Existence of a weak solution for a nonlinear parabolic problem with fractional derivates
Volume 30, Issue 3, pp 226--254
https://doi.org/10.22436/jmcs.030.03.04
Publication Date: January 12, 2023
Submission Date: May 31, 2022
Revision Date: September 22, 2022
Accteptance Date: December 19, 2022
Authors
R. A. Sanchez-Ancajima
- Dept. Mathematics Statistics and Informatics, Universidad Nacional de Tumbes, Peru.
L. J. Caucha
- Dept. Mathematics Statistics and Informatics, Universidad Nacional de Tumbes, Peru.
Abstract
The primary objective of this study was to demonstrate the existence and uniqueness of a weak solution for a nonlinear parabolic problem with fractional derivatives for the spatial and temporal variables on a one-dimensional domain. Using the Nehari manifold method and its relationship with the Fibering maps, the existence of a weak solution for the stationary case was demonstrated. Finally, using the Arzela-Ascoli theorem and Banach's fixed point theorem, the existence and uniqueness of a weak solution for the nonlinear parabolic problem were shown.
Share and Cite
ISRP Style
R. A. Sanchez-Ancajima, L. J. Caucha, Existence of a weak solution for a nonlinear parabolic problem with fractional derivates, Journal of Mathematics and Computer Science, 30 (2023), no. 3, 226--254
AMA Style
Sanchez-Ancajima R. A., Caucha L. J., Existence of a weak solution for a nonlinear parabolic problem with fractional derivates. J Math Comput SCI-JM. (2023); 30(3):226--254
Chicago/Turabian Style
Sanchez-Ancajima, R. A., Caucha, L. J.. "Existence of a weak solution for a nonlinear parabolic problem with fractional derivates." Journal of Mathematics and Computer Science, 30, no. 3 (2023): 226--254
Keywords
- Fractional calculus
- Nehari manifold
- Fibering maps
- weak Solution
MSC
References
-
[1]
R. Abeliuk, H. S. Wheater, Parameter Identification of Solute Transport Models for Unsaturated Soils, J. Hydrol., 117 (1990), 199–224
-
[2]
H. Brezis, Functional analysis, Sobolev spaces and partial differential equations, Springer, New York (2011)
-
[3]
K. J. Brown, The Nehari Manifold for a Semilinear Elliptic Equation Involving a Sublinear Term, Calc. Var. Partial Differ. Equ., 22 (2005), 483–494
-
[4]
K. J. Brown, Y. Zhang, The Nehari Manifold for a Semilinear Elliptic Equation With a Sign-Changing Weight Function, J. Differ. Equ, 193 (2003), 481–499
-
[5]
K. E. Bencala, D. M. McKnight, G.W. Zellweger, Characterization of Transport in An Acidic and Metal-r+Rich Mountain Stream Based on A Lithium Tracer Injection and Simulations of Transient Storage, Water Resour. Res., 26 (1990), 989– 1000
-
[6]
M. F. Causley, Asymptotic and Numerical Analysis of Time-Dependent Wave Propagation in Dispersive Dielectric Media That Exhibit Fractional relaxation, The State University of New Jersey, PhD thesis, (2011),
-
[7]
T. Chen, W. Liu, Solvability of fractional boundary value problem with p-Laplacian via critical point theory, Bound. Value Probl., 2016 (2016), 1–12
-
[8]
K. Diethelm, The Analysis of Fractional Differential Equations: An Application-Oriented Exposition Using Differential Operators of Caputo Type, Springer science & Business Media, (2010)
-
[9]
P. Dr´abek, A. Kufner, F. Nicolise, Quasilinear Elliptic Equations with Degenerations and Singularities, De Gruyter Ser. Nonlinear Anal. Appl., (1997),
-
[10]
L. C. Evans, Partial Differential Equations, Am. Math. Soc., (2010),
-
[11]
S. Gabriel, R. W. Lau, C. Gabriel, The dielectric properties of biological tissues: III. Parametric models for the dielectric spectrum of tissues, Phys. Med. Biol., 41 (1996), 24 pages
-
[12]
S. Goyal, K. Sreenadh, Nehari manifold for non-local elliptic operator with concave–convex nonlinearities and signchanging weight functions, Proc. Indian Acad. Sci. Math. Sci., 125 (2015), 545–558
-
[13]
M. Hartnett, A. M. Cawley, Mathematical Modelling of The Effects of Marine Aquaculture Developments on Certain Water Quality Parameters, In Water Pollution: Modelling, Measuring and Prediction, Springer, (1991), 279–295
-
[14]
F. Jiao, Y. Zhou, Existence of Solutions for a Class of Fractional Boundary Value Problems Via Critical Point Theory, Comput. Math. Appl., 62 (2011), 1181–1199
-
[15]
H. Jin, W. Liu, Eigenvalue problem for fractional differential operator containing left and right fractional derivatives, Adv. Differ. Equ., 2016 (2016), 1–12
-
[16]
Theory and Applications of Fractional Differential Equations, A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Elsevier Science B.V., Amsterdam (2006)
-
[17]
N. Laskin, Fractional Schr¨odinger equation, Phys. Rev. E, 66 (2002), 7 pages
-
[18]
C. T. Ledesma, Boundary Value Problem With Fractional p-Laplacian Operator, Adv. Nonlinear Anal., 5 (2016), 133– 146
-
[19]
C. E. T. Ledesma, M. C. M. Bonilla, Fractional Sobolev space with Riemann-Liouville fractional derivative and application to a fractional concave-convex problem, Adv. Oper. Theory, 6 (2021), 1–38
-
[20]
M. E. Londo˜ no-Lopez, Principio Fenomenol´ogico del Comportamiento Diel´ectrico de un Hidrogel de Alcohol Polivin´ılico- Phenomenological Principle Dielectrical Behaviour of Poly (vinyl alcohol) Hidrogel, Universidad Nacional de Colombia Sede Medell´ın Facultad de Minas Escuela de Ingenier´ıa de Materiales, Colombia ()
-
[21]
K. S. Miller, B. Ross, An Introduction to The Fractional Calculus and Fractional Differential Equations, John Wiley & Sons, New York (1993)
-
[22]
Z. Nehari, On a class of nonlinear second-order differential equations, Trans. Amer. Math. Soc., 95 (1960), 101–123
-
[23]
F. Navarrina, I. Colominas, M. Casteleiro, L. Cueto-Felgueroso, H. G´omez, J. Fe, A. Soage, A Numerical Model For High Impact Environmental Areas: Analysis of Hydrodynamic and Transport Phenomena at The Arosa Ria, Proceedings of the 8th Congress on Moving Boundary Problems, La Coru˜ na, 2005 (2005), 583–594
-
[24]
J. Patyn, E. Ledoux, A. Bonne, Geohydrological Research in Relation to Radioactive Waste Disposal in An Argillaceous Formation, J. Hydrol., 109 (1989), 267–285
-
[25]
H. Pu, L. Cao, Multiple Solutions for The Fractional Differential Equation With Concave-Convex Nonlinearities and Sign- Changing Weight Functions, Adv. Differ. Equ., 2017 (2017), 1–12
-
[26]
M. Qiu, L. Mei, Existence of weak solutions for nonlinear time-fractional p-Laplace problems, J. Appl. Math., 2014 (2014), 9 pages
-
[27]
H. H. G. Savenije, Salt Intrusion Model For High-Water Slack, Low-Water Slack, and Mean Tide on Spread Sheet, J. Hydrol., 107 (1989), 9–18
-
[28]
H. Schiessel, R. Metzler, A. Blumen, T. F. Nonnenmacher, Generalized Viscoelastic Models: Their Fractional Equations With Solutions, J. Phys. A: Math. Gen., 28 (1995), 19 pages
-
[29]
G. Tarantello, On nonhomogeneous elliptic equations involving critical Sobolev exponent, Ann. Inst. H. Poincar´e C Anal. Non Lin´eaire, 9 (1992), 281–304
-
[30]
C. Torres, N. Nyamoradi, Existence and Multiplicity Result For a Fractional p-Laplacian Equation With Combined Fractional Derivates, arXiv preprint arXiv:1703.02450, 2017 (2017), 17 pages
-
[31]
T.-F. Wu, The Nehari Manifold for a Semilinear Elliptic System Involving Sign-Changing Weight Functions, Nonlinear Anal., 68 (2008), 1733–1745
-
[32]
Y. Zhou, Basic Theory of Fractional Differential Equations, World Scientific, Xiangtan University, China (2014)
-
[33]
C. Z. Zhao, M. Werner, S. Taylor, R. R. Chalker, A. C. Jones, C. Zhao, Dielectric Relaxation of La-doped Zirconia Caused by Annealing Ambient, Nanoscale Res. Lett., 6 (2011), 1–6
-
[34]
Y. Zhou, J. R.Wang, L. Zhang, Basic theory of fractional differential equations, World Scientific Publishing, Singapore (2017)