Published: 2021-10-18

Page: 40-50

E. A. A. ZIADA *

Department of Basic Science, Nile Higher Institute for Engineering & Technology, Mansoura, Egypt.

*Author to whom correspondence should be addressed.


In this paper a nonlinear delay differential equation (NDDE) of arbitrary orders of Riemann- Liouville sense is studied. Adomian decomposition method (ADM) is used to solve this type of equations. The existence and stability of a unique solution will be proved. Convergence analysis of ADM is discussed. The maximum absolute truncated error of Adomian’s series solution is estimated. Stability of the solution is also discussed.

Keywords: Nonlinear delay differential equation, arbitrary orders, banach space, fixed point theorem, convergence analysis, stability, adomian decomposition method

How to Cite

ZIADA, E. A. A. (2021). ANALYTICAL SOLUTION FOR MULTI-TERM FRACTIONAL DELAY DIFFERENTIAL EQUATIONS. Asian Journal of Mathematics and Computer Research, 28(3), 40–50. Retrieved from


Download data is not yet available.


Miller KS, Ross B. An introduction to the fractional calculus and fractional differential equations. Wiley-Interscience, New York; 1993.

Podlubny I. Fractional Differential equations. Academic Press, New York; 1999.

Kilbas AA, Srivastava HM, Trujillo JJ. Theory and applications of fractional differential equations. Elsevier, New York; 2006.

Sh. A. Abd El-Salam, El-Sayed AMA. On the stability of some fractional-order non- autonomous systems. Electronic Journal of Qualitative Theory of Differential Equations. 2007;6:1-14.

El-Sayed AMA, Sh. A. Abd El-Salam. On the stability of a fractional-order differential equation with nonlocal initial condition. Electronic Journal of Qualitative Theory of Differential Equations. 2008;29:1-8.

Evans DJ, Raslan KR. The Adomian decomposition method for solving delay differential equation. International Journal of Computer Mathematics, (UK). 2005;82:49-54.

Zwillinger D. Handbook of differential equations. Academic Press; 1997.

Mensour B, Longtin A. Chaos control in multistable delay-differential equations and their singular limit maps. Pysical Review E. 1998;58:410-422.

Hefferan JM, Corless RM. Solving some delay differential equations with computer algebra. Applied Probability Trust. 2005;1-22.

El-Sayed AMA, El-Mesiry EM, El-Saka HAA. Numerical solution for multi-term fractional (arbitrary) orders differential equations. Comput. and Appl. Math. 2004;23(1):33-54.

El-Mesiry EM, El-Sayed AMA, El-Saka HAA. Numerical methods for multi-term fractional (arbitrary) orders differential equations, Appl. Math. and Comput. 2005;160(3):683-699.

Najeeb Alam Khan, Oyoon Abdul Razzaq, Asmat Ara, Fatima Riaz. Numerical So-lution of System of Fractional Differential Equations in Imprecise Environment; 2016. DOI: 10.5772/64150.

Abdon Atangana, Ernestine Alabaraoye. Solving a system of fractional partial dif-ferential equations arising in the model of HIV infection of CD4+ cells and attractor one-dimensional Keller-Segel equations. Advances in Difference Equations. 2013;94:1-14.

Hasanen A. Hammad, Manuel De la Sen. Tripled fixed point techniques for solving system of tripled-fractional differential equations. AIMS Mathematics. 2021;6(3):2330-2343.

Rida SZ, Arafa AAM. New method for solving linear fractional differential equations. International Journal of Differential Equations. 2011;1-8. DOI: 10.1155/2011/814132.

Daraghmeh A, Qatanani N, Saadeh A. Numerical solution of fractional differential equations. Applied Mathematics. 2020;11:1100-1115. DOI: 10.4236/am.2020.1111074.

Adomian G. Solving Frontier Problems of Physics: The Decomposition Method, Kluwer; 1995.

Adomian G. Stochastic system. Academic Press; 1983.

Adomian G. Nonlinear stochastic operator equations. Academic press, San Diego; 1986.

Adomian G. Nonlinear stochastic systems: theory and applications to physics. Kluwer; 1989.

Abbaoui K, Cherruault Y. Convergence of Adomian’s method applied to differential equations. Computers Math. Applic. 1994;28:103-109.

Cherruault Y, Adomian G, Abbaoui K, Rach R. Further remarks on convergence of decomposition method. International J. of Bio-Medical Computing. 1995;38:89-93.

Shawaghfeh NT. Analytical approximate solution for nonlinear fractional differential equations. J. Appl. Math. Comput. 2002;131:517-529.

El-kalla IL. Convergence of the Adomian method applied to a class of nonlinear integral equations. Applied Mathematics Letters. 2008;21:372-376.