Two Dimensional Wavelets Collocation Scheme for Linear and Nonlinear Volterra Weakly Singular Partial Integro-Differential Equations

  • Vijay Kumar Patel
  • Somveer Singh
  • Vineet Kumar SinghEmail author
  • Emran Tohidi
Original Paper


In this article, we have study a 2D Legendre and Chebyshev wavelets collocation scheme for solving a class of linear and nonlinear weakly singular Volterra partial integro-differential equations (PIDEs). The scheme is based on wavelets collocation for PIDEs with uniquely designed matrices over the Hilbert space defined on the domain \(\left( [0, 1]\times [0, 1]\right) \). Using piecewise approximation associated with 2D Legendre wavelet, 2D Chebyshev wavelet and its operational matrices, the considered PIDEs will be reduced into the corresponding system of linear and nonlinear algebraic equations. The corresponding linear and nonlinear system of equations solved by collocation scheme and well-known Newton–Raphson scheme at collocation points respectively. In addition, the convergence and error analysis of the numerical scheme is provided under several mild conditions. The numerical results are correlated with the exact solutions and the execution of the proposed scheme is determined by estimating the maximum absolute errors, \(l_{2}\hbox {-}norm\) errors and \(l_{\infty }\hbox {-}norm\) errors. The numerical result shows that the scheme is simply applicable, efficient, powerful and very precisely at small number of basis function. The main important applications of the proposed wavelets collocation scheme is that it can be applied on linear as well as nonlinear problems and can be applied on higher order partial differential equations too.


Partial integro-differential equations Legendre wavelet Chebyshev wavelet Collocation scheme Operational matrices 

Mathematics Subject Classification

35R09 60J60 34K37 65L60 42C40 



The corresponding author acknowledge the financial support from Science and Engineering Research Board (SERB) with sanction order No. YSS/2015/001017. The authors are very grateful to the editor and referees for their valuable comments and suggestion for the improvement of the paper.


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Copyright information

© Springer Nature India Private Limited 2018

Authors and Affiliations

  • Vijay Kumar Patel
    • 1
  • Somveer Singh
    • 1
  • Vineet Kumar Singh
    • 1
    Email author
  • Emran Tohidi
    • 2
  1. 1.Department of Mathematical SciencesIndian Institute of Technology (Banaras Hindu University)VaranasiIndia
  2. 2.Department of MathematicsKosar University of BojnordBojnordIran

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