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A fractional Tikhonov regularization method for an inverse backward and source problems in the time-space fractional diffusion equations

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  • Djennadi, Smina
  • Shawagfeh, Nabil
  • Abu Arqub, Omar

Abstract

In this research, we deal with two types of inverse problems for diffusion equations involving Caputo fractional derivatives in time and fractional Sturm-Liouville operator for space. The first one is to identify the source term and the second one is to identify the initial value along with the solution in both cases. These inverse problems are proved to be ill-posed in the sense of Hadamard whenever an additional condition at the final time is given. A new fractional Tikhonov regularization method is used for the reconstruction of the stable solutions. Under the a-priori and the a-posteriori parameter choice rules, the error estimates between the exact and its regularized solutions are obtained. To illustrate the validity of our study, we give numerical examples. A final note is utilized in the ultimate section.

Suggested Citation

  • Djennadi, Smina & Shawagfeh, Nabil & Abu Arqub, Omar, 2021. "A fractional Tikhonov regularization method for an inverse backward and source problems in the time-space fractional diffusion equations," Chaos, Solitons & Fractals, Elsevier, vol. 150(C).
    • Handle: RePEc:eee:chsofr:v:150:y:2021:i:c:s0960077921004811
    DOI: 10.1016/j.chaos.2021.111127
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    References listed on IDEAS

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    1. Abu Arqub, Omar & Maayah, Banan, 2019. "Modulation of reproducing kernel Hilbert space method for numerical solutions of Riccati and Bernoulli equations in the Atangana-Baleanu fractional sense," Chaos, Solitons & Fractals, Elsevier, vol. 125(C), pages 163-170.
    2. Abu Arqub, Omar & Al-Smadi, Mohammed, 2018. "Atangana–Baleanu fractional approach to the solutions of Bagley–Torvik and Painlevé equations in Hilbert space," Chaos, Solitons & Fractals, Elsevier, vol. 117(C), pages 161-167.
    3. Atangana, Abdon, 2016. "On the new fractional derivative and application to nonlinear Fisher’s reaction–diffusion equation," Applied Mathematics and Computation, Elsevier, vol. 273(C), pages 948-956.
    4. Phuong, Nguyen Duc & Tuan, Nguyen Huy & Hammouch, Zakia & Sakthivel, Rathinasamy, 2021. "On a pseudo-parabolic equations with a non-local term of the Kirchhoff type with random Gaussian white noise," Chaos, Solitons & Fractals, Elsevier, vol. 145(C).
    5. Arqub, Omar Abu & Maayah, Banan, 2018. "Numerical solutions of integrodifferential equations of Fredholm operator type in the sense of the Atangana–Baleanu fractional operator," Chaos, Solitons & Fractals, Elsevier, vol. 117(C), pages 117-124.
    6. Abu Arqub, Omar & Al-Smadi, Mohammed, 2020. "An adaptive numerical approach for the solutions of fractional advection–diffusion and dispersion equations in singular case under Riesz’s derivative operator," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 540(C).
    7. Arqub, Omar Abu & Maayah, Banan, 2019. "Fitted fractional reproducing kernel algorithm for the numerical solutions of ABC – Fractional Volterra integro-differential equations," Chaos, Solitons & Fractals, Elsevier, vol. 126(C), pages 394-402.
    8. Al-Smadi, Mohammed & Arqub, Omar Abu, 2019. "Computational algorithm for solving fredholm time-fractional partial integrodifferential equations of dirichlet functions type with error estimates," Applied Mathematics and Computation, Elsevier, vol. 342(C), pages 280-294.
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    Cited by:

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    2. Ahmad, Zubair & Ali, Farhad & Khan, Naveed & Khan, Ilyas, 2021. "Dynamics of fractal-fractional model of a new chaotic system of integrated circuit with Mittag-Leffler kernel," Chaos, Solitons & Fractals, Elsevier, vol. 153(P2).
    3. Jun Lu & Lianpeng Shi & Chein-Shan Liu & C. S. Chen, 2022. "Solving Inverse Conductivity Problems in Doubly Connected Domains by the Homogenization Functions of Two Parameters," Mathematics, MDPI, vol. 10(13), pages 1-17, June.
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    6. M. Asha & T. P. Surekha, 2023. "Development of OFDM technique for underwater communication in system on chip," International Journal of System Assurance Engineering and Management, Springer;The Society for Reliability, Engineering Quality and Operations Management (SREQOM),India, and Division of Operation and Maintenance, Lulea University of Technology, Sweden, vol. 14(3), pages 977-988, June.

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