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Solving the nonlinear integro-differential equation in complex plane with rationalized Haar wavelet

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  • Erfanian, Majid
  • Mansoori, Amin

Abstract

We investigate mixed nonlinear integro-differential equations (MNIDEs) in general, utilizing the concept of rationalized Haar (RH) wavelet. The complexity of the MNIDE solution is known to everyone. For this purpose, we present a numerical method by applying the RH wavelet to approximate solutions of the MNIDE of the second kind in the complex plane. At first, we describe a continuous integral operator . Also, under mild assumptions, the Banach fixed point theorem ensures that the integral operator has a unique solution. Moreover, we give a result for error and compute the rate of convergence. Employing an algorithm, we present some illustrative examples to demonstrate the performance of this approach.

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  • Erfanian, Majid & Mansoori, Amin, 2019. "Solving the nonlinear integro-differential equation in complex plane with rationalized Haar wavelet," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 165(C), pages 223-237.
    • Handle: RePEc:eee:matcom:v:165:y:2019:i:c:p:223-237
    DOI: 10.1016/j.matcom.2019.03.006
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    References listed on IDEAS

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    1. Faezeh Toutounian & Emran Tohidi & Stanford Shateyi, 2013. "A Collocation Method Based on the Bernoulli Operational Matrix for Solving High-Order Linear Complex Differential Equations in a Rectangular Domain," Abstract and Applied Analysis, Hindawi, vol. 2013, pages 1-12, April.
    2. Sharma, Vaishali & Setia, Amit & Agarwal, Ravi P., 2018. "Numerical solution for system of Cauchy type singular integral equations with its error analysis in complex plane," Applied Mathematics and Computation, Elsevier, vol. 328(C), pages 338-352.
    3. Yousefi, S. & Razzaghi, M., 2005. "Legendre wavelets method for the nonlinear Volterra–Fredholm integral equations," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 70(1), pages 1-8.
    4. Erfanian, M. & Gachpazan, M. & Beiglo, H., 2015. "Rationalized Haar wavelet bases to approximate solution of nonlinear Fredholm integral equations with error analysis," Applied Mathematics and Computation, Elsevier, vol. 265(C), pages 304-312.
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    Cited by:

    1. Dossan Baigereyev & Dinara Omariyeva & Nurlan Temirbekov & Yerlan Yergaliyev & Kulzhamila Boranbek, 2022. "Numerical Method for a Filtration Model Involving a Nonlinear Partial Integro-Differential Equation," Mathematics, MDPI, vol. 10(8), pages 1-24, April.
    2. Mart Ratas & Jüri Majak & Andrus Salupere, 2021. "Solving Nonlinear Boundary Value Problems Using the Higher Order Haar Wavelet Method," Mathematics, MDPI, vol. 9(21), pages 1-12, November.
    3. Beiglo, H. & Gachpazan, M., 2020. "Numerical solution of nonlinear mixed Volterra-Fredholm integral equations in complex plane via PQWs," Applied Mathematics and Computation, Elsevier, vol. 369(C).
    4. Kumar, Sunil & Kumar, Ranbir & Cattani, Carlo & Samet, Bessem, 2020. "Chaotic behaviour of fractional predator-prey dynamical system," Chaos, Solitons & Fractals, Elsevier, vol. 135(C).

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