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Regularization and Error Estimate for the Poisson Equation with Discrete Data

Author

Listed:
  • Nguyen Anh Triet

    (Faculty of Natural Sciences, Thu Dau Mot University, Thu Dau Mot City 820000, Binh Duong Province, Vietnam
    These authors contributed equally to this work.)

  • Nguyen Duc Phuong

    (Faculty of Fundamental Science, Industrial University of Ho Chi Minh City, Ho Chi Minh City 700000, Vietnam
    These authors contributed equally to this work.)

  • Van Thinh Nguyen

    (Department of Civil and Environmental Engineering, Seoul National University, Seoul 08826, South Korea
    These authors contributed equally to this work.)

  • Can Nguyen-Huu

    (Applied Analysis Research Group, Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City 700000, Vietnam
    These authors contributed equally to this work.)

Abstract

In this work, we focus on the Cauchy problem for the Poisson equation in the two dimensional domain, where the initial data is disturbed by random noise. In general, the problem is severely ill-posed in the sense of Hadamard, i.e., the solution does not depend continuously on the data. To regularize the instable solution of the problem, we have applied a nonparametric regression associated with the truncation method. Eventually, a numerical example has been carried out, the result shows that our regularization method is converged; and the error has been enhanced once the number of observation points is increased.

Suggested Citation

  • Nguyen Anh Triet & Nguyen Duc Phuong & Van Thinh Nguyen & Can Nguyen-Huu, 2019. "Regularization and Error Estimate for the Poisson Equation with Discrete Data," Mathematics, MDPI, vol. 7(5), pages 1-20, May.
    • Handle: RePEc:gam:jmathe:v:7:y:2019:i:5:p:422-:d:230082
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    References listed on IDEAS

    as
    1. Tuan, Nguyen Huy & Thang, Le Duc & Khoa, Vo Anh, 2015. "A modified integral equation method of the nonlinear elliptic equation with globally and locally Lipschitz source," Applied Mathematics and Computation, Elsevier, vol. 265(C), pages 245-265.
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