Fixed point approach to Basset problem

Year 2017, Volume: 5 Issue: 3, 175 - 181, 01.07.2017

Lale Cona

Abstract

In the present paper, a sufficient condition for existence and uniqueness of Basset problem is obtained. The theorem on existence and uniqueness is established. This approach permits us to use fixed point iteration method to solve problem for differential equation involving derivatives of nonlinear order.

Keywords

Fixed Point, initial Value Problem, Basset equation, Riemann-Liouville derivative

References

• Podlubny I: Mathematics in Science and Engineering 198. In Fractional Differential Equations. Academic Press, San Diego; 1999.
• Samko SG, Kilbas AA, Marichev OI: Fractional Integrals and Derivatives. Gordon & Breach, Yverdon; 1993.
• Basset AB: On the descent of a sphere in a viscous liquid. Q. J. Math. 1910, 42: 369-381.
• Ashyralyev A: Well-posedness of the Basset problem in spaces of smooth functions. Appl. Math. Lett. 2011, 24: 1176-1180. 10.1016/j.aml.2011.02.002
• Ashyralyev A, Well-posedness of fractional parabolic equations, Boundary Value Problems, 2013: 2013:31, 1-18.
• Baleanu D, Garra, R, Petras, I: A Fractional Variational Approach to the Fractional Basset-Type Equation, Reports on Mathematical Physics, 72 (1) 57-64, 2013.
• Petras I: Fractional-Order Nonlinear Systems: Modeling, Analysis and Simulation, Springer, New York (2011)
• Cakir Z: Stability of difference schemes for fractional parabolic PDE with the Dirichlet-Neumann conditions. Abstr. Appl. Anal. 2012., 2012: Article ID 463746
• Ashyralyev A, Cakir Z: On the numerical solution of fractional parabolic partial differential equations with the Dirichlet condition. Discrete Dyn. Nat. Soc. 2012., 2012: Article ID 696179.
• Ashyralyev A. , Z. Cakir; FDMfor fractional parabolic equations with the Neumann condition, Advances in Difference Equations, Vol. 2013, No. 120, doi:10.1186/1687-1847-2013-120, 2013.
• Ashyralyev A., A note on fractional derivatives and fractional powers of operators. JMAA, 357(2009) 232-236.
• Ashyralyev A., N. Emirov, Z. Cakir; Well-posedness of fractional parabolic differential and difference equations with Dirchlet-Neumann condition, Electronic Journal of Differential Equations, Vol. 2014, No. 97, pp. 1—17, 2014.
• Erwin Kreyszig, Advanced Engineering Mathematics, John Willey & Sons, New York,1993.