Abstract
To characterize the ultra-slow diffusion processes with time-dependent logarithmical law attenuation, the distributed order fractional diffusion equation is needed. This paper discusses an analysis of approximate controllability from the exterior of distributed order fractional diffusion problem with the fractional Laplace operator subject to the non-zero exterior condition. We first establish some well-posedness results, such as the existence, uniqueness and regularity of the solutions allowing the weighted function \(\mu \) that may be non-continuous. Especially, we show that the solutions can be represented by the series for the integral of a real-valued function. After giving the unique continuation property of the adjoint system, approximate controllability of the system is also included.
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References
Agrawal, O.P.: Fractional variational calculus in terms of Riesz fractional derivatives. J. Phys. A 40, 6287–303 (2007)
Amann, H.: Linear and Quasilinear Parabolic Problems, Abstract Linear Theory, vol. 1. Birkhauser, Berlin (1995)
Bobylev, A.V., Cercignani, C.: The inverse Laplace transform of some analytic functions with an application to the eternal solutions of the Boltzmann equation. Appl. Math. Lett. 15, 807–813 (2002)
Claus, B., Warma, M.: Realization of the fractional Laplacian with nonlocal exterior conditions via forms method. J. Evol. Equ. 20, 1597–1631 (2020)
Dipierro, S., Ros-Oton, X., Valdinoci, E.: Nonlocal problems with Neumann boundary conditions. Rev. Mat. Iberoam. 33(2), 377–416 (2017)
Ghosh, T., Salo, M., Uhlmann, G.: The Calderón problem for the fractional Schrödinger equation. Anal. PDE 13(2), 455–475 (2020)
Ghosh, T., Rüland, A., Salo, M., Uhlmann, G.: Uniqueness and reconstruction for the fractional Calderón problem with a single measurement. J. Funct. Anal. 279(1), 108505 (2020)
Grubb, G.: Fractional Laplacians on domains, a development of Hörmander’s theory of \(\mu \)-transmission pseudo-differential operators. Adv. Math. 268, 478–528 (2015)
Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and applications of fractional differential equations. In: North-Holland Mathematics Studies, vol. 204, Elsevier Science B.V., Amsterdam (2006)
Kochubei, A.N.: Distributed order calculus and equation of ultraslow diffusion. J. Math. Anal. Appl. 340(1), 252–281 (2008)
Kubica, A., Ryszewska, K.: Decay of solutions to parabolic-type problem with distributed order Caputo derivative. J. Math. Anal. Appl. 465(1), 75–99 (2018)
Kubica, A., Ryszewska, K.: Fractional diffusion equation with distributed-order Caputo derivative. J. Integral Equ. Appl. 31(2), 195–243 (2019)
Li, Z., Fujishiro, K., Li, G.S.: Uniqueness in the inversion of distributed orders in ultraslow diffusion equations. J. Comput. Appl. Math. 369, 112564 (2019)
Li, Z., Luchko, Y., Yamamoto, M.: Analyticity of solutions to a distributed order time-fractional diffusion equation and its application to an inverse problem. Comput. Math. Appl. 73, 1041–1052 (2016)
Li, Z., Kian, Y., Soccorsi, E.: Initial-boundary value problem for distributed order time-fractional diffusion equations. Asymptot. Anal. 115(1–2), 95–126 (2019)
Louis-Rose, C., Warma, M.: Approximate controllability from the exterior of space-time fractional wave equations. Appl. Math. Optim. 83, 207–250 (2021)
Luchko, Y.: Boundary value problems for the generalized time-fractional diffusion equation of distributed order. Fract. Calc. Appl. Anal. 12(4), 409–422 (2009)
Meerschaert, M.M., Scheffler, H.P.: Stochastic model for ultraslow diffusion. Stoch. Proc. Appl. 116, 1215–1235 (2006)
Peng, L., Zhou, Y., He, J.W.: The well-posedness analysis of distributed order fractional diffusion problems on \({\mathbb{R}}^N\). Monatshefte für Mathematik (2021). https://doi.org/10.1007/s00605-021-01631-8
Peng, L., Zhou, Y., Ahmad, B., Alsaedi, A.: The Cauchy problem for fractional Navier-Stokes equations in Sobolev spaces. Chaos Solitons Fractals 102, 218–228 (2017)
Peng, L., Zhou, Y., Ahmad, B.: The well-posedness for fractional nonlinear Schrödinger equations. Comput. Math. Appl. 77, 1998–2005 (2019)
Prüss, J.: Evolutionary Integral Equations and Applications, Monographs in Mathematics, vol. 87. Birkhäuser Verlag, Basel (1993)
Rundell, W., Zhang, Z.D.: Fractional diffusion: recovering the distributed fractional derivative from overposed data. Inverse Probl. 33, 035008 (2017)
Servadei, R., Valdinoci, E.: On the spectrum of two different fractional operators. Proc. Roy. Soc. Edinb. Sect. A 144(4), 831–855 (2014)
Sun, C.L., Liu, J.J.: An inverse source problem for distributed order time-fractional diffusion equation. Inverse Probl. 36(5), 055008 (2020)
Warma, M.: Approximate controllability from the exterior of space-time fractional diffusive equations. SIAM J. Control Optim. 57(3), 2037–2063 (2019)
Zhou, Y., He, J.W., Ahmad, B., Tuan, N.H.: Existence and regularity results of a backward problem for fractional diffusion equations. Math. Methods Appl. Sci. 42, 6775–6790 (2019)
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Project supported by National Natural Science Foundation of China (12001462, 12071396) and the General Project of Hunan Provincial Education Department of China (21C0083).
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Peng, L., Zhou, Y. The Analysis of Approximate Controllability for Distributed Order Fractional Diffusion Problems. Appl Math Optim 86, 22 (2022). https://doi.org/10.1007/s00245-022-09886-9
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DOI: https://doi.org/10.1007/s00245-022-09886-9
Keywords
- Distributed order fractional derivative
- Ultra-slow diffusion
- Well-posedness
- Approximate controllability