The first integral method for two fractional non-linear biological models

Olusola Kolebaje; Ebenezer Bonyah; Lateef Mustapha

doi:10.3934/dcdss.2019032

Article Contents

2019, Volume 12, Issue 3: 487-502. Doi: 10.3934/dcdss.2019032

This issue Previous Article Numerical analysis of coupled fractional differential equations with Atangana-Baleanu fractional derivative Next Article On the existence and uniqueness of the solution of an optimal control problem for Schrödinger equation

The first integral method for two fractional non-linear biological models

1.
Department of Physics, Adeyemi College of Education, Ondo, Nigeria
2.
Department of Mathematics Education, University of Education, Winneba, (Kumasi campus), Ghana
3.
Department of Physical Sciences, Al-Hikmah University, Ilorin, Nigeria

* Corresponding author: olusolakolebaje2008@gmail.com
* Corresponding author: olusolakolebaje2008@gmail.com

Received: June 2017

Revised: November 2017

Published: October 2018

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Abstract

Travelling wave solutions of the space and time fractional models for non-linear blood flow in large vessels and Deoxyribonucleic acid (DNA) molecule dynamics defined in the sense of Jumarie's modified Riemann-Liouville derivative via the first integral method are presented in this study. A fractional complex transformation was applied to turn the fractional biological models into an equivalent integer order ordinary differential equation. The validity of the solutions to the fractional biological models obtained with first integral method was achieved by putting them back into the models. We observed that introducing fractional order to the biological models changes the nature of the solution.

Keywords:

Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.

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Figure 1. Figure showing $P(z,t)$ with $\chi = z^{\sigma}/\Gamma(1+\sigma)+2 t^{\gamma}/\Gamma(1+\gamma)\sqrt{1-4 B_0^2}$ and $0\leq z, t \leq 20$ for (a) $\sigma = 1$ , $\gamma = 1$ , (b) $\sigma = 1$ , $\gamma = 0.5$ , (c) $\sigma = 0.5$ , $\gamma = 1$ , (d) $\sigma = 0.5$ , $\gamma = 0.5$

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Figure 2. Figure showing $P(z,t)$ with $\chi = z^{\sigma}/\Gamma(1+\sigma)-2 t^{\gamma}/\Gamma(1+\gamma)\sqrt{1-4 B_0^2}$ and $0\leq z, t \leq 20$ for (a) $\sigma = 1$ , $\gamma = 1$ , (b) $\sigma = 1$ , $\gamma = 0.5$ , (c) $\sigma = 0.5$ , $\gamma = 1$ , (d) $\sigma = 0.5$ , $\gamma = 0.5$

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Figure 3. Figure showing $\phi(x,t)$ (Eq. 65) with $B_0, l, k, Y, \lambda, \mu = 1$ , $h = 3.33$ , $\rho = 0.85$ , and $0\leq x, t \leq 5$ for (a) $\sigma = 1$ , $\gamma = 1$ , (b) $\sigma = 1$ , $\gamma = 0.5$ , (c) $\sigma = 0.5$ , $\gamma = 1$ , (d) $\sigma = 0.5$ , $\gamma = 0.5$

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Figure 4. Figure showing $\phi(x,t)$ (Eq. 66) with $B_0, l, k, Y, \lambda, \mu = 1$ , $h = 3.33$ , $\rho = 0.85$ , and $0\leq x, t \leq 5$ for (a) $\sigma = 1$ , $\gamma = 1$ , (b) $\sigma = 1$ , $\gamma = 0.5$ , (c) $\sigma = 0.5$ , $\gamma = 1$ , (d) $\sigma = 0.5$ , $\gamma = 0.5$

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Figure 5. Figure showing $\phi(x,t)$ (Eq. 78) with $B_0, l, k, Y, \lambda, \mu = 1$ , $h = 3.33$ , $\rho = 0.85$ , and $0\leq x, t \leq 5$ for (a) $\sigma = 1$ , $\gamma = 1$ , (b) $\sigma = 1$ , $\gamma = 0.5$ , (c) $\sigma = 0.5$ , $\gamma = 1$ , (d) $\sigma = 0.5$ , $\gamma = 0.5$

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Figure 6. Figure showing $\phi(x,t)$ (Eq. 79) with $B_0, l, k, Y, \lambda, \mu = 1$ , $h = 3.33$ , $\rho = 0.85$ , and $0\leq x, t \leq 5$ for (a) $\sigma = 1$ , $\gamma = 1$ , (b) $\sigma = 1$ , $\gamma = 0.5$ , (c) $\sigma = 0.5$ , $\gamma = 1$ , (d) $\sigma = 0.5$ , $\gamma = 0.5$

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