Convergence of the Adomian decomposition method for initial-value problems

Abdelrazec, Ahmed; Pelinovsky, Dmitry E.

doi:10.1002/num.20549

Cited by 89 publications

(43 citation statements)

References 29 publications

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“…[55][56][57][58] Elsewhere, [59] Fatoorehchi and Abolghasemi have developed a new improved algorithm to rapidly generate the Adomian polynomials of any desired analytic nonlinear operator.…”

Section: Accepted Articlementioning

confidence: 99%

An improved algorithm for calculation of the natural gas compressibility factor via the Hall‐Yarborough equation of state

et al. 2014

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Adomian decomposition method (ADM) is employed to devise a novel algorithm for calculating the compressibility factors of natural gases through this reliable equation of state. A convergence accelerator technique, namely the efficient Shanks transform, is also exploited to further improve our scheme in terms of computational speed. Unlike most of the previous numerical solution strategies, our algorithm does not require an initial guess as the starting point and is computationally efficient. The proposed algorithm is found to be superior over the common Newton-Raphson algorithm, where we have also demonstrated that the latter can easily lead to grossly erroneous solutions. For the sake of illustration, a number of real-world case study problems are solved by our algorithm and relevant comparisons are provided.

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Section: Accepted Articlementioning

confidence: 99%

An improved algorithm for calculation of the natural gas compressibility factor via the Hall‐Yarborough equation of state

et al. 2014

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“…We use the classical techniques to verify the convergence of the series (35), (36) and (37) in (Shah et al, 2016). We check the condition of convergence of the method by using the idea of the following theorem (Abdelrazec and Pelinovsky, 2011;Naghipour and Manafian, 2015). Theorem 4.1: Let be a Banach space and : → be a contractive nonlinear operator then there exit , ′ ∈ , ‖ ( ) − ( ′ )‖ ≤ ‖ − ′ ‖, 0 < < 1.…”

Section: Convergence Analysismentioning

confidence: 99%

Dynamical behavior of SIR epidemic model with non-integer time fractional derivatives: A mathematical analysis

Ahmad

2018

Int. j. adv. appl. sci.

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Protection of children from vaccine preventable diseases, such as measles is among primary goal for health worker. Measles is a highly contagious disease that can spread in a population depending on the number of peoples susceptible or infected and also depending on their dynamics in the community. The model monitors the temporal dynamics of a childhood disease in the presence of preventive vaccine. We presented a nonlinear time fractional model of measles in order to understand the outbreaks of this epidemic disease. The Caputo fractional derivative operator of order ∈ (0,1] is employed to obtain the system of fractional differential equations. The numerical solution of the time fractional model has been procured by employing Laplace Adomian decomposition method (LADM), qualitative and sensitivity analysis of the model was performed. Qualitative results shows that the model has endemic equilibrium which locally asymptotically stable for 0 > 1 and otherwise unstable. The convergence analysis and nonnegative solutions are verified for the proposed scheme. Simulation of different epidemiological classes at the effect of fractional parameter revealed that most individuals undergoing treatment join the recovered class. This method proves to be very efficient techniques for solving epidemic model to control infectious disease.

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“…We use the classical techniques to verify the convergence of the series (34-38). We check the condition of convergence of the method by using the idea of the following theorem (Abdelrazec and Pelinovsky, 2011;Naghipour and Manafian, 2015). and suppose that 0 ∈ ( ) where ( ) = { ′ ∈ : ‖ ′ − ‖ < } then we get …”

Section: Convergence Analysismentioning

confidence: 99%

Dynamical behavior of HIV immunology model with non-integer time fractional derivatives

Ashraf

2018

Int. j. adv. appl. sci.

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This paper introduces a mathematical model which describes the dynamics of the spread of HIV in the human body. Human immunodeficiency virus infection destroys the body immune system, increases the risk of certain pathologies, damages body organs such as the brain, kidney and heart or cause the death. Unfortunately, this infection disease currently has no cure to control the diseases. We propose a fractional order model in this paper to describe the dynamics of human immunodeficiency virus (HIV) infection. The Caputo fractional derivative operator of order ∈ (0,1] is employed to obtain the system of fractional differential equations. The basic reproductive number is derived for a general viral production rate which determines the local stability of the infection free equilibrium. The solution of the time fractional model has been procured by employing Laplace Adomian decomposition method (LADM) and the accuracy of the scheme is presented by convergence analysis Moreover, numerical simulation are performed to study the dynamical behavior of solution of the models. Simulations of different epidemiological classes at the effect of the fractional parameters revealed that most undergoing treatment join the recovered class. The results show the both viral production rate and death rate of infected cells play an important role the disease in the society.

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Convergence of the Adomian decomposition method for initial-value problems

Cited by 89 publications

References 29 publications

An improved algorithm for calculation of the natural gas compressibility factor via the Hall‐Yarborough equation of state

An improved algorithm for calculation of the natural gas compressibility factor via the Hall‐Yarborough equation of state

Dynamical behavior of SIR epidemic model with non-integer time fractional derivatives: A mathematical analysis

Dynamical behavior of HIV immunology model with non-integer time fractional derivatives

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