Hermite Operational Matrix for Solving Fractional Differential Equations

Yıl 2020, Cilt: 3 Sayı: 1, 87 - 90, 15.12.2020

Hatice Yalman Koşunalp , Mustafa Gülsu

Öz

This paper aims to solve the fractional differential equations (FDEs) with operational matrix method by Hermite polynomials in the sense of Caputo derivative. For this purpose, we attempt to re-define the FDEs with a set of algebraic equations with initial conditions which simplifies the complete problem. We achieve either exact or approximated solutions by solving these algebraic equations with the proposed method. To indicate the efficiency of the proposed method, various illustrative examples are solved.

Anahtar Kelimeler

Caputo derivative, Fractional Integration, Hermite, Operational Matrix

Kaynakça

• 1 K.S. Miller, B. Ross, (Eds.), Introduction to the Fractional Calculus and Fractional Differential Equations,, John Wiley and Sons, Inc., New York, 1993.
• 2 K.B. Oldham, J. Spanier, The Fractional Calculus, Theory and Appilcations of Differentiation and Integration to Arbitrary Order., Dover Publication, Mineola, 2006.
• 3 A. Plonka , J. Spanier, Recent Developments in dispersive kinetics., Progr. React. Kinet. Mech., 25(2)(2000), 109-127.
• 4 P. Allegrini, M. Buiatti, P. Grinolini, B.L. West Fractional Brownian Motion As a Nonstationary Process: Analternative Paradigm for dNA Sequences., Phys. Rev. E, 57(4)(1998), 558-567.
• 5 J. Bisquert, Fractional Diffusion in the Multiple-Trapping Regime and Revision of the Equivalence with the Continuous time Random Walk, Phys. Rev. Lett., 91(2003).
• 6 A. A. Kilbas, H. M. Srivastava, On matrix transformations between some sequence spaces and the hausdorff measure of noncompactness, Theory and Applications of Fractional Differential Equations, Elsevier, San Diego, 2006.
• 7 A. H. Bhrawy, A. S. Alofi, The Operational Matrix of Fractional Integration for Shifted Chebyshev Polynomials, Appl. Math. Lett., 26(2013), 25-31.
• 8 E. H. Doha, A. H. Bhrawy, S. S. Ezz-Eldien, A New Jacobi Operational Matrix: An Application for Solving Fractional Differential Equations, Appl. Math. Modell, 36(2013), 4931-4943.
• 9 A. H. Bhrawy, M. A. Alghamdi, A Shifted Jacobi –Gauss-Lobatto Collocation Method for Solving Nonlinear Fractional Langevin Equation, Bound. Value Probl. 62(2012).
• 10 M. H. Akrami, M. H, Atabekzadeh, G. H. Erjaee, The Operational Matrix of Fractional Integration for Shifted Legendre Polynomials, Iran. J. Sci. Technol., 37(4)(2013), 439-444.
• 11 R. Belgacem, A. Bokhari, A. Amir, Bernoulli Operational Matrix of Fractional Derivative for Solution of Fractional Differential Equations, Gen. Lett. Math., 5(1)(2018), 32-46.
• 12 F. Dusunceli, E. Celik, Numerical Solution for High-Order Linear Complex Differential Equations By Hermite Polynomials, I˘gdır Univ. J. Inst. Sci. Tech., 7(4)(2017), 189-201.