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A structure-preserving scheme for the Allen–Cahn equation with a dynamic boundary condition
Positivity, monotonicity, and convexity for convolution operators
1. | School of Mathematics and Statistics, UNSW Australia, Sydney, NSW 2052, Australia |
2. | Departamento de Matemática y Ciencia de la Computación, Universidad de Santiago de Chile, Las Sophoras 173, Santiago, Chile |
$ \begin{equation} a*u {\ge} v\notag \end{equation} $ |
$ a $ |
$ v $ |
$ a $ |
$ v $ |
$ u $ |
References:
[1] |
L. Abadias, C. Lizama, P. J. Miana and M. P. Velasco,
Cesàro sums and algebra homomorphisms of bounded operators, Israel J. Math., 216 (2016), 471-505.
doi: 10.1007/s11856-016-1417-3. |
[2] |
L. Abadias, C. Lizama, P. J. Miana and M. P. Velasco,
On well-posedness of vector-valued fractional differential-difference equations, Discr. Cont. Dyn. Systems, Series A, 39 (2019), 2679-2708.
doi: 10.3934/dcds.2019112. |
[3] |
L. Abadias and P. J. Miana,
Generalized Cesàro operators, fractional finite differences and Gamma functions, J. Funct. Anal., 274 (2018), 1424-1465.
doi: 10.1016/j.jfa.2017.10.010. |
[4] |
L. Erbe, C. S. Goodrich, J. Baoguo and A. Peterson,
Monotonicity results for delta fractional difference revisited, Math. Slovaca, 67 (2017), 895-906.
doi: 10.1515/ms-2017-0018. |
[5] |
J. Bonet, C. Fernández and R. Meise,
Characterization of the $\omega$-hypoelliptic convolution operators on ultradistributions, Ann. Acad. Sci. Fenn. Math., 25 (2000), 261-284.
|
[6] |
I. Bright,
Moving averages of ordinary differential equations via convolution, J. Differential Equations, 250 (2011), 1267-1284.
doi: 10.1016/j.jde.2010.10.011. |
[7] |
Y. Choi,
Injective convolution operators on $\ell^{\infty}(\Gamma)$ are surjective, Canad. Math. Bull., 53 (2010), 447-452.
doi: 10.4153/CMB-2010-053-5. |
[8] |
O. Constatin and J. Hargraves,
Monotone solutions to a nonlinear integral equation of convolution type, Nonlinear Anal. Real World Appl., 15 (2014), 38-41.
doi: 10.1016/j.nonrwa.2013.05.003. |
[9] |
R. Dahal and C. S. Goodrich,
A monotonicity result for discrete fractional difference operators, Arch. Math. (Basel), 102 (2014), 293-299.
doi: 10.1007/s00013-014-0620-x. |
[10] |
M. Darwish,
Monotonic solutions of a convolution functional-integral equation, Appl. Math. Comput., 219 (2013), 10777-10782.
doi: 10.1016/j.amc.2013.05.001. |
[11] |
O. Diekmann and H. Kaper,
On the bounded solutions of a nonlinear convolution equation, Nonlinear Anal., 2 (1978), 721-737.
doi: 10.1016/0362-546X(78)90015-9. |
[12] |
K. Diethelm,
Monotonicity of functions and sign changes of their Caputo derivatives, Fract. Calc. Appl. Anal., 19 (2016), 561-566.
doi: 10.1515/fca-2016-0029. |
[13] |
J. Dieudonné,
Sur le produit de composition. Ⅱ, J. Math. Pures Appl. (9), 39 (1960), 275-292.
|
[14] |
L. Ehrenpreis,
Solution of some problems of division. Ⅳ. Invertible and elliptic operators, Amer. J. Math., 82 (1960), 522-588.
doi: 10.2307/2372971. |
[15] |
C. Fernández, A. Galbis and D. Jornet,
Perturbations of surjective convolution operators, Proc. Amer. Math. Soc., 130 (2002), 2377-2381.
doi: 10.1090/S0002-9939-02-06359-1. |
[16] |
M. Gómez-Callado and E. Jordá,
Regularity of solutions of convolution equations, J. Math. Anal. Appl., 338 (2008), 873-884.
doi: 10.1016/j.jmaa.2007.05.053. |
[17] |
C. S. Goodrich,
A convexity result for fractional differences, Appl. Math. Lett., 35 (2014), 58-62.
doi: 10.1016/j.aml.2014.04.013. |
[18] |
C. S. Goodrich,
A note on convexity, concavity, and growth conditions in discrete fractional calculus with delta difference, Math. Inequal. Appl., 19 (2016), 769-779.
doi: 10.7153/mia-19-57. |
[19] |
C. S. Goodrich and C. Lizama, A transference principle for nonlocal operators using a convolutional approach: Fractional monotonicity and convexity, Israel J. Math., 236 (2020), 533-589.
doi: 10.1007/s11856-020-1991-2. |
[20] |
C. S. Goodrich and A. C. Peterson, Discrete Fractional Calculus, Springer International Publishing, 2015.
doi: 10.1007/978-3-319-25562-0. |
[21] |
A. Hanyga, On solutions of matrix-valued convolution equations, CM-derivatives and their applications in linear and nonlinar anisotropic viscoelasticity, Z. Angew. Math. Phys., 70 (2019), art. 103, 13 pp. |
[22] |
B. Jia, L. Erbe and A. Peterson,
Two monotonicity results for nabla and delta fractional differences, Arch. Math. (Basel), 104 (2015), 589-597.
doi: 10.1007/s00013-015-0765-2. |
[23] |
B. Jin, B. Li and Z. Zhou,
Discrete maximal regularity of time-stepping schemes for fractional evolution equations, Numer. Math., 138 (2018), 101-131.
doi: 10.1007/s00211-017-0904-8. |
[24] |
R. Kamocki,
A new representation formula for the Hilfer fractional derivative and its application, J. Comp. Appl. Math., 308 (2016), 39-45.
doi: 10.1016/j.cam.2016.05.014. |
[25] |
J. Kemppainen, J. Siljander, V. Vergara and R. Zacher,
Decay estimates for time-fractional and other non-local in time subdiffusion equations in $\Bbb{R}^d$, Math. Ann., 366 (2016), 941-979.
doi: 10.1007/s00208-015-1356-z. |
[26] |
A. N. Kochubei,
Distributed order calculus and equations of ultraslow diffusion, J. Math. Anal. Appl., 340 (2008), 252-281.
doi: 10.1016/j.jmaa.2007.08.024. |
[27] |
A. N. Kochubei,
General fractional calculus, evolution equations, and renewal processes, Integral Equations Oper. Theory, 71 (2011), 583-600.
|
[28] |
O. Lipovan,
Asymptotic properties of solutions to some nonlinear integral equations of convolution type, Nonlinear Anal., 69 (2008), 2179-2183.
doi: 10.1016/j.na.2007.07.056. |
[29] |
C. Lizama,
The Poisson distribution, abstract fractional difference equations, and stability, Proc. Amer. Math. Soc., 145 (2017), 3809-3827.
doi: 10.1090/proc/12895. |
[30] |
C. Lizama,
$\ell_p$-maximal regularity for fractional difference equations on UMD spaces, Math. Nachr., 288 (2015), 2079-2092.
doi: 10.1002/mana.201400326. |
[31] |
C. Lizama and M. Murillo-Arcila,
Maximal regularity in $\ell_p$ spaces for discrete time fractional shifted equations, J. Differential Equations., 263 (2017), 3175-3196.
doi: 10.1016/j.jde.2017.04.035. |
[32] |
G. Lv, H. Gao, J. Wei and J.-L. Wu,
BMO and Morrey-Campanato estimates for stochastic convolutions and Schauder estimates for stochastic parabolic equations, J. Differential Equations, 266 (2019), 2666-2717.
doi: 10.1016/j.jde.2018.08.042. |
[33] |
S. G. Samko and R. P. Cardoso,
Integral equations of the first kind of Sonine type, Int. J. Math. and Math. Sci., 57 (2003), 3609-3632.
|
[34] |
K. Schumacher,
Traveling-front solutions for integro-differential equations. Ⅰ., J. Reine Angew. Math., 316 (1980), 54-70.
doi: 10.1515/crll.1980.316.54. |
[35] |
K. Strom,
On convolutions of B-splines, J. Comp. Appl. Math., 55 (1994), 1-29.
doi: 10.1016/0377-0427(94)90182-1. |
[36] |
V. Vergara and R. Zacher,
Optimal decay estimates for time-fractional and other nonlocal subdiffusion equations via energy methods, SIAM J. Math. Anal., 47 (2015), 210-239.
doi: 10.1137/130941900. |
[37] |
Y. Wang, L. Liu and Y. Wu,
Positive solutions for a class of fractional boundary value problem with changing sign nonlinearity, Nonlinear Analysis, 74 (2011), 6434-6441.
doi: 10.1016/j.na.2011.06.026. |
[38] |
Z. Xu and C. Wu,
Monostable waves in a class of non-local convolution differential equation, J. Math. Anal. Appl., 462 (2018), 1205-1224.
doi: 10.1016/j.jmaa.2018.02.036. |
show all references
References:
[1] |
L. Abadias, C. Lizama, P. J. Miana and M. P. Velasco,
Cesàro sums and algebra homomorphisms of bounded operators, Israel J. Math., 216 (2016), 471-505.
doi: 10.1007/s11856-016-1417-3. |
[2] |
L. Abadias, C. Lizama, P. J. Miana and M. P. Velasco,
On well-posedness of vector-valued fractional differential-difference equations, Discr. Cont. Dyn. Systems, Series A, 39 (2019), 2679-2708.
doi: 10.3934/dcds.2019112. |
[3] |
L. Abadias and P. J. Miana,
Generalized Cesàro operators, fractional finite differences and Gamma functions, J. Funct. Anal., 274 (2018), 1424-1465.
doi: 10.1016/j.jfa.2017.10.010. |
[4] |
L. Erbe, C. S. Goodrich, J. Baoguo and A. Peterson,
Monotonicity results for delta fractional difference revisited, Math. Slovaca, 67 (2017), 895-906.
doi: 10.1515/ms-2017-0018. |
[5] |
J. Bonet, C. Fernández and R. Meise,
Characterization of the $\omega$-hypoelliptic convolution operators on ultradistributions, Ann. Acad. Sci. Fenn. Math., 25 (2000), 261-284.
|
[6] |
I. Bright,
Moving averages of ordinary differential equations via convolution, J. Differential Equations, 250 (2011), 1267-1284.
doi: 10.1016/j.jde.2010.10.011. |
[7] |
Y. Choi,
Injective convolution operators on $\ell^{\infty}(\Gamma)$ are surjective, Canad. Math. Bull., 53 (2010), 447-452.
doi: 10.4153/CMB-2010-053-5. |
[8] |
O. Constatin and J. Hargraves,
Monotone solutions to a nonlinear integral equation of convolution type, Nonlinear Anal. Real World Appl., 15 (2014), 38-41.
doi: 10.1016/j.nonrwa.2013.05.003. |
[9] |
R. Dahal and C. S. Goodrich,
A monotonicity result for discrete fractional difference operators, Arch. Math. (Basel), 102 (2014), 293-299.
doi: 10.1007/s00013-014-0620-x. |
[10] |
M. Darwish,
Monotonic solutions of a convolution functional-integral equation, Appl. Math. Comput., 219 (2013), 10777-10782.
doi: 10.1016/j.amc.2013.05.001. |
[11] |
O. Diekmann and H. Kaper,
On the bounded solutions of a nonlinear convolution equation, Nonlinear Anal., 2 (1978), 721-737.
doi: 10.1016/0362-546X(78)90015-9. |
[12] |
K. Diethelm,
Monotonicity of functions and sign changes of their Caputo derivatives, Fract. Calc. Appl. Anal., 19 (2016), 561-566.
doi: 10.1515/fca-2016-0029. |
[13] |
J. Dieudonné,
Sur le produit de composition. Ⅱ, J. Math. Pures Appl. (9), 39 (1960), 275-292.
|
[14] |
L. Ehrenpreis,
Solution of some problems of division. Ⅳ. Invertible and elliptic operators, Amer. J. Math., 82 (1960), 522-588.
doi: 10.2307/2372971. |
[15] |
C. Fernández, A. Galbis and D. Jornet,
Perturbations of surjective convolution operators, Proc. Amer. Math. Soc., 130 (2002), 2377-2381.
doi: 10.1090/S0002-9939-02-06359-1. |
[16] |
M. Gómez-Callado and E. Jordá,
Regularity of solutions of convolution equations, J. Math. Anal. Appl., 338 (2008), 873-884.
doi: 10.1016/j.jmaa.2007.05.053. |
[17] |
C. S. Goodrich,
A convexity result for fractional differences, Appl. Math. Lett., 35 (2014), 58-62.
doi: 10.1016/j.aml.2014.04.013. |
[18] |
C. S. Goodrich,
A note on convexity, concavity, and growth conditions in discrete fractional calculus with delta difference, Math. Inequal. Appl., 19 (2016), 769-779.
doi: 10.7153/mia-19-57. |
[19] |
C. S. Goodrich and C. Lizama, A transference principle for nonlocal operators using a convolutional approach: Fractional monotonicity and convexity, Israel J. Math., 236 (2020), 533-589.
doi: 10.1007/s11856-020-1991-2. |
[20] |
C. S. Goodrich and A. C. Peterson, Discrete Fractional Calculus, Springer International Publishing, 2015.
doi: 10.1007/978-3-319-25562-0. |
[21] |
A. Hanyga, On solutions of matrix-valued convolution equations, CM-derivatives and their applications in linear and nonlinar anisotropic viscoelasticity, Z. Angew. Math. Phys., 70 (2019), art. 103, 13 pp. |
[22] |
B. Jia, L. Erbe and A. Peterson,
Two monotonicity results for nabla and delta fractional differences, Arch. Math. (Basel), 104 (2015), 589-597.
doi: 10.1007/s00013-015-0765-2. |
[23] |
B. Jin, B. Li and Z. Zhou,
Discrete maximal regularity of time-stepping schemes for fractional evolution equations, Numer. Math., 138 (2018), 101-131.
doi: 10.1007/s00211-017-0904-8. |
[24] |
R. Kamocki,
A new representation formula for the Hilfer fractional derivative and its application, J. Comp. Appl. Math., 308 (2016), 39-45.
doi: 10.1016/j.cam.2016.05.014. |
[25] |
J. Kemppainen, J. Siljander, V. Vergara and R. Zacher,
Decay estimates for time-fractional and other non-local in time subdiffusion equations in $\Bbb{R}^d$, Math. Ann., 366 (2016), 941-979.
doi: 10.1007/s00208-015-1356-z. |
[26] |
A. N. Kochubei,
Distributed order calculus and equations of ultraslow diffusion, J. Math. Anal. Appl., 340 (2008), 252-281.
doi: 10.1016/j.jmaa.2007.08.024. |
[27] |
A. N. Kochubei,
General fractional calculus, evolution equations, and renewal processes, Integral Equations Oper. Theory, 71 (2011), 583-600.
|
[28] |
O. Lipovan,
Asymptotic properties of solutions to some nonlinear integral equations of convolution type, Nonlinear Anal., 69 (2008), 2179-2183.
doi: 10.1016/j.na.2007.07.056. |
[29] |
C. Lizama,
The Poisson distribution, abstract fractional difference equations, and stability, Proc. Amer. Math. Soc., 145 (2017), 3809-3827.
doi: 10.1090/proc/12895. |
[30] |
C. Lizama,
$\ell_p$-maximal regularity for fractional difference equations on UMD spaces, Math. Nachr., 288 (2015), 2079-2092.
doi: 10.1002/mana.201400326. |
[31] |
C. Lizama and M. Murillo-Arcila,
Maximal regularity in $\ell_p$ spaces for discrete time fractional shifted equations, J. Differential Equations., 263 (2017), 3175-3196.
doi: 10.1016/j.jde.2017.04.035. |
[32] |
G. Lv, H. Gao, J. Wei and J.-L. Wu,
BMO and Morrey-Campanato estimates for stochastic convolutions and Schauder estimates for stochastic parabolic equations, J. Differential Equations, 266 (2019), 2666-2717.
doi: 10.1016/j.jde.2018.08.042. |
[33] |
S. G. Samko and R. P. Cardoso,
Integral equations of the first kind of Sonine type, Int. J. Math. and Math. Sci., 57 (2003), 3609-3632.
|
[34] |
K. Schumacher,
Traveling-front solutions for integro-differential equations. Ⅰ., J. Reine Angew. Math., 316 (1980), 54-70.
doi: 10.1515/crll.1980.316.54. |
[35] |
K. Strom,
On convolutions of B-splines, J. Comp. Appl. Math., 55 (1994), 1-29.
doi: 10.1016/0377-0427(94)90182-1. |
[36] |
V. Vergara and R. Zacher,
Optimal decay estimates for time-fractional and other nonlocal subdiffusion equations via energy methods, SIAM J. Math. Anal., 47 (2015), 210-239.
doi: 10.1137/130941900. |
[37] |
Y. Wang, L. Liu and Y. Wu,
Positive solutions for a class of fractional boundary value problem with changing sign nonlinearity, Nonlinear Analysis, 74 (2011), 6434-6441.
doi: 10.1016/j.na.2011.06.026. |
[38] |
Z. Xu and C. Wu,
Monostable waves in a class of non-local convolution differential equation, J. Math. Anal. Appl., 462 (2018), 1205-1224.
doi: 10.1016/j.jmaa.2018.02.036. |
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