METHOD OF LINES FOR FRACTIONAL HYBRID INTEGRO-DIFFERENTIAL EQUATIONS
Keywords:
Strong solution; Riemann-Liouville fractional derivative; Existence and uniqueness; method of lines.Abstract
In this manuscript, we study an initial boundary value problem for a fractional hybrid integro-differential equations in a Banach algebra. We establish the existence and uniqueness of a strong solution of the problem. The results are obtained by using method of lines, popularly known as method of semi-discretization or Rothe’s method. An example is given in the final section to demonstrate how the theoretical results can be applied.
References
V. Lakshmikantham, V. Devi, Theory of fractional differential equations in Banach space,
Eur. J. Pure Appl. Math. 1, 38-45, 2008.
V. Lakshmikantham, A. Vatsala, Basic theory of fractional differential equations, Nonlinear
Anal. TMA 69, 2677-2682, 2008.
K. Oldham, J. Spanier, The Fractional Calculus, Academic Press New York, 1974.
I. Podlubny, Fractional Differential Equations. Mathematics in Sciences and Engineering, vol.
, Academic Press San Diego, 1999.
A. Ashyralyev, Well-posedness of the Basset problem in spaces of smooth functions, Appl.
Math. Lett. vol. 24, no. 7, pp. 1176–1180, 2011.
A. B. Basset, On the descent of a sphere in a viscous liquid, Quart. J. Math., vol. 42, pp.
–381, 1910.
T. M. Sabri Thabet and B. Machindra Dhakne, On abstract fractional integro-differential
equations via measure of non-compactness, Adv. Fixed Point Theory, vol. 6, no. 2, pp. 175
, 2016.
J. Zhao, J. Xiao, and J. Neville Ford, Collocation methods for fractional integro-differential
equations with weakly singular kernels, Numer. Algor., vol. 65, no. 4, pp. 723–743, 2014.
Z. Guo, M. Liu, and D. Wang, Solutions of nonlinear fractional integrodifferential equations
with boundary conditions, Bull. TICMI, vol. 16, no. 2, pp. 58–65, 2012.
F. Awawdeh, E. A. Rawashdeh, and H. M. Jaradat, Analytic solution of fractional integro
differential equations, Ann. Univ. Craiova Math. Comput. Sci., vol. 38, no. 1, pp. 1–10, 2011.
D. Bahuguna and V. Raghavendra, Rothe’s method to parabolic integro-differential equations
via abstract integro-differential equations, Appl. Anal., vol. 33, no. 3-4, pp. 153-167, 1989.
D. Bahuguna, Quasilinear integrodifferential equations in Banach spaces, Nonlinear Anal.
Theory Appl., vol. 24, no. 2, pp. 175–183, 1995.
A. Chaoui and H. Ahmed, On the solution of a fractional diffusion integrodifferential equation
with Rothe time discretization, Numer. Funct. Anal. Optim., vol. 39, no. 6, pp. 643–654, 2018.
A. Guezzane-Lakoud and D. Belakroum, Rothe’s method for telegraph equation with integral
conditions, Nonlinear Anal., vol. 70, pp. 3842–3853, 2009.
K. Rektorys, The Method of Discretization in Time and Partial Differential Equations, D.
Reidel Publishing Company, 1982.
J. Kacur, Method of Rothe in Evolution Equations. Lecture Notes in Mathematics, vol. 1192,
Springer, 1985, pp. 23–34.
D. Bahuguna, Rothe’s method to strongly damped wave equations, Acta Applic. Math., vol.
, no. 2, pp. 185–196, 1995.
J. Kim, Semidiscretization method for three-dimensional motion of a Bingham fluid, Siam
J. Math. Anal., vol. 21, no. 1, pp. 53–75, Jan. 1990.
S. Agarwal and D. Bahuguna, Method of semidiscretization in time to nonlinear retarded
differential equations with nonlocal history conditions, Int. J. Math. Math. Sci., vol. 2004, no.
, pp. 1943–1956, 2004.
J. M. Holte, Discrete Gronwall lemma and applications, MAA-NCS Meeting at the University
of North Dakota, Oct. 24, 2009.
Y. Hu, C. Li, and H. Li, The finite difference method for Caputo-type parabolic equation with
fractional Laplacian: One-dimension case, Chaos Solitons Fractals, vol. 102, pp. 319–326,
A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations,
New York: Springer-Verlag, 1983.
C. Li and Z. Fanhai, Finite difference methods for fractional differential equations, Int. J.
Bifurcation Chaos, vol. 22, no. 04, pp. 1230014, 2012.
D. Bahuguna and V. Raghavendra, Application of Rothe’s method to nonlinear Schr¨odinger
type equations, Appl. Anal., vol. 31, no. 1–2, pp. 149–160, 1988.
I. Podlubny, Fractional Differential Equations, San Diego: Academic Press, 1999.
T. Kato, Nonlinear semigroups and evolution equations, J. Math. Soc. Japan, vol. 19, no. 4,
pp. 508–520, 1967.
B. C. Dhage, On α-condensing mappings in Banach algebras, Math. Student 63 (1994) 146
RAJIB HALOI AND MONUJ GOGOI*
B. C. Dhage, Fixed point theorems in ordered Banach algebras and applications, Panamer.
Math. J. 9 (4) (1999) 93-102.
B. C. Dhage, A nonlinear alternative in Banach algebras with applications to functional
differential equations, Nonlinear Funct. Appl. 8 (2004) 563-575.
B. C. Dhage, V. Lakshmikantham, Quadratic perturbations of periodic boundary value prob
lems of second order ordinary differential equations, Differ. Equ. Appl. 2 (4) (2010) 465–486.
E. Rothe, Two-dimensional parabolic boundary value problems as a limiting case of one
dimensional boundary value problems, Math. Ann. 102 (1930), 650–670.
D.Bahuguna, Rothe’s method to strongly damped wave equations, Acta Applicandae Mathe
maticae, 38(1995), 185-196.
D.Bahuguna and V. Raghavendra, Rothe’s method to parabolic integrodifferential equation
via abstract integrodifferential equation, Appl. Anal., 33(1989), 153-167.
A.Jaiswal and D.Bahuguna, A second order evolution equation with a lower order fractional
term in a Banach space, AIP Conference Proceedings, 2095 (2019), 030001, 1-13.
B. C. Dhage, V. Lakshmikantham, Basic results on hybrid differential equations, Nonlinear
Anal. Hybrid 4 (2010) 414-424.
O.A. Ladyzenskaja and N.N. Ural’ceva, Boundary problems for linear and quasilinear para
bolic equations, Am. Math. Soc. Transl. Ser., 2, 47(1956), 217-299.
K. Rektorys, The Method of Discretization in time and Partial Differential Equations,
D.Reidel Publishing Company, 1982.
Y.Zhou, J.Wong, L.Zhang, Basic theory of fractional differential equations, World Scientific
Publishing Company (2016).
D.Bahuguna, Anjali Jaiswal, Application of Rothe’s method to fractional differential equa
tions, vol.7, no. 3, 399-407, 2019
