## Numerical comparison of methods for solving linear differential equations of fractional order.(English)Zbl 1137.65450

Summary: We implement relatively new analytical techniques, the variational iteration method and the Adomian decomposition method, for solving linear differential equations of fractional order. The two methods in applied mathematics can be used as alternative methods for obtaining analytic and approximate solutions for different types of fractional differential equations. In these schemes, the solution takes the form of a convergent series with easily computable components. This paper will present a numerical comparison between the two methods and a conventional method such as the fractional difference method for solving linear differential equations of fractional order. The numerical results demonstrate that the new methods are quite accurate and readily implemented.

### MSC:

 65R20 Numerical methods for integral equations 45J05 Integro-ordinary differential equations 26A33 Fractional derivatives and integrals 65L05 Numerical methods for initial value problems involving ordinary differential equations 34A30 Linear ordinary differential equations and systems
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### References:

  Basset, A.B., On the descent of a sphere in a viscous liquid, Quart J math, 42, 369-381, (1910) · JFM 41.0826.01  Bagley, R.L.; Torvik, P.J., On the appearance of the fractional derivative in the behavior of real materials, J appl mech, 51, 294-298, (1994) · Zbl 1203.74022  Shawagfeh, N.T., Analytical approximate solutions for nonlinear fractional differential equations, Appl math comput, 131, 517-529, (2002) · Zbl 1029.34003  Podlubny, I., Fractional differential equations, (1999), Academic Press New York · Zbl 0918.34010  Gao, X.; Yu, J., Synchronization of two coupled fractional-order chaotic oscillators, Chaos, solitons & fractals, 26, 1, 141-145, (2005) · Zbl 1077.70013  Lu, J.G., Chaotic dynamics and synchronization of fractional-order arneodo’s systems, Chaos, solitons & fractals, 26, 4, 1125-1133, (2005) · Zbl 1074.65146  Lu, J.G.; Chen, G., A note on the fractional-order Chen system, Chaos, solitons & fractals, 27, 3, 685-688, (2006) · Zbl 1101.37307  He, J.H., Variational iteration method for delay differential equations, Commun nonlinear sci numer simulat, 2, 4, 235-236, (1997)  He, J.H., Approximate analytical solution for seepage flow with fractional derivatives in porous media, Comput meth appl mech eng, 167, 57-68, (1998) · Zbl 0942.76077  He, J.H., Approximate solution of non linear differential equations with convolution product nonlinearities, Comput meth appl mech eng, 167, 69-73, (1998) · Zbl 0932.65143  He, J.H., Variational iteration method—a kind of non-linear analytical technique: some examples, Int J nonlinear mech, 34, 699-708, (1999) · Zbl 1342.34005  He, J.H., Variational iteration method for autonomous ordinary differential systems, Appl math comput, 114, 115-123, (2000) · Zbl 1027.34009  He, J.H.; Wan, Y.Q.; Guo, Q., An iteration formulation for normalized diode characteristics, Int J circ theory appl, 32, 6, 629-632, (2004) · Zbl 1169.94352  Adomian, G., A review of the decomposition method in applied mathematics, J math anal appl, 135, 501-544, (1988) · Zbl 0671.34053  Adomian, G., Solving frontier problems of physics: the decomposition method, (1994), Kluwer Academic Publishers Boston · Zbl 0802.65122  Shawagfeh, N.; Kaya, D., Comparing numerical methods for the solutions of systems of ordinary differential equations, Appl math lett, 17, 323-328, (2004) · Zbl 1061.65062  Momani, S.; Al-Khaled, K., Numerical solutions for systems of fractional differential equations by the decomposition method, Appl math comput, 162, 3, 1351-1365, (2005) · Zbl 1063.65055  Momani S. Numerical simulation of a dynamic system containing fractional derivatives. Accepted for presentation at the International Symposium on Nonlinear Dynamics, Shanghai, China, December 20-21; 2005.  Marinca, V., An approximate solution for one-dimensional weakly nonlinear oscillations, Int J nonlinear sci numer simulat, 3, 2, 107-110, (2002) · Zbl 1079.34028  Draˇgaˇnescu, G.E.; Caˇpaˇlnaˇsan, V., Nonlinear relaxation phenomena in polycrystalline solids, Int J nonlinear sci numer simulat, 4, 3, 219-226, (2003)  Liu, H.M., Generalized variational principles for ion acoustic plasma waves by he’s semi-inverse method, Chaos, solitons & fractals, 23, 2, 573-576, (2005) · Zbl 1135.76597  Hao, T.H., Search for variational principles in electrodynamics by Lagrange method, Int J nonlinear sci numer simulat, 6, 2, 209-210, (2005) · Zbl 1401.78004  Momani, S.; Abuasad, S., Application of he’s variational iteration method to Helmholtz equation, Chaos, solitons & fractals, 27, 5, 1119-1123, (2006) · Zbl 1086.65113  Odibat, Z.; Momani, S., Application of variational iteration method to nonlinear differential equations of fractional order, Int J nonlinear sci numer simulat, 6, 1, 27-34, (2005) · Zbl 1401.65087  Luchko Y, Gorneflo R. The initial value problem for some fractional differential equations with the Caputo derivative. Preprint series A08-98, Fachbereich Mathematik und Informatik, Freie Universitat Berlin, 1998.  Miller, K.S.; Ross, B., An introduction to the fractional calculus and fractional differential equations, (1993), John Wiley and Sons Inc. New York · Zbl 0789.26002  Oldham, K.B.; Spanier, J., The fractional calculus, (1974), Academic Press New York · Zbl 0428.26004  Caputo, M., Linear models of dissipation whose Q is almost frequency independent. part II, J roy austral soc, 13, 529-539, (1967)  Inokuti, M.; Sekine, H.; Mura, T., General use of the Lagrange multiplier in non-linear mathematical physics, (), 156-162  He, J.H., Semi-inverse method of establishing generalized principles for fluid mechanics with emphasis on turbomachinery aerodynamics, Int J turbo jet-engines, 14, 1, 23-28, (1997)  He, J.H., Variational theory for linear magneto-electro-elasticity, Int J nonlinear sci numer simulat, 2, 4, 309-316, (2001) · Zbl 1083.74526  He, J.H., Generalized variational principles in fluids (in Chinese), (2003), Science and Culture publishing house of China Hong Kong, p. 222-30  He, J.H., Variational principle for nano thin film lubrication, Int J nonlinear sci numer simulat, 4, 3, 313-314, (2003)  He, J.H., Variational principle for some nonlinear partial differential equations with variable coefficients, Chaos, solitons & fractals, 19, 4, 847-851, (2004) · Zbl 1135.35303  Liu, H.M., Variational approach to nonlinear electrochemical system, Int J nonlinear sci numer simulat, 5, 1, 95-96, (2004)
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