Legendre multi-wavelets. We also use the linear Legendre multi-wavelets basis for the proposed method. In this technique, the unknown function is approximated by truncated linear Legendre multi-wavelets series. The newly developed numerical technique is applied to both linear and nonlinear benchmark test models from the literature including models with discontinues and non-differentiable exact solutions. The numerical results are compared with the other existing numerical techniques from literature. The comparison among the discussed methods with Monte Carlo method, rationalized Haar functions method, Operational matrix with block-pulse functions method, Bernstein operational matrices method and P-order spline direct method divulge that the present technique is authentic and valid for other physical and engineering problems. The proposed numerical method handle the discontinuity and non-differentiability in the exact solutions very well, whereas the other existence numerical methods are failed to capture these problems. Further, more the comparison of exact and approximate in term of L norms which demonstrate the accuracy, flexibility and robustness of the newly proposed numerical method.

Original languageEnglish
Pages (from-to)5099-5109
Number of pages11
JournalAlexandria Engineering Journal
Issue number6
Publication statusPublished - Dec 2020


  • Linear Legendre multi-wavelets
  • Nonlinear Fredholm integral equations
  • Nonlinear Volterra integral equations
  • Nonlinear Volterra-Fredholm integral equations

ASJC Scopus subject areas

  • Engineering(all)


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