Solving smoking epidemic model of fractional order using a modified homotopy analysis transform method

P Veeresha,Haci Mehmet Baskonus,D G Prakasha

doi:10.1007/s40096-019-0284-6

P Veeresha, Haci Mehmet Baskonus + Show 1 more

Open Access

https://doi.org/10.1007/s40096-019-0284-6

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Journal: Mathematical Sciences	Publication Date: Apr 25, 2019
Citations: 70	License type: open-access

Affiliation: Karnatak University, Harran University

Abstract

The pivotal aim of the present work is to obtain an approximated analytical solution for the fractional smoking epidemic model with the aid of a novel technique called q-homotopy analysis transform method (q-HATM). The considered nonlinear mathematical model has been effectively employed to elucidate the evolution of smoking in a population and its impact on public health in a community. We find some new approximate solutions in a series form, which converges rapidly, and the proposed algorithm provides auxiliary parameters, which are very reliable and feasible in controlling the convergence of obtained approximate solutions. Further, we present novel simulations for all cases of results to validate the applicability and effectiveness of proposed scheme. The outcomes of the study reveal that the q-HATM is computationally very effective to analyse nonlinear fractional differential equations arises in daily life problems.

Highlights

In 1766, Swiss mathematician and physicist Bernoulli [1] established and nurtured the idea of mathematical modelling for spread of disease, which gave birth to the start of modern epidemiology
Ross [2] presented the modelling of infectious disease in the beginning of twentieth century and explains the nature of epidemic models by using the law of mass action
Epidemic models have been extensively employed to study epidemiological processes which include transmission of contagious diseases. This kind of model has been applied to study the dissemination of social habits, such as the alcohol consumption [3], obesity epidemics [4], cocaine consumption [5], smoking habit [6]

Summary

Introduction

In 1766, Swiss mathematician and physicist Bernoulli [1] established and nurtured the idea of mathematical modelling for spread of disease, which gave birth to the start of modern epidemiology. We consider the system of five nonlinear differential equations describing the smoking epidemic model. To present the fundamental idea of proposed method [36, 37], consider the nonlinear non-homogeneous partial differential equation of fractional order: Dμt v(x, t) + v(x, t) + v(x, t) = f (x, t), n − 1 < μ ≤ n,. (13) where Dt v(x, t) represents the Caputo fractional derivative of the function v(x, t), and , respectively, specifies the linear and nonlinear differential operator, and f (x, t) represents the source term. On choosing the auxiliary linear operator, the initial guess is v0(x, t), n, and ħ , and the series (19) converges at q = n1 ; it gives one of the solutions of the original nonlinear equation of the form.

Define the nonlinear operator as

Numerical results and discussion

Conclusion