A numeric–analytic method for approximating a giving up smoking model containing fractional derivatives

Ertürk, Vedat Suat; Zaman, Gul; Momani, Shaher

doi:10.1016/j.camwa.2012.02.002

Cited by 84 publications

(43 citation statements)

References 13 publications

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“…Comparing these two fractional derivatives, one easily arrives at the fact that Caputo derivative of a constant is equal to zero, which is not the case for the Riemann-Liouville derivative [19]. The main concern of the paper thus focuses on the Caputo derivative of order α > 0, which is rather applicable in real application [20,21]. Fractional calculus has previously been used in epidemiological studies [22,23,24,25].…”

Section: Introductionmentioning

confidence: 99%

Numerical solution and stability analysis of a nonlinear vaccination model with historical ejects

Rostamy¹

2017

HJMS

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In this paper, we extend the classical vaccination epidemic model from a deterministic framework to a model with historical effects by formulating it as a system of fractional-order differential equations (FDEs). The basic reproduction number R0 of the resulting fractional model is computed and it is shown that if R0 is less than one, the disease-free equilibrium is locally asymptotically stable. Particularly, we analytically calculate a certain threshold-value for R0 and present the existence conditions of endemic equilibrium. By using stability analysis, we prove stability and α-stability of the endemic equilibrium points. The proposed model is applied on Pertussis disease and the fractional nonlinear system of the model is solved by applying multi-step generalized differential transform method (MSGDTM). Our results show that historical effects play an important role on the disease spreading.

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Section: Introductionmentioning

confidence: 99%

Numerical solution and stability analysis of a nonlinear vaccination model with historical ejects

Rostamy¹

2017

HJMS

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“…In [10], the authors studied the giving up smoking dynamics using a fractional order model; approximate solutions via Laplace Adomian decomposition method were obtained. The multi-step generalized differential transform method was employed in [9] to obtain accurate solutions to a giving up smoking model of fractional order. The giving up smoking dynamics models have been extended to the scope of fractional derivatives using power law and exponential decay law.…”

Section: Introductionmentioning

confidence: 99%

Mathematical modeling of the smoking dynamics using fractional differential equations with local and nonlocal kernel

Morales‐Delgado¹,

Gómez‐Aguilar²,

Taneco-Hernández³

et al. 2018

J. Nonlinear Sci. Appl.

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In this paper, we analyze the fractional modeling of the giving up the smoking using the definitions of Liouville-Caputo and Atangana-Baleanu-Caputo fractional derivatives. Applying the homotopy analysis method and the Laplace transform with polynomial homotopy, the analytical solution of the smoking dynamics has obtained. Furthermore, using an iterative scheme by the Laplace transform, and the Atangana-Baleanu fractional integral, special solutions of the model are obtained. Uniqueness and existence of the solutions by the fixed-point theorem and Picard-Lindelof approach are studied. Finally, some numerical simulations are carried out for illustrating the results obtained.

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“…Several methods have been proposed to analytically and numerically solve nonlinear fractional order differential equations including initial value problems (IVPs) and boundary value problems (BVPs). These methods include the Adomian decomposition method (ADM) [9][10][11][12][13][14], the multistep generalized differential transform method (MSGDTM) [15], the Adams-Bashforth-Moulton type predictor-corrector scheme [16], and the Haar wavelet method [17].…”

Section: Introductionmentioning

confidence: 99%

New Modified Adomian Decomposition Recursion Schemes for Solving Certain Types of Nonlinear Fractional Two-Point Boundary Value Problems

Sirisubtawee

Kaewta

2017

International Journal of Mathematics and Mathematical Sciences

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We apply new modified recursion schemes obtained by the Adomian decomposition method (ADM) to analytically solve specific types of two-point boundary value problems for nonlinear fractional order ordinary and partial differential equations. The new modified recursion schemes, which sometimes utilize the technique of Duan's convergence parameter, are derived using the DuanRach modified ADM. The Duan-Rach modified ADM employs all of the given boundary conditions to compute the remaining unknown constants of integration, which are then embedded in the integral solution form before constructing recursion schemes for the solution components. New modified recursion schemes obtained by the method are generated in order to analytically solve nonlinear fractional order boundary value problems with a variety of two-point boundary conditions such as Robin and separated boundary conditions. Some numerical examples of such problems are demonstrated graphically. In addition, the maximal errors (ME ) or the error remainder functions (ER ( )) of each problem are calculated.

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A numeric–analytic method for approximating a giving up smoking model containing fractional derivatives

Cited by 84 publications

References 13 publications

Numerical solution and stability analysis of a nonlinear vaccination model with historical ejects

Numerical solution and stability analysis of a nonlinear vaccination model with historical ejects

Mathematical modeling of the smoking dynamics using fractional differential equations with local and nonlocal kernel

New Modified Adomian Decomposition Recursion Schemes for Solving Certain Types of Nonlinear Fractional Two-Point Boundary Value Problems

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