Caputo and Atangana-Baleanu-Caputo Fractional Derivative Applied to Garden Equation

Abstract

In this study, the garden equation which is a nonlinear partial differential equation is discussed. First, we will expand the garden equation to the Caputo derivative and Atangana-Baleanu fractional derivative in the sense of Caputo. Then, we will then demonstrate the existence of the new equation with the help of the fixed point theorem. Finally, we will examine uniqueness solution for the two fractional operators.

Keywords

• Caputo fractional derivative,Atanagana-Baleanu-Caputo fractional derivative,Fixed-Point theorem,Garden equation,Existence and Uniqueness

Kaynakça

1. I. Podlubny, Fractional Differential Equations, vol. 198 of Mathematics in Science and Engineering, Academic Press, San Diego, Calif, USA, 1999.
2. M. Caputo, Linear models of dissipation whose Q is almost frequency independent, Part II, Geophys. J. R. Astr. Soc. 13 (1967) 529--539.
3. A. Atangana, D. Baleanu, New fractional derivatives with non-local and non-singular kernel: Theory and application to heat transfer model, Therm. Sci. 20 (2016) 763–-769.
4. T. Yamamoto, X. Chen, An existence and nonexistence theorem for solutions of nonlinear systems and its application to algebraic equations, Journal of computational and applied mathematics 30 (1990) 87--97.
5. J. H. He, Approximate solution of nonlinear differential equations with convolution product nonlinearities. Computer methods in applied mechanics and engineering, 167(1-2), (1998) 69-73.
6. S. Liao, An optimal homotopy-analysis approach for strongly nonlinear differential equations. Communications in Nonlinear Science and Numerical Simulation, 15(8), (2010) 2003-2016.
7. N. A. Kudryashov, Simplest equation method to look for exact solutions of nonlinear differential equations. Chaos, Solitons $\&$ Fractals, 24(5), (2005) 1217-1231.