Surveys in Mathematics and its Applications

ISSN 1842-6298 (electronic), 1843-7265 (print)
 Volume 15 (2020), 153 -- 168

Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 International License.

EXISTENCE RESULTS OF SELF-SIMILAR SOLUTIONS TO THE CAPUTO-TYPE'S SPACE-FRACTIONAL HEAT EQUATION

Bilal Basti and Noureddine Benhamidouche

Abstract. This paper investigates the problem of existence and uniqueness of solutions under the self-similar forms to the space-fractional heat equation. By applying the properties of Banach's fixed point theorems, Schauder's fixed point theorem and the nonlinear alternative of Leray-Schauder type, we establish several results on the existence and uniqueness of self-similar solutions to this equation.

2020 Mathematics Subject Classification: 35R11; 35A01; 34A08; 35C06; 34K37.
Keywords: Space-fractional heat equation; Self-similar solutions; Existence; Uniqueness.

Full text

References

  1. R. P. Agarwal, M. Meehan, D. O'Regan, Fixed Point Theory and Applications, Cambridge University Press, Cambridge, 2001. MR1825411. Zbl 0960.54027.

  2. Y. Arioua, N. Benhamidouche, Boundary value problem for Caputo-Hadamard fractional differential equations, Surveys in Mathematics and its Applications 12(2017), 103--115. MR3744769. Zbl 1399.34005.

  3. Y. Arioua, B. Basti, N. Benhamidouche, Initial value problem for nonlinear implicit fractional differential equations with Katugampola derivative, Applied Mathematics E-Notes 19(2019), 397--412. MR3980748. Zbl 1426.34006.

  4. Z. Bai, H. Lü, Positive solutions for boundary value problem of nonlinear fractional differential equation, J. Math. Anal. Appl. 311(2005), 495--505. MR2168413. Zbl 1079.34048.

  5. B. Basti, Y. Arioua, N. Benhamidouche, Existence and uniqueness of solutions for nonlinear Katugampola fractional differential equations, Journal of Mathematics and Applications, 42(2019), 35--61. Zbl 1425.34007.

  6. E. Buckwar, Y. Luchko, Invariance of a partial differential equation of fractional order under the Lie group of scaling transformations, J. Math. Anal. Appl. 227(1998), No. 1, 81-97. MR1652906. Zbl 0932.58038.

  7. M. El-Shahed, Positive solutions for boundary value problem of nonlinear fractional differential equation, Abstract and Applied Analysis 2007, 1--8. MR2353784. Zbl 1149.26012.

  8. A. Granas, J. Dugundji, Fixed Point Theory, Springer-Verlag, New York, 2003. MR1987179. Zbl 1025.47002.

  9. T. H. Gronwall, Note on the derivatives with respect to a parameter of the solutions of a system of differential equations, Annals of Math. (2) 20(1918--1919), 292--296. MR1502565. JFM 47.0399.02.

  10. A. A. Kilbas, H. H. Srivastava, J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier Science B.V, Amsterdam, 2006. MR2218073. Zbl 1092.45003.

  11. Y. Luchko, R. Gorenflo, Scale-invariant solutions of a partial differential equation of fractional order, Fract. Calc. Appl. Anal. 1(1) (1998), 63–78. MR1662409. Zbl 0940.45001.

  12. Y. Luchko, J. J. Trujillo, Caputo-type modification of the Erdélyi-Kober fractional derivative, Fractional Calculus and Applied Analysis, 10(3) (2007), 249–267. MR2382781. Zbl 1152.26304.

  13. I. Podlubny, Fractional Differential Equations, Mathematics in Science and Engineering, Academic Press, New York, 1999. MR1658022. Zbl 0924.34008.

  14. A. D. Polyanin, V. F. Zaitsev, Handbook of nonlinear partial differential equations, Boca Raton, FL: Chapman & Hall/CRC, 2004. MR2042347. Zbl 1053.35001.

  15. S. G. Samko, A. A. Kilbas, O. I. Marichev, Fractional Integral and Derivatives (Theory and Applications), Gordon and Breach, Switzerland, 1993. MR1347689. Zbl 0818.26003.



Bilal Basti
Laboratory for Pure and Applied Mathematics, University of Mohamed Boudiaf M'sila, Algeria,
Professor in Department of Mathematics, University of Ziane Achour Djelfa, 17000, Algeria.
e-mail: bilalbasti@gmail.com; b.basti@univ-djelfa.dz

Noureddine Benhamidouche
Laboratory for Pure and Applied Mathematics, University of Mohamed Boudiaf M'sila, Algeria.
e-mail: nbenhamidouche@univ-msila.dz

http://www.utgjiu.ro/math/sma