Difference Scheme Method for Fractional Differential Equation Defined by Caputo Derivative

Year 2018, Volume: 2 Issue: 2, 53 - 56, 10.01.2019

Mahmut Modanlı

Abstract

This study gives numerical
solution of the fractional order partial differential equation defined by
Caputo fractional derivative. Laplace transform method is used for the exact
solution of this equation depend on intial-boundary value problem. The
difference schemes are constructed for this equation. The stability of this
difference schemes is proved. Error analysis is performed by comparing the
exact solution with the approximate solution. The effectiveness of the method
is shown from the error analysis table.

Keywords

Fractional differential equation with Caputo derivative; Laplace method; Stability estimates; The difference method

References

• m1:C.Celik, M.Duman, Crank-Nicholson method for the fractional equation with the Riezs fractional derivative. Journal of computational physics, 231:1743-1750, 2012. m2 : I.I.Gorial, Numerical methods for fractional reaction-dispersion equation with Riesz space fractional derivative. Eng. and Tech. Journal, 29:709-715, 2011. m3 : H.Jafari,V.D.Gejii. Solving linear and nonlinear fractional diffusion and wave equations by adomian decomposition. Appl. Math. and Comput., 180:488-497, 2006. h1 : A.Atangana, I.Koca,. Chaos in a simple nonlinear system with Atangana--Baleanu derivatives with fractional order. Chaos, Solitons & Fractals, 89, 447-454. h2 : R.T.Alkahtani, (2016). Chua's circuit model with Atangana--Baleanu derivative with fractional order. Chaos, Solitons & Fractals, 89, 547-551. h3 : K.M.Owolabi, A.Atangana, (2018). Chaotic behaviour in system of noninteger-order ordinary differential equations. Chaos, Solitons & Fractals, 115, 362-370. h4 : A.Atangana, D.Baleanu, New fractional derivatives with nonlocal and non-singular kernel: theory and application to heat transfer model. arXiv preprint arXiv:1602.03408(2016). h5 : A.Atangana, D.Baleanu, New fractional derivatives with nonlocal and non-singular kernel: Theory and application to heat transfer model, Thermal Science, 20,763-769, 2016. http://dx.doi.org/10.2298/TSCI160111018A. h6 : R.T.Alqahtani, Atangana-Baleanu derivative with fractional order applied to the model of groundwater within an unconfined aquifer. f1 : F. Dusunceli, E. Celik, (2017). Fibonacci matrix Polynomial Method For LinearComplex Differential Equations, Asian Journal of Mathematics and Computer Research, .15(3): 229-238. f2 : F. Dusunceli, E. Celik, (2015). An Effective Tool: Numerical Solutions by Legendre Polynomials for High-Order Linear Complex Differential Equations, BritishJournal of Applied Science & Technology, . 8(4): 348-355. md1 : M. Modanli, A. Akgül, 2017. Numerical solution of fractional telegraph differential equations by theta-method: The European Physical Journal Special Topics, 226.16-18 (2017): 3693-3703. md3 : M.Modanlı, (2018). Two numerical methods for fractional partial differential equation with nonlocal boundary value problem: Advances in Difference Equations, 2018(1), 333. f3 : F.Dusunceli, (2018). Solutions for the Drinfeld-Sokolov Equation Using an IBSEFM Method, MSU Journal of Science, 6(1), 505-510. DOI : 10.18586/msufbd.403217. f4 : F. Dusunceli, E. Celik, (2017). Numerical Solution For High-Order LinearComplex Differential Equations with Variable Coefficients, Numerical Methods for PartialDifferential Equations, DOI: 10.1002/num.22222. (SCI-E)