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An Efficient Modified Haar Wavelet Collocation Method for Numerical Solution of Two-Dimensional Elliptic PDEs

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Abstract

In this work, we present a new method for solving elliptic partial differential equations using Haar wavelet. This work improves the earlier work (Aziz et al. in Appl Math Model 37:676–694, 2013) in terms of efficiency and contains an extension to nonlinear elliptic partial differential equations as well. In this paper the earlier algorithm (Aziz et al. in Appl Math Model 37:676–694, 2013) has been modified by starting the approximation with a fourth order mixed derivative rather than approximation of the second order derivatives with respect to x and y separately which results in a more efficient algorithm than the earlier algorithm. The use of Kronecker tensor products makes the new algorithm robust and easier to implement in a programming language. A distinguishing feature of the new method is that it can be applied to a variety of boundary conditions with a little modification of the program. The method is tested on several benchmark linear as well as nonlinear models. The numerical results show convergence, simple applicability and efficiency of the method.

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Authors and Affiliations

  1. Department of Mathematics, University of Peshawar, Peshawar, Pakistan

    Imran Aziz

  2. Department of Basic Sciences, University of Engineering and Technology, Peshawar, Pakistan

    Siraj-ul-Islam

  3. Department of Mathematics, South Asian University, Akbar Bhawan, New Delhi, India

    Siraj-ul-Islam

Authors
  1. Imran Aziz
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  2. Siraj-ul-Islam
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Correspondence to Imran Aziz.

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Aziz, I., Siraj-ul-Islam An Efficient Modified Haar Wavelet Collocation Method for Numerical Solution of Two-Dimensional Elliptic PDEs. Differ Equ Dyn Syst 25, 347–360 (2017). https://doi.org/10.1007/s12591-015-0262-x

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  • DOI: https://doi.org/10.1007/s12591-015-0262-x

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