Analytical solution of non-integer extra-ordinary differential equation via Adomian decomposition method

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DOI:

https://doi.org/10.26637/mjm401/016

Abstract

In the present paper, we obtain the analytical solution of the linear extraordinary fractional equations with constant coefficients by Adomian decomposition method under nonhomgeneous initial value condition, this method is a powerful method which consider the approximate solution as an infinite series usually converges to the exact solution.

Keywords:

Extraordinary Fractional differential equation, Adomian decomposition method

Mathematics Subject Classification:

34A08, 74H10, 26A33
  • Ahcene Merad Departement of Mathematics, Laboratory of Dynamical Systems and Control, Faculty of Sciences, Larbi Ben M’hidi University, Oum El Bouaghi, 04000, Algeria.
  • Samir Hadid Department of Mathematics and Science, College of Education and Basic Science, Ajman University of Science and Technology, UAE.
  • Pages: 126-135
  • Date Published: 01-01-2016
  • Vol. 4 No. 01 (2016): Malaya Journal of Matematik (MJM)

F. Abdelwahid, A Mathematical model of Adomian polynomials, Appl. Math. Comput., 141 (2003), no. 2-3, $474-453$. DOI: https://doi.org/10.1016/S0096-3003(02)00266-7

M. Abramowitz and I.A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, National Bureau of Standards Applied Mathematics Series, vol.55,U.S.Government Printing Office, District of Columbia, 1964. DOI: https://doi.org/10.1115/1.3625776

G. Adomian, Nonlinear Stochastic Operator Equations, Academic Press, Florida, 1986. DOI: https://doi.org/10.1016/B978-0-12-044375-8.50012-5

G. Adomian, Nonlinear Stochastic Theory and Applications to Physics, Mathematics and Applications, Vol. 46, Kluwer Academic Publishers Group, Dordrecht, 1989.

G. Adomian, Solving Frontier Problems of Physics: The Decomposition Method, Fundamental theories of physics, Vol. 60, Kluwer Academic Publishers Group, Dordrecht, 1990.

G. Adomian and R. Rach, On linear and nonlinear integro-differential equations, J. Math. Anal. Appl. 113 (1986), no 1, 199-201. DOI: https://doi.org/10.1016/0022-247X(86)90343-4

S. Abbas, Pecompseudo almost automorphic solutions of fractional order neutral differential equation, Semigroup Forum, Volume 81, Number 3 (2010), 393-404. DOI: https://doi.org/10.1007/s00233-010-9227-0

R. P. Agarwal, Benchohra, M.; Hamani, S.; Boundary value problems for fractional differential equations, Georgian Math. J., 16, 3 (2009), 401-411. DOI: https://doi.org/10.1515/GMJ.2009.401

R.P. Agarwal, Y. Zhou and Y. He, Existence of fractional neutral functional differential equations Comp. Math. Appl., 59, 3 (2010), 1095-1100. DOI: https://doi.org/10.1016/j.camwa.2009.05.010

E. Ahmed, A. M. A. El-Sayed and H. A. A. El-Saka, Equilibrium points, stability and numerical solutions of fractional-order predator-prey and rabies models. J. Math. Anal. Appl., Vol. 325 (2007), no. 1, $542-553$. DOI: https://doi.org/10.1016/j.jmaa.2006.01.087

H.L. Arora and F. Abdelwahid, Solution of non-integer order differential equations via the Adomian decomposition method, Appl. Math. Lett. 6 (1993),no.1,21–23 DOI: https://doi.org/10.1016/0893-9659(93)90140-I

R.N. Bracewell, The Fourier Transform and Its Applications, McGraw-Hill, NewYork,(1978).

J. Banas and T. Zajac, A new approach to the theory of functional integral equations of fractional order $J$. Math. Anal. Appl., 375, 2 (2011), 375-387. DOI: https://doi.org/10.1016/j.jmaa.2010.09.004

A. Bashir and N. Sotiris, A fully Hadamard type Integral BVP of a Coupled System Frac. Calc. Appl. Anal., Vol.17, No.2 (2014), pp. 348-360. DOI: https://doi.org/10.2478/s13540-014-0173-5

V. Daftardar-Gdjji and A. Babakhani, Analysis of a system of fractional differential equations, J. Math. Anal. Appl. 293 (2004), 511-522. DOI: https://doi.org/10.1016/j.jmaa.2004.01.013

D. Shantanu; Functional Fractional Calculus for System Identification and Controls, Springer, 2007.

D. Delbosco and L. Rodino, Existence and uniqueness for a nonlinear fractional differential equation, J. Math. Anal. Appl., 204 (1996), 609-625. DOI: https://doi.org/10.1006/jmaa.1996.0456

J. Deng and L. Ma, Existence and uniqueness of solutions of initial value problems for nonlinear fractional differential equations, Appl. Math. Lett., 23, 6 (2010), 676-680. DOI: https://doi.org/10.1016/j.aml.2010.02.007

K. Diethelm, Neville J. Ford; Analysis of fractional differential equations, J. Math. Anal. Appl., 265 (2002), 229-248. DOI: https://doi.org/10.1006/jmaa.2000.7194

K. Diethelm, The analysis of fractional differential equations, Springer, 2010. DOI: https://doi.org/10.1007/978-3-642-14574-2

A. M. A. El-Sayed, Fractional order differential equation, Kyungpook Math. J., Vol. 28,2 (1988), 18-22.

A. M. A. El-Sayed, On the fractional differential equations. Appl. Math. Comput., Vol. 49 (1992), no. 2-3, 205-213 DOI: https://doi.org/10.1016/0096-3003(92)90024-U

A. M. A. El-Sayed, Nonlinear functional-differential equations of arbitrary orders. Nonlinear Anal. 33 (1998), no. 2, 181-186. DOI: https://doi.org/10.1016/S0362-546X(97)00525-7

A. M. A. El-Sayed, El-Mesiry, A. E. M.; El-Saka, H. A. A.; On the fractional-order logistic equation, Appl. Math. Lett., 20, 7 (2007), 817-823. DOI: https://doi.org/10.1016/j.aml.2006.08.013

K. M. Furati, Tatar, Nasser-eddine; Long time behavior for a nonlinear fractional model, J. Math. Anal. Appl., 332, 1 (2007), 441-454. DOI: https://doi.org/10.1016/j.jmaa.2006.10.027

A.J. George and A. Chakrabarti, The Adomian method applied to some extraordinary differential equations, Appl. Math. Lett. 8 (1995),no.3,91-97 DOI: https://doi.org/10.1016/0893-9659(95)00036-P

C. Giannantoni,; The problem of the initial conditions and their physical meaning in linear differential equations of fractional order, Appl. Math. Comp., 141, 1 (2003), 87-102. DOI: https://doi.org/10.1016/S0096-3003(02)00323-5

R. Hilfer, Applications of fractional calculus in physics. World Scientific Publishing Co., Inc., River Edge, NJ, 2000. DOI: https://doi.org/10.1142/3779

N. Heymans and I. Podlubny, Physical interpretation of initial conditions for fractional differential equations with Riemann-Liouville fractional derivatives. Rheol Acta, 45 (2006), 765-771. DOI: https://doi.org/10.1007/s00397-005-0043-5

S. B. Hadid, Local and global existence theorems on differential equations of non-integer order, J. Fract. Calc., 7 (1995), 101-105.

S. B. Hadid and S. Momani, On the existence of maximal and minimal solutions of differential equations of non-integer order, J. Fract. Calc. 9 (1996), 41-44.

I. W. Rabha and S. Momani, On the existence and uniqueness of solutions of a class of fractional differential equations, J. Math. Anal. Appl. 334 (2007), 1-10. DOI: https://doi.org/10.1016/j.jmaa.2006.12.036

B. B. Jaimini, N. Shrivastava and H. M. Srivastava, The integral analogue of the Leibniz rule for fractional calculus and its applications involving functions of several variables, Comput. Math. Appl., 41, 1-2 (2001), $149-155$. DOI: https://doi.org/10.1016/S0898-1221(01)85013-6

A. A. Kilbas, H. M.Srivastava and J. J.Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, Volume 204, 2006.

V. Lakshmikantham and A. S.Vatsala, Basic theory of fractional differential equations, Nonlinear Analysis: TMA, 69, $8(2008), 2677-2682$. DOI: https://doi.org/10.1016/j.na.2007.08.042

V. Lakshmikantham and A. S.Vatsala, Theory of fractional differential inequalities and applications. Communications in Applied Analysis, Vol-II (2007), 395-402.

V. Lakshmikantham, Theory of fractional functional differential equations, Nonlinear Analysis: TMA, 69, $10(2008), 3337-3343$. DOI: https://doi.org/10.1016/j.na.2007.09.025

J. A. T. Machado, V. Kiryakova and F. Mainardi, Recent history of fractional calculus, Commun. in Nonl. Sci. Numerical Simul., 16, 3 (2011), 1140-1153. DOI: https://doi.org/10.1016/j.cnsns.2010.05.027

F. Mainardi, Some basic problems in continuum and statistical mechanics, in Fractals and Fractional Calculus in Continuum Mechanics, Springer, New York, 1997. DOI: https://doi.org/10.1007/978-3-7091-2664-6_7

S. A. Murad and S.B.Hadid, Existence and Uniqueness Theorem for FDE with Integral BoundaryCondition", Journal of Fractional Calculus and Applications, Vol. 3, July 2012, No. 6, pp. 1-9.

K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley and Sons, Inc., New, York, 1993.

K.B Oldham and J. Spanier, The Fractional Calculus, Mathematics in Science and Engineering, vol.111, Academic Press, NewYork,1974.

I. Podlubny, Fractional Differential Equations. Academic Press, London, 1999.

I. Podlubny, Geometric and physical interpretation of fractional integration and frac- tional differentiation. Dedicated to the 60th anniversary of Prof. Francesco Mainardi. Fract. Calc. Appl. Anal., $5,4(2002), 367-386$.

I. Podlubny, I. Petras, B. M.Vinagre and P. O'Leary,; Dorcak, L.; Analogue realizations of fractional-order controllers. Fractional order calculus and its applications. Nonlinear Dynam. 29, 1-4 (2002), $281-296$.

V. P. Rubanik, Oscillations of quasilinear systems with retardation. Nauka, Moscow, 1969.

W. Rudin, Real and Complex Analysis, McGraw Hill, New York, 1966.

K. Schmitt, Nonlinear analysis and differential equations, an introduction, notes, 2004.

D. R. Smart, Fixed point theorems. Cambridge Uni. Press., Cambridge, 1980.

S. G. Samko, A. A.Kilbas and O.I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach, 1993.

J. Sabatier, O. P. Agrawal and J. A. T. Machado,; Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering, Springer, 2007. DOI: https://doi.org/10.1007/978-1-4020-6042-7

N. Ya and R. S. Hong, Regularity of mild solution to Fractional Cauchy problem with Riemann-Liouville Fractional derivative", Electronic J. of D. Es., Vol. 2014 (2014), No. 184, pp. 1-13.

C. Yu and G. Gao, Existence of fractional differential equations. J. Math. Anal. Appl., 310, 1 (2005), $26-29$. DOI: https://doi.org/10.1016/j.jmaa.2004.12.015

C. Yu and G.G. Gao, Some results on a class of fractional functional differential equations, Commun. Appl. Nonlinear Anal., 11, $3(2004), 67-75$.

X. Zhang, Some results of linear fractional order time-delay system Appl. Math. Comp., 197, 1 (2008), 407-411. DOI: https://doi.org/10.1016/j.amc.2007.07.069

S. Zhang, Positive solutions for boundary-value problems of nonlinear fractional differential equations. Electron. J. Diff. Eqns. 2006 (2006), No. 36, 1-12.

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Published

01-01-2016

How to Cite

Ahcene Merad, and Samir Hadid. “Analytical Solution of Non-Integer Extra-Ordinary Differential Equation via Adomian Decomposition Method”. Malaya Journal of Matematik, vol. 4, no. 01, Jan. 2016, pp. 126-35, doi:10.26637/mjm401/016.