Three Regularization Methods for A Homogeneous Time Fractional Diffusion Equation with Space - Dependent Diffusivity Coefficient
Main Article Content
Abstract
We studied an inverse problem for a time fractional diffusion equation with space - dependent diffusivity coefficient in the nonhomogeneous case. This problem is ill – posed in Hadamard’s sense. So, a regularization is essential. Using three regularization methods, we get the estimation of the error between the regularized solution and the exact solution
Keywords:
Time fractional diffusion equation, inverse problem, regularization, space - dependent coefficient, error estimate.
References
[1] J. J. Liu, M. Yamamoto, A Backward Problem for the Time – Fractional Diffusion Equation, Applicable Analysis, Vol. 89, 2010, pp. 1769-1788, https://doi.org/10.1080/00036810903479731.
[2] K. Sakamoto, M. Yamamoto, Initial Value/Boundary Value Problemsf Fractional Diffusion – Wave Equations and Applications to Some Inverse Problems, Journal of Mathematical Analysis and Applications, Vol. 382, Iss. 1, 2011, pp. 426-447, https://doi.org/10.1016/j.jmaa.2011.04.058.
[3] N. H. Tuan, L. N. Huynh, T. B. Ngoc, Y. Zhou, On a Backward Problem for Nonlinearfractional Diffusion Equations, Applied Mathematics Letters, Vol. 92, 2019, pp. 76-84, https://doi.org/10.1016/j.aml.2018.11.015.
[4] K. Sakamoto, M. Yamamoto, Inverse Source Problem With A Final Overdetermination For A Fractional Diffusion Equation, Mathematical Control and Related Field, Vol. 1, No. 4, 2011, pp. 509-518, https://doi.org/10.3934/mcrf.2011.1.509.
[5] H. Cheng, C. L. Fu, An Iteration Regularization for A Time – Fractional Inverse Diffusion Problem, Applied Mathematical Modelling, Vol. 36, Iss. 11, 2012, pp. 5642-5649, https://doi.org/10.1016/j.apm.2012.01.016.
[6] G. H. Zheng, T. Wei, Spectral, Regularization Method for Solving A Time – Fractional Inverse Diffusion Problem Applied Mathematics and Computation, Vol. 218, Iss. 2, 2011, pp. 396-405, https://doi.org/10.1016/j.amc.2011.05.076.
[7] G. H. Zheng, T. Wei, A New Regularization Method For Solving A Time – Fractional Inverse Diffusion Problem, Journal of Mathematical Analysis and Applications, Vol. 378, Iss. 2, 2011, pp. 418-431, https://doi.org/10.1016/j.jmaa.2011.01.067.
[8] L. V. C. Hoan, H. D. Binh, T. B. Ngoc, A Truncation Regularization Method for A Time – Fractional Diffusion Equation with An In – homogeneous Source, ITM Web of Conferences, Vol. 20, 2018, pp. 1-9.
[9] N. H. Tuan, L. V. C. Hoan, S. Tatar, An Inverse Problem for An Inhomogeneous Time – Fractional Diffusion Equation: A Regularization Method and Error Estimate, Computational and Applied Mathematics, Vol. 38, 2019.
[10] N. H. Tuan, M. Kirane, L. V. C. Hoan, B. Bin – Mohsin, A regularization Method for Time – Fractional Linear Inverse Diffusion Problems, Electronic Journal of Differential Equations, Vol. 2016, No. 290, 2016, pp. 1-18.
[2] K. Sakamoto, M. Yamamoto, Initial Value/Boundary Value Problemsf Fractional Diffusion – Wave Equations and Applications to Some Inverse Problems, Journal of Mathematical Analysis and Applications, Vol. 382, Iss. 1, 2011, pp. 426-447, https://doi.org/10.1016/j.jmaa.2011.04.058.
[3] N. H. Tuan, L. N. Huynh, T. B. Ngoc, Y. Zhou, On a Backward Problem for Nonlinearfractional Diffusion Equations, Applied Mathematics Letters, Vol. 92, 2019, pp. 76-84, https://doi.org/10.1016/j.aml.2018.11.015.
[4] K. Sakamoto, M. Yamamoto, Inverse Source Problem With A Final Overdetermination For A Fractional Diffusion Equation, Mathematical Control and Related Field, Vol. 1, No. 4, 2011, pp. 509-518, https://doi.org/10.3934/mcrf.2011.1.509.
[5] H. Cheng, C. L. Fu, An Iteration Regularization for A Time – Fractional Inverse Diffusion Problem, Applied Mathematical Modelling, Vol. 36, Iss. 11, 2012, pp. 5642-5649, https://doi.org/10.1016/j.apm.2012.01.016.
[6] G. H. Zheng, T. Wei, Spectral, Regularization Method for Solving A Time – Fractional Inverse Diffusion Problem Applied Mathematics and Computation, Vol. 218, Iss. 2, 2011, pp. 396-405, https://doi.org/10.1016/j.amc.2011.05.076.
[7] G. H. Zheng, T. Wei, A New Regularization Method For Solving A Time – Fractional Inverse Diffusion Problem, Journal of Mathematical Analysis and Applications, Vol. 378, Iss. 2, 2011, pp. 418-431, https://doi.org/10.1016/j.jmaa.2011.01.067.
[8] L. V. C. Hoan, H. D. Binh, T. B. Ngoc, A Truncation Regularization Method for A Time – Fractional Diffusion Equation with An In – homogeneous Source, ITM Web of Conferences, Vol. 20, 2018, pp. 1-9.
[9] N. H. Tuan, L. V. C. Hoan, S. Tatar, An Inverse Problem for An Inhomogeneous Time – Fractional Diffusion Equation: A Regularization Method and Error Estimate, Computational and Applied Mathematics, Vol. 38, 2019.
[10] N. H. Tuan, M. Kirane, L. V. C. Hoan, B. Bin – Mohsin, A regularization Method for Time – Fractional Linear Inverse Diffusion Problems, Electronic Journal of Differential Equations, Vol. 2016, No. 290, 2016, pp. 1-18.