Application of Rothe’s method to fractional differential equations
Downloads
DOI:
https://doi.org/10.26637/MJM0703/0006Abstract
In this paper we consider an initial value problem for a fractional differential equation formulated in a Banach space $X$ where the fractional derivative is Riemann-Liouville type of order $0<\alpha<1$. We establish the existence and uniqueness of a strong solution of the problem by applying the method of semi-discretization in time, also known as the method of lines or more popularly as Rothe's method. The dual space $X^*$ of $X$ is assumed to be uniformly convex. In the final section, we illustrate the applicability of the theoretical results with the help of an example.
Keywords:
Riemann-Liouville fractional derivative, Rothe’s method, Basset problem, accretive operator, strong solution.Mathematics Subject Classification:
Mathematics- Pages: 399-407
- Date Published: 01-07-2019
- Vol. 7 No. 03 (2019): Malaya Journal of Matematik (MJM)
A. Ashyralyev, Well-posedness of the Basset problem in spaces of smooth functions, Applied Mathematics Letters, 24( 2011), 1176-1180.
A. Ashyralyev and P.E. Sobolevskii, Well-Posedness of Parabolic Difference Equations, Operator Theory: Advances and Applications, Basel, Boston, Berlin: Birkhauser Verlag, 1994.
D. Bahuguna, Rothe's method to strongly damped wave equations, Acta Applicandae Mathematicae, 38( 1995), $185-196$.
D. Bahuguna and A. Jaiswal, Rothe time discretization method for fractional integro-differential equations, International Journal for Computational Methods in Engineering Science and Mechanics, DOI: $10.1080 / 15502287.2019 .1600075$, (2019).
D. Bahuguna and V. Raghavendra, Rothe's method to parabolic integrodifferential equation via abstract integrodifferential equation, Appl. Anal., 33( 1989) , 153167.
A. B. Basset, On the descent of a sphere in a viscous liquid, Quart. J. Math., 42(1910), 369-381.
L. Cona, Fixed point approach to Basset problem, New Trends in Mathematical Sciences, 5, No. 3(2017), 175181.
M.S El-Azab, Solution of nonlinear transport diffusion problem by linearisation, Appl. Math. Comput., $192(2007), 205-2015$.
G.H. Gao, Z. Sun and H.W. Zhang, A new fractional numerical differentiation formula to approximate the Caputo fractional derivative and its applications, Journal of computational Physics, 259( 2014), 33-50.
V. Govindaraj and K. Balachandran, Stability of Basset Equation, Journal of Fractional Calculus and Applications, Vol. 5(3S) No. 20,(2014), pp. 1-15.
E. Hernandez, D. O. Regan and K. Balachandran, Existence results for abstract fractional differential equations with nonlocal conditions via resolvent operators, Indagationes Mathematicae, 24,1(2013), 68-82.
${ }^{[12]}$ E. Hernandez, D. O. Regan and K. Balachandran, On recent developments in the theory of abstract differential equations with fractional derivatives, Nonlinear Analysis: Theory Methods and Applications, 73, 10(2010), 34623471.
Y. Hu, C. Li and H. Li, The finite difference method for Caputo-type parabolic equation with fractional Laplacian: One-dimension case, Chaos, Solitons and Fractals, 102( 2017), 319-326.
A. Jaiswal and D. Bahuguna, A second order evolution equation with a lower order fractional term in a $mathrm{Ba}-$ nach space, AIP Conference Proceedings, 2095 (2019), 030001, 1-13.
J. Kacur, Method of Rothe in evolution equations, Lecture Notes in Mathematics Springer, Berlin, 1192( 1985), $23-$ 34.
O.A. Ladyzenskaja and N.N. Ural'ceva, Boundary problems for linear and quasilinear parabolic equations, $mathrm{Am}$. Math. Soc. Transl. Ser., 2, 47(1956), 217-299.
C. Li et al., Numerical approaches to fractional calculus and fractional ordinary differential equation Journal of Computational Physics, 230(2011),3352-3368.
Y. Lin and X. Chuanju, Finite difference/spectral approximations for the time-fractional diffusion equation, Journal of Computational Physics, 225(2007), 1533-1552.
K. Li and J. Junxiong, Existence and uniqueness of mild solutions for abstract delay fractional differential equations, Computer and Mathematics with Applications, 62,3(2011), 1398-1404.
C. Li and F. Zeng, Finite Difference Methods for fractional differential equations, International Journal of Bifurcation and Chaos, 22, No.4, 1230014( 2012), 1-28.
C. Lizama and G. M. N' N'Guérékata, Mild solutions for abstract fractional differential equations, Applicable Analysis, 92,8(2013),1731-1754.
R. Magin, Fractional Calculus in bioengeenering, Crit.Rev. Biomed. Eng., 32(1)(2004), 1-104.
F. Mainardi, Fractal and Fractional in Continuum Mechanics, Springer , New York, 1997.
K. Nishimoto,Fractional Calculus and Its Applications, Nihon University, Koriyama, 1990.
K.B. Oldham, Fractional Differential Equations in electrochemistry, Adv.Eng.Softw., 41(1)( 2010), 9-12.
M. Y. Ongun, D. Arslan and R. Garrappa, Nonstandard finite difference schemes for a fractional order Brusselator system, Advances in Differential Equations, 102(2013).
A. Pazy, Semigroup of linear operators and application to partial differential equations, Springer-Verlag, New York, 1983.
I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999.
K. Rektorys, On application of direct variational methods to the solution of parabolic boundary value problems of arbitrary order in space variables, Czechoslov. Math. J., 21(1971), 318-339.
K. Rektorys, The Method of Discretization in time and Partial Differential Equations, D. Reidel Publishing Company, 1982.
E. Rothe, Zweidimensionale parabolische Randwertaufgaben als Grenz fall eindimonsionaler RandwertaufgabenMath. Ann., 102(1930), 650-670 .
Y. Zhou, J. Wang and L. Zhang, Basic theory of fractional differential equations, World Scientific, 2016.
- NA
Similar Articles
- A.L. Pathak, K.K. Dixit, Saurabh Porwal, R. Tripathi , On the coefficients of some classes of multivalent functions related to complex order , Malaya Journal of Matematik: Vol. 7 No. 03 (2019): Malaya Journal of Matematik (MJM)
You may also start an advanced similarity search for this article.
Metrics
Published
How to Cite
Issue
Section
License
Copyright (c) 2019 MJM
This work is licensed under a Creative Commons Attribution 4.0 International License.
