A MESH-FREE TECHNIQUE OF NUMERICAL SOLUTION OF NEWLY DEFINED CONFORMABLE DIFFERENTIAL EQUATIONS

Year 2016, Volume: 4 Issue: 2, 149 - 157, 01.10.2016

FUAT Usta

Abstract

Motivated by the recently defined conformable derivatives proposed in [2], we introduced a new approach of solving the conformable ordinary differential equation with the mesh-free numerical method. Since radial basis function collocation technique has outstanding feature in comparison with the other numerical methods, we use it to solve non-integer order of differential equation. We subsequently present the results of numerical experimentation to show that our algorithm provide successful consequences.

Keywords

Conformable derivative , mesh-less method , radial basis functions , collocation technique

References

• [1] T. Abdeljawad, On conformable fractional calculus, Journal of Computational and Applied Mathematics 279 (2015) 57-66.
• [2] Douglas R. Anderson and Darin J. Ulness, Newly dened conformable derivatives, Advances in Dynamical Systems and Applications Vol:10, No.2 (2015), 109-137.
• [3] C. Franke and R. Schaback, Solving partial differential equations by collocation using radial basis functions, Applied Mathematics and Computation 93 (1998) 73-82.
• [4] R. L. Hardy, Theory and applications of the multiquadric biharmonic method. 20 years of discovery 1968-1988, Computers and Mathematics with Applications 19(8-9) (1990) 163{208.
• [5] E. J. Kansa, Multiquadricsa scattered data approximation scheme with applications to computational luid-dynamics. I. Surface approximations and partial derivative estimates, Computers and Mathematics with Applications 19(8-9) (1990) 127{145.
• [6] U.N. Katugampola, A new fractional derivative with classical properties, Journal of the American Math.Soc., 2014, in press, arXiv:1410.6535.
• [7] R. Khalil, M. Al horani, A. Yousef and M. Sababheh, A new denition of fractional derivative, Journal of Computational Applied Mathematics, 264 (2014), 65-70.
• [8] A. A. Kilbas, H.M. Srivastava and J.J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier B.V., Amsterdam, Netherlands, 2006.
• [9] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego CA, 1999.
• [10] M. J. D. Powell, The theory of radial basis function approximation in 1990, Oxford University Press, New York, 1992.
• [11] S.G. Samko, A.A. Kilbas and O.I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordonand Breach, Yverdon et alibi, 1993.
• [12] Y. Zhang, A nite difference method for fractional partial dierential equation, Applied Mathematics and Computation, Vol:215, No.2 (2009), 524-529.