Analytical solutions of some integral fractional differential–difference equations

Liu, Jian‐Gen; Yang, Xiao‐Jun; Feng, Yi‐Ying

doi:10.1142/s0217984920500098

Cited by 24 publications

(9 citation statements)

References 22 publications

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“…So, it is expected that the fractional logistic equation will give a more suitable description for the population growth. Other fractional models, including the generalized time-fractional diffusion equation, quadratic integro-differential equations, integral fractional differential–difference equations, fractional stochastic heat equation, fractional integro-partial differential equations, and fractional-order HIV/AIDS model, can be found in [11] , [12] , [13] , [14] , [15] , [16] .…”

Section: Introductionmentioning

confidence: 99%

Modified analytical approach for generalized quadratic and cubic logistic models with Caputo-Fabrizio fractional derivative

Djeddi

Hasan

Al‐Smadi

et al. 2020

Alexandria Engineering Journal

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In this paper, a modified reproducing kernel algorithm is proposed to solve a class of quadratic and cubic logistic equations with Caputo-Fabrizio fractional derivative in Hilbert space. These equations are the generalization of Verhulst’s model which describes population growth taking in account that individuals will compete for limited resources. A novel reproducing kernel function is constructed to create an orthogonal system and to calculate the analytical and approximate solutions in the desirable Sobolev space. The stability, convergence, and complexity of the proposed approach are discussed. Furthermore, the effects of the Caputo-Fabrizio fractional derivatives are studied in solving the population growth model comparing with those of the classical Caputo derivatives. The main motivation for using the proposed technique is high accuracy and low computational cost compared to other existing methods especially when involving fractional differentiation operators. In this orientation, the effectiveness, applicability, and feasibility of this technique are verified by numerical examples. In a numerical viewpoint, the obtained results indicate that the suggested intelligent method has many advantages in accuracy and stability using the new Caputo-Fabrizio derivative.

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Section: Introductionmentioning

confidence: 99%

Modified analytical approach for generalized quadratic and cubic logistic models with Caputo-Fabrizio fractional derivative

Djeddi

Hasan

Al‐Smadi

et al. 2020

Alexandria Engineering Journal

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show abstract

“…In a previous research [21], Asaduzzman et al, have studied the special cases of Equation (1) by considering α = 2. In this paper, we have extracted the general solutions of MAE according to α as being in the solutions of Equations (16), (18), and (32). Moreover, we have also investigated other values of α such as complex and rations in the coefficients of Equations (19), (21), (23), (25) and (29).…”

Section: Comparison and Discussionmentioning

confidence: 99%

“…More general properties of auxiliary and sub-equation function methods have been comprehensively introduced in the literature [28,29]. Moreover, there are many published methods for solving similar equations using different techniques and methods [30][31][32][33][34][35][36][37][38][39][40][41][42][43][44][45][46][47][48][49].…”

Section: Introductionmentioning

confidence: 99%

Extraction Complex Properties of the Nonlinear Modified Alpha Equation

Baskonus

Ercan

2021

Fractal Fract

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This paper applies one of the special cases of auxiliary method, which is named as the Bernoulli sub-equation function method, to the nonlinear modified alpha equation. The characteristic properties of these solutions, such as complex and soliton solutions, are extracted. Moreover, the strain conditions of solutions are also reported in detail. Observing the figures plotted by considering various values of parameters of these solutions confirms the effectiveness of the approximation method used for the governing model.

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“…To this aim, we feel motivated to carry out the rigorous study of this pandemic from fractional variant. This is because, it can be noted that, in the recent past, multifarious scientists have shown that the fractional calculus can more precisely explain natural phenomena than the integer calculus 8‐28 . In light of this fact, the noninteger calculus becomes increasingly significant and common in the realistic cases modeling, those with memory effects in particular.…”

Section: Introductionmentioning

confidence: 99%

Analysis of the fractional corona virus pandemic via deterministic modeling

Tuan

Tri

Băleanu

2020

Math Methods in App Sciences

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With every passing day, one comes to know that cases of the corona virus disease are increasing. This is an alarming situation in many countries of the globe. So far, the virus has attacked as many as 188 countries of the world and 5 549 131 (27 May 2020) human population is affected with 348 224 deaths. In this regard, public and private health authorities are looking for manpower with modeling skills and possible vaccine. In this research paper, keeping in view the fast transmission dynamics of the virus, we have proposed a new mathematical model of eight mutually distinct compartments with the help of memory-possessing operator of Caputo type. The fractional order parameter of the model has been optimized so that smallest error can be attained while comparing simulations and the real data set which is considered for the country Pakistan. Using Banach fixed point analysis, it has been shown that the model has a unique solution whereas its basic reproduction number  0 is approximated to be 6.5894. Disease-free steady state is shown to be locally asymptotically stable for  0 < 0, otherwise unstable. Nelder-Mead optimization algorithm under MATLAB Toolbox with daily real cases of the virus in Pakistan is employed to obtain best fitted values of the parameters for the model's validation. Numerical simulations of the model have come into good agreement with the practical observations wherein social distancing, wearing masks, and staying home have proved to be the most effective measures in order to prevent the virus from further spread.

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Analytical solutions of some integral fractional differential–difference equations

Cited by 24 publications

References 22 publications

Modified analytical approach for generalized quadratic and cubic logistic models with Caputo-Fabrizio fractional derivative

Modified analytical approach for generalized quadratic and cubic logistic models with Caputo-Fabrizio fractional derivative

Extraction Complex Properties of the Nonlinear Modified Alpha Equation

Analysis of the fractional corona virus pandemic via deterministic modeling

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