GEGENBAUER WAVELETS OPERATIONAL MATRIX METHOD FOR FRACTIONAL DIFFERENTIAL EQUATIONS
0
조회
0
다운로드
0
인용
0
참고문헌
키워드
초록
In this article we introduce a numerical method, named Ge- genbauer wavelets method, which is derived from conventional Gegen- bauer polynomials, for solving fractional initial and boundary value prob- lems. The operational matrices are derived and utilized to reduce the lin- ear fractional differential equation to a system of algebraic equations. We perform the convergence analysis for the Gegenbauer wavelets method. We also combine Gegenbauer wavelets operational matrix method with quasilinearization technique for solving fractional nonlinear differential equation. Quasilinearization technique is used to discretize the nonlinear fractional ordinary differential equation and then the Gegenbauer wavelet method is applied to discretized fractional ordinary differential equations. In each iteration of quasilinearization technique, solution is updated by the Gegenbauer wavelet method. Numerical examples are provided to illustrate the efficiency and accuracy of the methods.
Abstract
In this article we introduce a numerical method, named Ge- genbauer wavelets method, which is derived from conventional Gegen- bauer polynomials, for solving fractional initial and boundary value prob- lems. The operational matrices are derived and utilized to reduce the lin- ear fractional differential equation to a system of algebraic equations. We perform the convergence analysis for the Gegenbauer wavelets method. We also combine Gegenbauer wavelets operational matrix method with quasilinearization technique for solving fractional nonlinear differential equation. Quasilinearization technique is used to discretize the nonlinear fractional ordinary differential equation and then the Gegenbauer wavelet method is applied to discretized fractional ordinary differential equations. In each iteration of quasilinearization technique, solution is updated by the Gegenbauer wavelet method. Numerical examples are provided to illustrate the efficiency and accuracy of the methods.
참고문헌 (37)
- A. Ali, "Hermite wavelets method for boundary value problems", Int. J. Mod. Appl. Phys, 3(1), pp.38~47 (2013)
- A. Arikoglu, "Solution of fractional differential equations by using differen-tial transform method", Chaos Solitons Fractals, 34(5), pp.1473~1481 (2007)
- E. Babolian, "Numerical solution of differential equations by using Chebyshev wavelet operational matrix of integration", Appl. Math. Comput, 188(1), pp.417~426 (2007)
- R. E. Bellman, "Functional equations in the theory of dynamic programming. II. Non-linear differential equations", Proc. Natl. Acad. Sci, 41, pp.482~485 (1955)
- R. E. Bellman, "Functional equations in the theory of dynamic programming. V. Positivity and quasilinearity", Proc. Natl. Acad. Sci, 41, pp.743~746 (1955)
- R. E. Bellman, "Quasilinearization and nonlinear boundary-value prob-lems" (1965), American Elsevier Publishing Company
- C. Chen, "Haar wavelet method for solving lumped and distributed-parameter systems", IEE P.-Contr. Theor. Appl, 144, pp.87~94 (1997)
- I. Daubechies, "Orthonormal bases of compactly supported wavelets", Comm. Pure Appl. Math, 41(7), pp.909~996 (1988)
- M. Dehghan, "Numerical solution of nonlinear system of second-order boundary value problems using cubic B-spline scaling functions", Int. J. Comput. Math, 85(9), pp.1455~1461 (2008)
- J. V. Devi, "Generalized quasilinearization for fractional differential equations", Comput. Math. Appl, 59(3), pp.1057~1062 (2010)
- J. V. Devi, "Quasilinearization for fractional differential equations", Com-mun. Appl. Anal, 12(4), pp.407~418 (2008)
- K. T. Elgindy, "Solving boundary value problems, integral, and integro-differential equations using Gegenbauer integration matrices", J. Comput. Appl. Math, 237(1), pp.307~325 (2013)
- A. E. M. El-Mesiry, "Numerical methods for multi-term fractional (arbitrary) orders differential equations", Appl. Math. Comp, 160(3), pp.683~699 (2005)
- S. A. El-Wakil, "Adomian decomposition method for solving fractional nonlinear differential equations", Appl. Math. Comput, 182(1), pp.313~324 (2006)
- N. Engheta, "On fractional calculus and fractional multipoles in electromagnetism", IEEE T. Antenn. Propag, 44(4), pp.554~566 (1996)
- I. Hashim, "Homotopy analysis method for fractional IVPs", Commun. Nonlinear Sci. Numer. Simul, 14(3), pp.674~684 (2009)
- E. Hesameddini, "The Chebyshev wavelet method for numerical solutions of a fractional oscillator", Int. J. Appl. Math. Research, 1(4), pp.493~509 (2012)
- A. Kilicman, "Kronecker operational matrices for fractional cal-culus and some applications", Appl. Math. Comp, 187(1), pp.250~265 (2007)
- V. V. Kulish, "Application of fractional calculus to fluid mechanics", J. Fluids Eng, 124, pp.803~806 (2002)
- C. Lederman, "Mathematical justification of a non-linear integrodifferential equation for the propagation of spherical flames", Ann. Mat. Pura Appl, 183(2), pp.173~239 (2004)
- U. Lepik, "Solving PDEs with the aid of two-dimensional Haar wavelets", Comput. Math. Appl, 61(7), pp.1873~1879 (2011)
- Y. Li, "Solving a nonlinear fractional differential equation using Chebyshev wavelets", Commun. Nonlinear Sci. Numer. Simul, 15(9), pp.2284~2292 (2010)
- R. L. Magin, "Fractional calculus models of complex dynamics in biological tissues", Com-put. Math. Appl, 59(5), pp.1586~1593 (2010)
- F. Mainardi, "Fractals and Fractional Calculus in Continuum Mechanics", pp.291~348 (1997), Springer-Verlag
- M. Meerschaert, "Finite difference approximations for two-sided space-fractional partial differential equations", Appl. Numer. Math, 56(1), pp.80~90 (2006)
- R. N. Mohapatra, "An improved quasilinearization method for second order nonlinear boundary value problems", J. Math. Anal. Appl, 214(1), pp.55~62 (1997)
- C. A. Monje, "Fractional-order systems and controls, Advances in Industrial Control" (2010), Springer
- M. Razzaghi, "The Legendre wavelets operational matrix of integration", Internat. J. Systems Sci, 32(4), pp.495~502 (2001)
- M. Razzaghi, "Legendre wavelets method for constrained optimal control problems", Math. Methods Appl. Sci, 25(7), pp.529~539 (2002)
- M. Rehman, "A numerical method for solving boundary value problems for fractional differential equations", Appl. Math. Model, 36(3), pp.894~907 (2012)
- U. Saeed, "Haar wavelet–quasilinearization technique for fractional non-linear differential equations", Appl. Math. Comput, 220, pp.630~648 (2013)
- U. Saeed, "Wavelet-Galerkin quasilinearization method for nonlinear boundary value prob-lems", Abstr. Appl. Anal, 2014, pp.10~ (2014)
- U. Saeed, "Hermite wavelet method for fractional delay differential equations", J. Difference Equations, 2014, pp.8~ (2014)
- L. R. Soares, "New compactly supported scaling and wavelet functions derived from Gegenbauer polynomials", Electrical and Com-puter Engineering, Canadian Conference, 4, pp.2347~2350 (2004)[DOI]
- S. G. Venkatesh, "The Legendre wavelet method for solving initial value problems of Bratu-type", Comput. Math. Appl, 63(8), pp.1287~1295 (2012)
- Y. Wang, "The second kind Chebyshev wavelet method for solving fractional differential equations", Appl. Math. Comp, 218(17), pp.8592~8601 (2012)
- Y.Wang, "Solving two-point boundary value problems using combined homotopy perturbation method and Greens function method", Appl. Math. Comput, 212(2), pp.366~376 (2009)