Abstract
In this paper, we apply a Landweber iterative regularization method to determine a space-dependent source for a time-space fractional diffusion-wave equation from the final measurement. In general, this problem is ill-posed, and a Landweber iterative regularization method is used to obtain the regularization solution. Under the a priori parameter choice rule and the a posteriori parameter choice rule, we give the error estimates between the regularization solution and the exact solution, respectively. Some numerical results in the one-dimensional and two-dimensional cases show the utility of the used regularization method.
Funding source: National Natural Science Foundation of China
Award Identifier / Grant number: 11961044
Funding source: Natural Science Foundation of Gansu Province
Award Identifier / Grant number: 21JR7RA214
Funding statement: The project is supported by the National Natural Science Foundation of China (No. 11961044), the Doctor Fund of Lan Zhou University of Technology, the Natural Science Foundation of Gansu Province (No. 21JR7RA214).
References
[1] A. Chatterjee, Statistical origins of fractional derivatives in viscoelasticity, J. Sound. Vib. 284 (2005), no. 3–5, 1239–1245. 10.1016/j.jsv.2004.09.019Search in Google Scholar
[2] W. Chen, L. Ye and H. Sun, Fractional diffusion equations by the Kansa method, Comput. Math. Appl. 59 (2010), no. 5, 1614–1620. 10.1016/j.camwa.2009.08.004Search in Google Scholar
[3] H. Cheng and C.-L. Fu, An iteration regularization for a time-fractional inverse diffusion problem, Appl. Math. Model. 36 (2012), no. 11, 5642–5649. 10.1016/j.apm.2012.01.016Search in Google Scholar
[4] B. I. Henry, T. A. Langlands and S. L. Wearne, Fractional cable models for spiny neuronal dendrites, Phys. Rev. Lett. 100 (2008), no. 12, 128–103. 10.1103/PhysRevLett.100.128103Search in Google Scholar PubMed
[5] R. Hilfer, On fractional diffusion and continuous time random walks, Phys. A 329 (2003), no. 1–2, 35–40. 10.1016/S0378-4371(03)00583-1Search in Google Scholar
[6] M. I. Ismailov and M. Çiçek, Inverse source problem for a time-fractional diffusion equation with nonlocal boundary conditions, Appl. Math. Model. 40 (2016), no. 7–8, 4891–4899. 10.1016/j.apm.2015.12.020Search in Google Scholar
[7] J. Jia and K. Li, Maximum principles for a time-space fractional diffusion equation, Appl. Math. Lett. 62 (2016), 23–28. 10.1016/j.aml.2016.06.010Search in Google Scholar
[8] A. A. Kilbas, H. M. Slodicka and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier Science, Amsterdam, 2006. Search in Google Scholar
[9] D. Lesnic, S. O. Hussein and B. T. Johansson, Inverse space-dependent force problems for the wave equation, J. Comput. Appl. Math. 306 (2016), 10–39. 10.1016/j.cam.2016.03.034Search in Google Scholar
[10] G. Li, D. Yao, H. Jiang and X. Jia, Numerical inversion of a time-dependent reaction coefficient in a soil-column infiltrating experiment, CMES Comput. Model. Eng. Sci. 74 (2011), no. 2, 83–107. Search in Google Scholar
[11] Z. Liu, G. Y. Yang, N. He and X. Y. Tan, Landweber iterative algorithm based on regularization in electromagnetic tomography for multiphase flow measurement, Flow. Meas. Instrum. 27 (2012), 53–58. 10.1016/j.flowmeasinst.2012.04.011Search in Google Scholar
[12] A. Lopushansky and H. Lopushanska, Inverse source Cauchy problem for a time fractional diffusion-wave equation with distributions, Electron. J. Differential Equations 2017 (2017), Paper No. 182. Search in Google Scholar
[13] R. Magin, X. Feng and D. Baleanu, Solving the fractional order Bloch equation, Concept. Magn. Reson. A. 34 (2009), 16–23. 10.1002/cmr.a.20129Search in Google Scholar
[14] H. T. Nguyen, D. L. Le and V. T. Nguyen, Regularized solution of an inverse source problem for a time fractional diffusion equation, Appl. Math. Model. 40 (2016), no. 19–20, 8244–8264. 10.1016/j.apm.2016.04.009Search in Google Scholar
[15] I. Podlubny, Fractional Differential Equations, Math. Sci. Eng. 198, Academic Press, San Diego, 1999. Search in Google Scholar
[16] K. Sakamoto and M. Yamamoto, Initial value/boundary value problems for fractional diffusion-wave equations and applications to some inverse problems, J. Math. Anal. Appl. 382 (2011), no. 1, 426–447. 10.1016/j.jmaa.2011.04.058Search in Google Scholar
[17] O. Scherzer, Convergence criteria of iterative methods based on Landweber iteration for solving nonlinear problems, J. Math. Anal. Appl. 194 (1995), no. 3, 911–933. 10.1006/jmaa.1995.1335Search in Google Scholar
[18] K. Šišková and M. Slodička, Recognition of a time-dependent source in a time-fractional wave equation, Appl. Numer. Math. 121 (2017), 1–17. 10.1016/j.apnum.2017.06.005Search in Google Scholar
[19] S. Tatar, R. Tinaztepe and S. Ulusoy, Determination of an unknown source term in a space-time fractional diffusion equation, J. Fract. Calc. Appl. 6 (2015), no. 1, 83–90. Search in Google Scholar
[20] S. Tatar, R. Tınaztepe and S. Ulusoy, Simultaneous inversion for the exponents of the fractional time and space derivatives in the space-time fractional diffusion equation, Appl. Anal. 95 (2016), no. 1, 1–23. 10.1080/00036811.2014.984291Search in Google Scholar
[21] S. Tatar and S. Ulusoy, An inverse source problem for a one-dimensional space-time fractional diffusion equation, Appl. Anal. 94 (2015), no. 11, 2233–2244. 10.1080/00036811.2014.979808Search in Google Scholar
[22] L. Wang, X. Han, J. Liu and J. Chen, An improved iteration regularization method and application to reconstruction of dynamic loads on a plate, J. Comput. Appl. Math. 235 (2011), no. 14, 4083–4094. 10.1016/j.cam.2011.02.034Search in Google Scholar
[23] T. Wei, L. Sun and Y. Li, Uniqueness for an inverse space-dependent source term in a multi-dimensional time-fractional diffusion equation, Appl. Math. Lett. 61 (2016), 108–113. 10.1016/j.aml.2016.05.004Search in Google Scholar
[24] T. Wei and Z. Q. Zhang, Stable numerical solution to a Cauchy problem for a time fractional diffusion equation, Eng. Anal. Bound. Elem. 40 (2014), 128–137. 10.1016/j.enganabound.2013.12.002Search in Google Scholar
[25] Z. Wei, Q. Li and J. Che, Initial value problems for fractional differential equations involving Riemann–Liouville sequential fractional derivative, J. Math. Anal. Appl. 367 (2010), no. 1, 260–272. 10.1016/j.jmaa.2010.01.023Search in Google Scholar
[26] X. Xiong and X. Xue, Fractional Tikhonov method for an inverse time-fractional diffusion problem in 2-dimensional space, Bull. Malays. Math. Sci. Soc. 43 (2020), no. 1, 25–38. 10.1007/s40840-018-0662-5Search in Google Scholar
[27] F. Yang, C. Fu and X. Li, Identifying an unknown source in space-fractional diffusion equation, Acta Math. Sci. Ser. B (Engl. Ed.) 34 (2014), no. 4, 1012–1024. 10.1016/S0252-9602(14)60065-5Search in Google Scholar
[28] F. Yang and C.-L. Fu, The quasi-reversibility regularization method for identifying the unknown source for time fractional diffusion equation, Appl. Math. Model. 39 (2015), no. 5–6, 1500–1512. 10.1016/j.apm.2014.08.010Search in Google Scholar
[29] F. Yang, C.-L. Fu and X.-X. Li, A mollification regularization method for unknown source in time-fractional diffusion equation, Int. J. Comput. Math. 91 (2014), no. 7, 1516–1534. 10.1080/00207160.2013.851787Search in Google Scholar
[30] F. Yang, C.-L. Fu and X.-X. Li, The inverse source problem for time-fractional diffusion equation: Stability analysis and regularization, Inverse Probl. Sci. Eng. 23 (2015), no. 6, 969–996. 10.1080/17415977.2014.968148Search in Google Scholar
[31] F. Yang, X. Liu and X.-X. Li, Landweber iterative regularization method for identifying the unknown source of the modified Helmholtz equation, Bound. Value Probl. 2017 (2017), Paper No. 91. 10.1186/s13661-017-0823-8Search in Google Scholar
[32] F. Yang, Y.-P. Ren and X.-X. Li, Landweber iteration regularization method for identifying unknown source on a columnar symmetric domain, Inverse Probl. Sci. Eng. 26 (2018), no. 8, 1109–1129. 10.1080/17415977.2017.1384825Search in Google Scholar
[33] F. Yang, Y.-P. Ren and X.-X. Li, The quasi-reversibility method for a final value problem of the time-fractional diffusion equation with inhomogeneous source, Math. Methods Appl. Sci. 41 (2018), no. 5, 1774–1795. 10.1002/mma.4705Search in Google Scholar
[34] F. Yang, Y.-R. Sun, X.-X. Li and C.-Y. Huang, The quasi-boundary value method for identifying the initial value of heat equation on a columnar symmetric domain, Numer. Algorithms 82 (2019), no. 2, 623–639. 10.1007/s11075-018-0617-9Search in Google Scholar
[35] F. Yang, N. Wang, X.-X. Li and C.-Y. Huang, A quasi-boundary regularization method for identifying the initial value of time-fractional diffusion equation on spherically symmetric domain, J. Inverse Ill-Posed Probl. 27 (2019), no. 5, 609–621. 10.1515/jiip-2018-0050Search in Google Scholar
[36] F. Yang, Y. Zhang and X.-X. Li, Landweber iterative method for identifying the initial value problem of the time-space fractional diffusion-wave equation, Numer. Algorithms 83 (2020), no. 4, 1509–1530. 10.1007/s11075-019-00734-6Search in Google Scholar
[37] M. Yang and J. Liu, Solving a final value fractional diffusion problem by boundary condition regularization, Appl. Numer. Math. 66 (2013), 45–58. 10.1016/j.apnum.2012.11.009Search in Google Scholar
[38] Z. Q. Zhang and T. Wei, An optimal regularization method for space-fractional backward diffusion problem, Math. Comput. Simulation 92 (2013), 14–27. 10.1016/j.matcom.2013.04.008Search in Google Scholar
[39] G. H. Zheng and T. Wei, A new regularization method for a Cauchy problem of the time fractional diffusion equation, Adv. Comput. Math. 36 (2012), no. 2, 377–398. 10.1007/s10444-011-9206-3Search in Google Scholar
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