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May  2022, 21(5): 1673-1690. doi: 10.3934/cpaa.2022039

## Analysis of one-sided 1-D fractional diffusion operator

 1 Department of Mathematics and Statistics, University of Nevada Reno, NV 89557, USA 2 Department of Mathematics, Sakarya University, 54050, Sakarya, Turkey

* Corresponding author

Received  November 2021 Revised  January 2022 Published  May 2022 Early access  February 2022

This work establishes the parallel between the properties of classic elliptic PDEs and the one-sided 1-D fractional diffusion equation, that includes the characterization of fractional Sobolev spaces in terms of fractional Riemann-Liouville (R-L) derivatives, variational formulation, maximum principle, Hopf's Lemma, spectral analysis, and theory on the principal eigenvalue and its characterization, etc. As an application, the developed results provide a novel perspective to study the distribution of complex roots of a class of Mittag-Leffler functions and, furthermore, prove the existence of real roots.

Citation: Yulong Li, Aleksey S. Telyakovskiy, Emine Çelik. Analysis of one-sided 1-D fractional diffusion operator. Communications on Pure and Applied Analysis, 2022, 21 (5) : 1673-1690. doi: 10.3934/cpaa.2022039
##### References:
 [1] T. Aleroev and E. Kekharsaeva, Boundary value problems for differential equations with fractional derivatives, Integral Trans. Spec. Funct., 28 (2017), 900-908.  doi: 10.1080/10652469.2017.1381844. [2] T. Aleroev and H. Aleroeva, Problems of Sturm-Liouville type for differential equations with fractional derivatives, in Handbook of Fractional Calculus with Applications, De Gruyter, Berlin, 2019. [3] M. Al-Qurashi and L. Ragoub, Lyapunov-type inequality for a Riemann-Liouville fractional differential boundary value problem, Hacet. J. Math. Stat., 47 (2018), 1447-1452. [4] M. M. Džrbašjan, A boundary value problem for a Sturm-Liouville type differential operator of fractional order, Izv. Akad. Nauk Armjan. SSR Ser. Mat., 5 (1970), 71-96. [5] V. J. Ervin and J. P. Roop, Variational formulation for the stationary fractional advection dispersion equation, Numer. Methods Partial Differ. Equ., 22 (2006), 558-576.  doi: 10.1002/num.20112. [6] V. J. Ervin and J. P. Roop, Variational solution of fractional advection dispersion equations on bounded domains in ${\mathbb{R}}^d$, Numer. Methods Partial Differ. Equ., 23 (2007), 256-281.  doi: 10.1002/num.20169. [7] V. J. Ervin, N. Heuer and J. P. Roop, Regularity of the solution to 1-D fractional order diffusion equations, Math. Comp., 87 (2018), 2273-2294.  doi: 10.1090/mcom/3295. [8] V. J. Ervin, Regularity of the solution to fractional diffusion, advection, reaction equations in weighted Sobolev spaces, J. Differ. Equ., 278 (2021), 294-325.  doi: 10.1016/j.jde.2020.12.034. [9] L. C. Evans, Partial Differential Equations, 2$^{nd}$ edition, American Mathematical Society, Providence, RI, 2010. [10] R. A. C. Ferreira, A Lyapunov-type inequality for a fractional boundary value problem, Fract. Calc. Appl. Anal., 16 (2013), 978-984.  doi: 10.2478/s13540-013-0060-5. [11] G. B. Folland, Real Analysis, 2$^{nd}$ edition, John Wiley & Sons, Inc., New York, 1999. [12] V. Ginting and Y. Li, On the fractional diffusion-advection-reaction equation in ${\mathbb{R}}$, Fract. Calc. Appl. Anal., 22 (2019), 1039-1062.  doi: 10.1515/fca-2019-0055. [13] R. Gorenflo, A. A. Kilbas, F. Mainardi and S. Rogosin, Mittag-Leffler Functions, Related Topics and Applications, 2$^{nd}$ edition, Springer, Berlin, 2020. [14] G. C. Hsiao and W. L. Wendland, Boundary Integral Equations, 2$^{nd}$ edition, Springer, Cham, 2021. [15] L. Jia, H. Chen and V. J. Ervin, Existence and regularity of solutions to 1-D fractional order diffusion equations, Electron. J. Differ. Equ., (2019), 21 pp. [16] B. Jin, R. Lazarov, J. Pasciak and W. Rundell, Variational formulation of problems involving fractional order differential operators, Math. Comp., 84 (2015), 2665-2700.  doi: 10.1090/mcom/2960. [17] A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier Science B.V., Amsterdam, 2006. [18] Y. Li, On the decomposition of solutions: from fractional diffusion to fractional Laplacian, Fract. Calc. Appl. Anal., 24 (2021), 1571-1600.  doi: 10.1515/fca-2021-0066. [19] Y. Li, Integral representation bound of the true solution to the BVP of double-sided fractional diffusion advection reaction equation, Rend. Circ. Mat. Palermo, II. Ser, (2021), 22 pp. doi: 10.1007/s12215-021-00592-z. [20] R. Meise and D. Vogt, Introduction to Functional Analysis, The Clarendon Press, Oxford University Press, New York, 1997. [21] N. I. Muskhelishvili, Singular Integral Equations. Boundary Problems of Function Theory and Their Application to Mathematical Physics, P. Noordhoff N. V., Groningen, 1953. [22] B. E. Petersen, Introduction to the Fourier Transform & Pseudodifferential Operators, Pitman (Advanced Publishing Program), Boston, MA, 1983. [23] W. Rudin, Real and Complex Analysis, 3$^{nd}$ edition, McGraw-Hill Book Co., New York, 1987. [24] S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives, Theory and Applications, Gordon and Breach Science Publishers, Yverdon, 1993. [25] I. Tikhonov and Y. S. Éidel'man, Inverse scattering transform for differential equations in Banach space and the distribution of zeros of an entire Mittag-Leffler type function, Differ. Uravn., 38 (2002), 637-644. [26] S. Yang, H. Chen, V. J. Ervin and H. Wang, Solvability and approximation of two-side conservative fractional diffusion problems with variable-coefficient based on least-squares, Appl. Math. Comput., 406 (2021), 21 pp. doi: 10.1016/j.amc.2021.126229. [27] 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.  doi: 10.1016/j.apnum.2020.02.019. [28] X. Zheng, V. J. Ervin and H. Wang, Numerical approximations for the variable coefficient fractional diffusion equations with non-smooth data, Comput. Methods Appl. Math., 20 (2020), 573-589.  doi: 10.1515/cmam-2019-0038. [29] 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 (2021), 22 pp. doi: 10.1007/s10915-020-01366-y.

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##### References:
 [1] T. Aleroev and E. Kekharsaeva, Boundary value problems for differential equations with fractional derivatives, Integral Trans. Spec. Funct., 28 (2017), 900-908.  doi: 10.1080/10652469.2017.1381844. [2] T. Aleroev and H. Aleroeva, Problems of Sturm-Liouville type for differential equations with fractional derivatives, in Handbook of Fractional Calculus with Applications, De Gruyter, Berlin, 2019. [3] M. Al-Qurashi and L. Ragoub, Lyapunov-type inequality for a Riemann-Liouville fractional differential boundary value problem, Hacet. J. Math. Stat., 47 (2018), 1447-1452. [4] M. M. Džrbašjan, A boundary value problem for a Sturm-Liouville type differential operator of fractional order, Izv. Akad. Nauk Armjan. SSR Ser. Mat., 5 (1970), 71-96. [5] V. J. Ervin and J. P. Roop, Variational formulation for the stationary fractional advection dispersion equation, Numer. Methods Partial Differ. Equ., 22 (2006), 558-576.  doi: 10.1002/num.20112. [6] V. J. Ervin and J. P. Roop, Variational solution of fractional advection dispersion equations on bounded domains in ${\mathbb{R}}^d$, Numer. Methods Partial Differ. Equ., 23 (2007), 256-281.  doi: 10.1002/num.20169. [7] V. J. Ervin, N. Heuer and J. P. Roop, Regularity of the solution to 1-D fractional order diffusion equations, Math. Comp., 87 (2018), 2273-2294.  doi: 10.1090/mcom/3295. [8] V. J. Ervin, Regularity of the solution to fractional diffusion, advection, reaction equations in weighted Sobolev spaces, J. Differ. Equ., 278 (2021), 294-325.  doi: 10.1016/j.jde.2020.12.034. [9] L. C. Evans, Partial Differential Equations, 2$^{nd}$ edition, American Mathematical Society, Providence, RI, 2010. [10] R. A. C. Ferreira, A Lyapunov-type inequality for a fractional boundary value problem, Fract. Calc. Appl. Anal., 16 (2013), 978-984.  doi: 10.2478/s13540-013-0060-5. [11] G. B. Folland, Real Analysis, 2$^{nd}$ edition, John Wiley & Sons, Inc., New York, 1999. [12] V. Ginting and Y. Li, On the fractional diffusion-advection-reaction equation in ${\mathbb{R}}$, Fract. Calc. Appl. Anal., 22 (2019), 1039-1062.  doi: 10.1515/fca-2019-0055. [13] R. Gorenflo, A. A. Kilbas, F. Mainardi and S. Rogosin, Mittag-Leffler Functions, Related Topics and Applications, 2$^{nd}$ edition, Springer, Berlin, 2020. [14] G. C. Hsiao and W. L. Wendland, Boundary Integral Equations, 2$^{nd}$ edition, Springer, Cham, 2021. [15] L. Jia, H. Chen and V. J. Ervin, Existence and regularity of solutions to 1-D fractional order diffusion equations, Electron. J. Differ. Equ., (2019), 21 pp. [16] B. Jin, R. Lazarov, J. Pasciak and W. Rundell, Variational formulation of problems involving fractional order differential operators, Math. Comp., 84 (2015), 2665-2700.  doi: 10.1090/mcom/2960. [17] A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier Science B.V., Amsterdam, 2006. [18] Y. Li, On the decomposition of solutions: from fractional diffusion to fractional Laplacian, Fract. Calc. Appl. Anal., 24 (2021), 1571-1600.  doi: 10.1515/fca-2021-0066. [19] Y. Li, Integral representation bound of the true solution to the BVP of double-sided fractional diffusion advection reaction equation, Rend. Circ. Mat. Palermo, II. Ser, (2021), 22 pp. doi: 10.1007/s12215-021-00592-z. [20] R. Meise and D. Vogt, Introduction to Functional Analysis, The Clarendon Press, Oxford University Press, New York, 1997. [21] N. I. Muskhelishvili, Singular Integral Equations. Boundary Problems of Function Theory and Their Application to Mathematical Physics, P. Noordhoff N. V., Groningen, 1953. [22] B. E. Petersen, Introduction to the Fourier Transform & Pseudodifferential Operators, Pitman (Advanced Publishing Program), Boston, MA, 1983. [23] W. Rudin, Real and Complex Analysis, 3$^{nd}$ edition, McGraw-Hill Book Co., New York, 1987. [24] S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives, Theory and Applications, Gordon and Breach Science Publishers, Yverdon, 1993. [25] I. Tikhonov and Y. S. Éidel'man, Inverse scattering transform for differential equations in Banach space and the distribution of zeros of an entire Mittag-Leffler type function, Differ. Uravn., 38 (2002), 637-644. [26] S. Yang, H. Chen, V. J. Ervin and H. Wang, Solvability and approximation of two-side conservative fractional diffusion problems with variable-coefficient based on least-squares, Appl. Math. Comput., 406 (2021), 21 pp. doi: 10.1016/j.amc.2021.126229. [27] 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.  doi: 10.1016/j.apnum.2020.02.019. [28] X. Zheng, V. J. Ervin and H. Wang, Numerical approximations for the variable coefficient fractional diffusion equations with non-smooth data, Comput. Methods Appl. Math., 20 (2020), 573-589.  doi: 10.1515/cmam-2019-0038. [29] 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 (2021), 22 pp. doi: 10.1007/s10915-020-01366-y.
First real root of Mittag Leffler function $E_{1+\mu,1+\mu}[z]$ with different values of $\mu$. The pictures are generated with functions 'mlf' and 'intersections' from Matlab Exchange
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