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Two-dimensional wavelets scheme for numerical solutions of linear and nonlinear Volterra integro-differential equations

Author

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  • Behera, S.
  • Saha Ray, S.

Abstract

In this article, an effective approach has been proposed to obtain the approximate solutions of linear and nonlinear two-dimensional Volterra integro-differential equations. First, the two-dimensional wavelets are introduced, and using it the operational matrices of integration, differentiation, and product have been constructed. Then, by utilizing the properties and matrices of wavelets along with the collocation point, the matrix form of Volterra integro-differential equations has been derived. This approach reduces the two-dimensional linear and nonlinear Volterra integro-differential equations into the system of linear and nonlinear algebraic equations respectively. The convergence analysis and error analysis have been extensively studied by the help of two-dimensional wavelets approximation. Some illustrative examples are examined to clarify the accuracy and effectiveness of the proposed scheme. The graphical representations obtained by two proposed wavelets have been plotted to justify the applicability and validity of the method.

Suggested Citation

  • Behera, S. & Saha Ray, S., 2022. "Two-dimensional wavelets scheme for numerical solutions of linear and nonlinear Volterra integro-differential equations," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 198(C), pages 332-358.
    • Handle: RePEc:eee:matcom:v:198:y:2022:i:c:p:332-358
    DOI: 10.1016/j.matcom.2022.02.018
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    References listed on IDEAS

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    1. Sahu, P.K. & Saha Ray, S., 2015. "Legendre spectral collocation method for Fredholm integro-differential-difference equation with variable coefficients and mixed conditions," Applied Mathematics and Computation, Elsevier, vol. 268(C), pages 575-580.
    2. Behera, S. & Ray, S. Saha, 2020. "An operational matrix based scheme for numerical solutions of nonlinear weakly singular partial integro-differential equations," Applied Mathematics and Computation, Elsevier, vol. 367(C).
    3. Postnikov, Eugene B. & Lebedeva, Elena A. & Lavrova, Anastasia I., 2016. "Computational implementation of the inverse continuous wavelet transform without a requirement of the admissibility condition," Applied Mathematics and Computation, Elsevier, vol. 282(C), pages 128-136.
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