Abstract
At present, three numerical solution methods have mainly been used to solve fractional-order chaotic systems in the literature: frequency domain approximation, predictor–corrector approach and Adomian decomposition method (ADM). Based on the literature, ADM is capable of dealing with linear and nonlinear problems in a time domain. Also, the Adomian decomposition method (ADM) is among the efficient approaches for solving linear and non-linear equations. Numerical solution method is one of the critical problems in theoretical research and in the applications of fractional-order systems. The solution is decomposed into an infinite series and the integral transformation to a differential equation is implemented in this work. Furthermore, the solution can be thought of as an infinite series that converges to an exact solution. The aim of this study is to combine the Adomian decomposition approach with a different integral transformation, including Laplace, Sumudu, Natural, Elzaki, Mohand, and Kashuri-Fundo. The study's key finding is that employing the combined method to solve fractional ordinary differential equations yields good results. The main contribution of our study shows that the combined numerical methods considered produce excellent numerical performance for solving fractional ordinary differential equations. Therefore, the proposed combined method has practical implications in solving fractional order differential equations in science and social sciences, such as finding analytical and numerical solutions for secure communication system, biological system, financial risk models, physics phenomenon, neuron models and engineering application.
Highlights
The Adomian decomposition method (ADM) has many applications in science and technology [1]-[5], such as biological population model [6], SEIR epidemic models [7], arm robot [8], mobile robot navigation [9], financial risk chaotic system [10], dissipative magnetic Jeffrey biofluid [11], two-disk dynamo system [12], communication system based on Internet of Thing (IoT) [13], electronic circuit [14], electrostatic micro-actuators system [15], Simulated Rocket Motor [16] and field programmable gate array [17]
Based on the background study of previous research, this paper presents review the combination of Adomian decomposition methods with other integral transforms, such as Sumudu, Laplace, Natural, Elzaki, Mohand, and Kashuri-Fundo
Consider fractional ordinary differential equation defined in equation (14), Using definition 2.6 on equation (14), we obtain the solution using the Natural Decomposition method as follows: yy(0) uuαα uuαα yy(ss, uu) = ss + ssαα NN[gg(tt)] + ssαα NN[NNNN(tt)] +
Summary
The ADM has many applications in science and technology [1]-[5], such as biological population model [6], SEIR epidemic models [7], arm robot [8], mobile robot navigation [9], financial risk chaotic system [10], dissipative magnetic Jeffrey biofluid [11], two-disk dynamo system [12], communication system based on Internet of Thing (IoT) [13], electronic circuit [14], electrostatic micro-actuators system [15], Simulated Rocket Motor [16] and field programmable gate array [17]. Doğan [21] presented combined Laplace Transform and ADM for solution of the ordinary differential equations (ODEs) They found that the Combined Adomian decomposition- Laplace transform can be applied in the linear and non-linear systems of ODEs. The Sumudu Decomposition Method has very important applications in mathematical fields, such as fractional Bratu-type differential equations [22], Volterra integrodifferential equations [23], Klein-Gordon equations [24], Space-Fractional Telegraph Equations [25], fractional Harry Dym equation [26], Riccati equation of variable fractional order [27] and Conformable Fractional Fitzhugh–Nagumo Model [28]. Based on the background study of previous research, this paper presents review the combination of Adomian decomposition methods with other integral transforms, such as Sumudu, Laplace, Natural, Elzaki, Mohand, and Kashuri-Fundo
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