Figure 1.
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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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 |
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.
Keywords:
spectral,
elliptic operator,
principal eigenvalue,
maximum principle,
Hopf's Lemma,
fractional derivative,
Mittag-Leffler function.
Mathematics Subject Classification:
Primary: 34B05, 34L10; Secondary: 34B09.
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. |
show all references
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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