Raising the regularity of generalized Abel equations in fractional Sobolev spaces with homogeneous boundary conditions. (English) Zbl 1491.45001
Summary: The generalized (or coupled) Abel equations exist in many BVPs of fractional-order differential equations and play a key role during the process of converting weak solutions to the true solutions. Motivated by the analysis of double-sided fractional diffusion-advection-reaction equations, this article develops the mapping properties of generalized Abel operators \({\alpha_a}D_x^{- s} + {\beta_x}D_b^{- s}\) in fractional Sobolev spaces, where \(0 < \alpha\), \(\beta\), \(\alpha + \beta = 1\), \(0 < s < 1\) and \(_aD_x^{- s}\), \(_xD_b^{- s}\) are fractional Riemann-Liouville integrals. It is mainly concerned with the regularity property of \(({{\alpha_a}D_x^{- s} + {\beta_x}D_b^{- s}})u = f\) by taking into account homogeneous boundary conditions. Namely, we investigate the regularity behavior of \(u(x)\) while letting \(f(x)\) become smoother and imposing homogeneous boundary restrictions \(u(a) = u(b) = 0\).
MSC:
| 45A05 | Linear integral equations |
| 45E10 | Integral equations of the convolution type (Abel, Picard, Toeplitz and Wiener-Hopf type) |
| 45P05 | Integral operators |
| 26A33 | Fractional derivatives and integrals |
References:
| [1] | A. Almeida, L. Castro, and F.-O. Speck (editors), Advances in harmonic analysis and operator theory, Operator Theory: Advances and Applications 229, Springer, 2013. · Zbl 1258.00012 · doi:10.1007/978-3-0348-0516-2 |
| [2] | V. J. Ervin, “Regularity of the solution to fractional diffusion, advection, reaction equations in weighted Sobolev spaces”, J. Differential Equations 278 (2021), 294-325. · Zbl 1456.35212 · doi:10.1016/j.jde.2020.12.034 |
| [3] | V. J. Ervin and J. P. Roop, “Variational formulation for the stationary fractional advection dispersion equation”, Numer. Methods Partial Differential Equations 22:3 (2006), 558-576. · Zbl 1095.65118 · doi:10.1002/num.20112 |
| [4] | V. J. Ervin and J. P. Roop, “Variational solution of fractional advection dispersion equations on bounded domains in \[\mathbb R^d\]”, Numer. Methods Partial Differential Equations 23:2 (2007), 256-281. · Zbl 1117.65169 · doi:10.1002/num.20169 |
| [5] | V. J. Ervin, N. Heuer, and J. P. Roop, “Regularity of the solution to 1-D fractional order diffusion equations”, Math. Comp. 87:313 (2018), 2273-2294. · Zbl 1394.65145 · doi:10.1090/mcom/3295 |
| [6] | F. D. Gakhov, Boundary value problems, Dover Publications, New York, 1990. · Zbl 0830.30026 |
| [7] | V. Ginting and Y. Li, “On the fractional diffusion-advection-reaction equation in \[\mathbb R\]”, Fract. Calc. Appl. Anal. 22:4 (2019), 1039-1062. · Zbl 1442.34012 · doi:10.1515/fca-2019-0055 |
| [8] | Z. Hao, G. Lin, and Z. Zhang, “Error estimates of a spectral Petrov-Galerkin method for two-sided fractional reaction-diffusion equations”, Appl. Math. Comput. 374 (2020), 125045, 13. · Zbl 1433.65211 · doi:10.1016/j.amc.2020.125045 |
| [9] | A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and applications of fractional differential equations, North-Holland Mathematics Studies 204, Elsevier Science B.V., Amsterdam, 2006. · Zbl 1092.45003 |
| [10] | Y. Li, On fractional differential equations and related questions, Ph.D. thesis, University of Wyoming, 2019, available at https://www.proquest.com/docview/2278078937. |
| [11] | Y. Li, “On the skewed fractional diffusion advection reaction equation on the interval”, preprint, 2020. |
| [12] | N. I. Muskhelishvili, Singular integral equations: boundary problems of function theory and their application to mathematical physics, P. Noordhoff N. V., Groningen, 1953. · Zbl 0051.33203 |
| [13] | J. Sabatier, O. P. Agrawal, and J. A. T. Machado (editors), Advances in fractional calculus, Springer, 2007. · Zbl 1116.00014 · doi:10.1007/978-1-4020-6042-7 |
| [14] | S. G. Samko, Methods of inversion of potential type operators and equations with involutive operators and their applications, (Russian) Doctor of Science Thesis, Steklov Math. Inst., 1978. |
| [15] | S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional integrals and derivatives, Gordon and Breach Science Publishers, Yverdon, 1993. · Zbl 0818.26003 |
| [16] | V. E. Tarasov, Fractional dynamics, Springer, 2010. · Zbl 1214.81004 · doi:10.1007/978-3-642-14003-7 |
| [17] | L. von Wolfersdorf, “Zur Lösung der verallgemeinerten Abelschen Integralgleichung mit konstanten Koeffizienten”, Z. Angew. Math. Mech. 49 (1969), 759-761. · Zbl 0188.17801 · doi:10.1002/zamm.19690491212 |
| [18] | X. Zheng, V. J. Ervin, and H. Wang, “Wellposedness of the two-sided variable coefficient Caputo flux fractional diffusion equation and error estimate of its spectral approximation”, Appl. Numer. Math. 153 (2020), 234-247. · Zbl 1477.65127 · doi:10.1016/j.apnum.2020.02.019 |
| [19] | X. Zheng, V. J. Ervin, and H. Wang, “Optimal Petrov-Galerkin spectral approximation method for the fractional diffusion, advection, reaction equation on a bounded interval”, J. Sci. Comput. 86:3 (2021), Paper No. 29, 22 · Zbl 1465.65154 · doi:10.1007/s10915-020-01366-y |
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