---

Discontinuity, Nonlinearity, and Complexity 11(4) (2022) 723--734 | DOI:10.5890/DNC.2022.12.010

Amel Labadla, Abderrazek Chaoui, Manal Djaghout

Department of Mathematics, Faculty of Sciences, University 8 May 1945, B.P.401, 24000, Guelma, Algeria

Download Full Text PDF

 

Abstract

In this article, we prove the existence, uniqueness and some stability results for fractional integro-differential equation of reconstruction of the unknown time-dependent boundary function $\gamma \left( t\right) $ from an additional integral measurement $\theta \left( t\right) =\int_{\Omega }I^{1-\alpha }\left( u\left( t, x\right) \right) dx$ by the use of Rothe time discretization. Numerical experiments are given to illustrate the results.

References

1. [1]	Anh, V.V. and Leonenko, N.N. (2001), Spectral analysis of fractional kinetic equations withrandom data, J. Stat. Phys., 104, 1349-1387.

2. [2]	Metzler, R. and Klafter, J. (2000), The random walk's guide to anomalous diffusion: A fractional dynamics approach, Phys. Rep., 339, 1-77.

3. [3]	Chaoui, A. and Lakoud, A.G. (2015), Solution to an integrodifferential equation with integral codition, Appl. Math. comput., 266, 903-908.

4. [4]	Chaoui, A. and Rezgui, N. (2018), Solution to fractional pseudoparabolic equation with fractional integral condition, Rend. Circ. Mat. Palermo, II. Ser, 67(2), 205-213.

5. [5]	Bahuguna, D., Abbas, S., and Dabas, J. (2008), Partial functional differential equation with an integral condition and applications to population dynamics, Nonliear Analysis, 69 , 2623-2635.

6. [6]	El-Azab, M.S. (2007), Solution of nonlinear transport diffusion problem by linearisation, Appl. Math. Comput., 192, 205-215.

7. [7]	Petersson, L.E. and Kuliev, K.(2007), An extension of rothe's method to non-cylindrical domains, Application of Mathematics, 52(5), 365-389.

8. [8]	Bagley, R.L. (1983), Theorical basis for the application of fractional calculus to viscoelasticity, Journal of Rheology, 27(3) , 201-210.

9. [9]	Engeita, N. (1996), On fractional calculus and fractional mulipoles in electromagnetism, IEEE Trans., 44(4), 554-566.

10. [10]	Agrawl, O.P., Sabatier, J., and Machado, J.A.T. (2007), Advances in fractional calculus, Springer-Verlag.

11. [11]	Magin, R. (2004), Fractional calculus in bioengeenering, Crit. Rev. Biom. Eng., 32(1), 1-104.

12. [12]	Mainardi, F. (1997), Fractal and fractional calculus in cintinuum mechanics, Springer, New York.

13. [13]	Nishimoto, K. (1990), Fractional calculus and its applications, Nihon University, Koriyama.

14. [14]	Oldham, K.B. (2009), Fractional differential equations in electrochemistry, Advances in Engineering Software.

15. [15]	Podlubny, I. (1999), Fractional Differential Equations, Academic Press, San Diego.

16. [16]	Raheem, A. and Bahuguna, D. (2013), Existence and uniqueness of solution for a fractional differential equation by Rothe method, Nonl. Evol. Equ. Appl., 43(6), 43-54.

17. [17]	Oldham, K.B. and Spanier, J. (1974), The fractional calculus, Academic press, New york.

18. [18]	Mainardi, F. and Paradisi, P. (1997), Model of diffusion waves in viscoelasticity based on fractal calculus, IEEE, New York, 4961-4966.

19. [19]	EL-Borai, M. (2002)., Some probability densities and fundamental solutions of fractional evolution equations, Chaos Solitons of fractals, 14 , 433-440.

20. [20]	Kacur, J. (1985), Method of roth in evolution equations, Teubner Texte zur Mathematik, 80.

21. [21]	Evans, L.C. (1990), Weak convergence methods for nonlinear partial differential equations, American Mathematical Society, Providence, 74, 205-2015.