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The Numerical Validation of the Adomian Decomposition Method for Solving Volterra Integral Equation with Discontinuous Kernels Using the CESTAC Method

Author

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  • Samad Noeiaghdam

    (Industrial Mathematics Laboratory, Baikal School of BRICS, Irkutsk National Research Technical University, 664074 Irkutsk, Russia
    Department of Applied Mathematics and Programming, South Ural State University, Lenin Prospect 76, 454080 Chelyabinsk, Russia)

  • Denis Sidorov

    (Industrial Mathematics Laboratory, Baikal School of BRICS, Irkutsk National Research Technical University, 664074 Irkutsk, Russia
    Energy Systems Institute of Siberian Branch of Russian Academy of Science, 664033 Irkutsk, Russia)

  • Abdul-Majid Wazwaz

    (Department of Mathematics, Saint Xavier University, Chicago, IL 60655, USA)

  • Nikolai Sidorov

    (Institute of Mathematics and Information Technologies, Irkutsk State University, 1 Karl Marx Str., 664003 Irkutsk, Russia)

  • Valery Sizikov

    (Faculty of Software Engineering and Computer Systems, ITMO University, 49 Kronverksky Prospect, 197101 Saint Petersrburg, Russia)

Abstract

The aim of this paper is to present a new method and the tool to validate the numerical results of the Volterra integral equation with discontinuous kernels in linear and non-linear forms obtained from the Adomian decomposition method. Because of disadvantages of the traditional absolute error to show the accuracy of the mathematical methods which is based on the floating point arithmetic, we apply the stochastic arithmetic and new condition to study the efficiency of the method which is based on two successive approximations. Thus the CESTAC method (Controle et Estimation Stochastique des Arrondis de Calculs) and the CADNA (Control of Accuracy and Debugging for Numerical Applications) library are employed. Finding the optimal iteration of the method, optimal approximation and the optimal error are some of advantages of the stochastic arithmetic, the CESTAC method and the CADNA library in comparison with the floating point arithmetic and usual packages. The theorems are proved to show the convergence analysis of the Adomian decomposition method for solving the mentioned problem. Also, the main theorem of the CESTAC method is presented which shows the equality between the number of common significant digits between exact and approximate solutions and two successive approximations.This makes in possible to apply the new termination criterion instead of absolute error. Several examples in both linear and nonlinear cases are solved and the numerical results for the stochastic arithmetic and the floating-point arithmetic are compared to demonstrate the accuracy of the novel method.

Suggested Citation

  • Samad Noeiaghdam & Denis Sidorov & Abdul-Majid Wazwaz & Nikolai Sidorov & Valery Sizikov, 2021. "The Numerical Validation of the Adomian Decomposition Method for Solving Volterra Integral Equation with Discontinuous Kernels Using the CESTAC Method," Mathematics, MDPI, vol. 9(3), pages 1-15, January.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:3:p:260-:d:488608
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    References listed on IDEAS

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    1. Vignes, J., 1993. "A stochastic arithmetic for reliable scientific computation," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 35(3), pages 233-261.
    2. Saelao, Jeerawan & Yokchoo, Natsuda, 2020. "The solution of Klein–Gordon equation by using modified Adomian decomposition method," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 171(C), pages 94-102.
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    Cited by:

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    2. Bashir, Azhar & Seadawy, Aly R. & Ahmed, Sarfaraz & Rizvi, Syed T.R., 2022. "The Weierstrass and Jacobi elliptic solutions along with multiwave, homoclinic breather, kink-periodic-cross rational and other solitary wave solutions to Fornberg Whitham equation," Chaos, Solitons & Fractals, Elsevier, vol. 163(C).
    3. Seadawy, Aly R. & Rizvi, Syed T.R. & Ahmed, Sarfaraz, 2022. "Multiple lump, generalized breathers, Akhmediev breather, manifold periodic and rogue wave solutions for generalized Fitzhugh-Nagumo equation: Applications in nuclear reactor theory," Chaos, Solitons & Fractals, Elsevier, vol. 161(C).

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