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

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  • 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.

    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) $\sigma = 0.5$ , $\gamma = 0.5$

    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) $\sigma = 0.5$ , $\gamma = 0.5$

    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) $\sigma = 0.5$ , $\gamma = 0.5$

    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) $\sigma = 0.5$ , $\gamma = 0.5$

    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) $\sigma = 0.5$ , $\gamma = 0.5$

    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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