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

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.


1.
Introduction. It is no new knowledge that most processes and phenomena that arise in plasma physics, mathematical physics, quantum mechanics, fluid mechanics, solid state physics, hydrodynamics, bio-genetics, chemical kinematics, etc., can be described by non-linear evolution equations. However, in recent times, it has been found that many of these physical, chemical and biological processes are governed by non-linear evolution equations of non-integer or fractional order [31,28,17,36].
Various attempts by different authors at obtaining the exact solutions to these non-linear evolution equations in other to better understand the phenomena they describe have stimulated the discovery of an impressive array of methods. These methods include the inverse scattering transform [2], the Backlund transform [29,34], the Darboux transform [27], the Hirota bilinear method [18], the tanh-function method [11], the sine-cosine method [26,38], the exp-function method [16], the generalized Riccati equation [39], the Homogeneous balance method [12], the (G /G) expansion method [35,41], the modified simple equation method [19,42,44].
The first integral method introduced by Feng [13] is a very effective tool for generating exact soliton and periodic solutions to non-linear evolution equations.
Some useful properties of the modified Riemann-Liouville derivative are listed below [20,21]: D σ x (f (x)g(x)) = g(x)D σ x f (x) + f (x)D σ x g(x) (3) This paper is organized as follows. In Section 2, the first integral method is introduced and explained. In Section 3, the first integral method is applied to obtain solutions to two space and time fractional biological models with non-linear blood flow in large vessels model and Deoxyribonucleic acid (DNA) molecule dynamics model presented in Section 3.1 and 3.2 respectively. Finally, a conclusion is given in Section 4.

2.
Description of the first integral method. Here, we provide a brief explanation of the first integral method for finding travelling wave solutions of non-linear fractional evolution equations. Suppose the non-linear fractional evolution equation is in the following form: ℘ is a polynomial of u(x, t) and its derivatives (integer and fractional) with respect to x and t. σ and γ are parameters that describe the order of the space and time derivatives respectively.
Theorem 2.1. Fractional complex transformation. To transform Eq. 4 into a nonlinear ordinary differential equation (ODE) of integer order by applying a fractional complex transformation proposed by Li and He [25]: where V is an arbitrary constant and Eq. 4 reduces to a non-linear integer order ODE of the form P (u, u , u , u . . .) = 0 Next, we introduce new independent variables X(ξ) and Y (ξ) such that This leads to a system of non-linear ordinary differential equations: According to the qualitative theory of ordinary differential equations [9], if we can find the integrals to 8 under the same condition, then the general solutions to 8 can be solved directly. Generally, it is very difficult to realize this even for one first integral, because for a given plane autonomous system, there is neither a systematic theory that informs us on how to obtain its first integrals, nor is there a logical way for telling us what these first integrals are. We apply the Division theorem to obtain one first integral to equation 8 which reduces equation 6 to a first-order integrable ordinary differential equation. This equation is then solved to obtain the exact solution to equation 4.
Theorem 2.2. Division Theorem [13]. Suppose that P (y, w), Q(y, w) are polynomials in C[y, w] and P (y, w) is irreducible in C[y, w]. If Q(y, w) vanishes at all zero points of P (y, w), then there exists a polynomial G(y, w) in C[y, w] such that The Division theorem follows immediately from the Hilbert-Nullstellensatz Theorem of commutative algebra [6].

Application.
3.1. Model for blood flow in large vessels. Hashimuze [15] and Yomosa [40] working independently showed that blood flow is expected to behave like a nonlinear wave or soliton due to its pulsatile nature and other physiological factors [7,8,23]. We consider the blood flow in the vessel to be Newtonian, viscous, homogeneous and incompressible. Therefore, the dynamics of blood flow in large vessels is governed by the Navier-Stokes and the continuity equations as [30,10,23] ∂A ∂t where z is the axial coordinate, t is time, W is the axial component of the fluid velocity, A is the cross-sectional area of the blood vessel and P is the pressure inside the vessel. For the wall dynamics, applying Newton's second law on a portion of the vessel wall gives the pressure as [30,10,23].
The space and time fractional model for non-linear blood flow in large vessels is given by where σ and γ are parameters that describe the order of the space and time derivatives respectively. Here we apply the first integral method to this space and time fractional model. Substituting Eq. 11 into Eq. 13, we have We make the fractional complex transformation W (z, t) = W (ξ), A(z, t) = A(ξ) by combining the independent variables z and t into one Then, Eqs. 14 and 15 becomes Integrating Eqs. 16 and 17 once with respect to ξ yields where K 1 and K 2 are constants of integration. From Eq. 18, it is possible to express W as a function of A. Substituting this into Eq. 19 and taking K 2 = 0 leads to Using equation 7, we get According to the first integral method, we suppose that X(ξ) and Y (ξ) are nontrivial solutions of 21 and 22, and a i (X) (i = 0, 1, 2, . . . , m) are polynomials of X and a m (X) = 0. Eq. 23 is the first integral to 21 and 22. Due to the Division Theorem, there exists a polynomial Here, we take two different cases, corresponding to when m = 1 and m = 2 in Eq. 25.
Case 1. Suppose that m = 1, by equating the coefficients of Y i (i = 2, 1, 0) on both sides of Eq. 25, we havė Since a i (X) (i = 0, 1, 2, . . . , m) are polynomials, then from Eq. 26 we deduce that a 1 (X) is a constant and h(X) = 0. For simplicity, take a 1 (X) = 1, then Eq. 27 becomesȧ Balancing the degrees of g(X) and a 0 (X) in Eq. 30, we conclude that deg(g(X)) = 0 only. Suppose that g(X) = B 0 , and B 0 = 0, then we get a 0 (X) as Substituting a 0 (X), a 1 (X) and g(X) in Eq. 31 and setting all the coefficients of X i (i = 0, 1, 2, . . .) to zero, we obtain a system of non-linear algebraic equations and by solving it, we obtain Using Eq. 32 in Eq. 23, we obtain the first integral to Eqs. 21 and 22 as Solving Eq. 33, we get the exact solutions to the space and time fractional model for non-linear blood flow in large vessels as where C is a constant of integration and the value of B 0 is neither 0 or 1/2.
Case 2. Suppose that m = 2, by equating the coefficients of Y i (i = 3, 2, 1, 0) on both sides of Eq. 25, we haveȧ a 0 (X) 4V 2 X 2 (X + 1) Since a i (X) (i = 0, 1, 2) are polynomials, then from Eq. 36, we deduce that a 2 (X) is a constant and h(X) = 0. For simplicity, we take a 2 (X) = 1, then Eqs. 37 -39 becomesȧ a 0 (X) 4V 2 X 2 (X + 1) Balancing the degrees of g(X) and a 0 (X) in Eq. 42, we conclude that deg(g(X)) = 0 only. Suppose that g(X) = B 0 , and B 0 = 0, then we obtain a 1 (X) and a 0 (X) as Substituting a 0 (X), a 1 (X) and g(X) in Eq. 42 and setting all the coefficients of X i (i = 0, 1, 2, . . .) to zero, we obtain a system of non-linear algebraic equations and by solving it, we obtain Using Eq. 45 in Eq. 23, we obtain the first integral to Eqs. 21 and 22 as Solving Eq. 46, we get the exact solutions to the space and time fractional model for non-linear blood flow in large vessels as and C is a constant of integration. The solution obtained in Eq. 47 is complex in nature.

3.2.
Deoxyribonucleic (DNA) model. The Deoxyribonucleic acid (DNA) molecule which is the carrier of information for life and reproduction of organisms consists of two long elastic poly-nucleotide chains or strands, connected to each other by an elastic membrane representing the hydrogen bonds between the pair of bases in the two chains. Recently, the investigation of DNA dynamics has successfully predicted inherent non-linear structures, which are responsible for forming localized waves that transport energy without dissipation [37,32,24,3,4,43]. We consider a non-linear model of DNA molecule out-of-phase motion developed by [24] based on an earlier lattice model presented in [32]. The model is written in the form [24,4]: where Here ρ, λ and Y are respectively the mass density, the area of transverse crosssection and the Young's modulus of each strand; h is the distance between the two strands, µ is the rigidity of the elastic membrane and l 0 is the height of the membrane in the equilibrium position. Recently, Knyazev and Knyazev [22] constructed solutions to Eq. 48 by using the Hirota bilinear method, Alka et. al [4] presented Riccati generalized solitary wave solutions to Eq. 48 via the elliptic equation method and Zayed and Arnous [43] applied the generalized Riccati equation mapping method to find many travelling wave solutions to the model. In this section, we apply the first integral method to the space and time fractional DNA model given by

OLUSOLA KOLEBAJE, EBENEZER BONYAH AND LATEEF MUSTAPHA
We make the fractional complex transformation φ(x, t) = φ(ξ) by combining the independent variables x and t into one variable ξ = x σ /Γ(1 + σ) − V t γ /Γ(1 + γ). Then Eq. 49 becomes According to the first integral method, we suppose that X(ξ) and Y (ξ) are nontrivial solutions of 52 and 53, and a i (X) (i = 0, 1, 2, . . . , m) are polynomials of X and a m (X) = 0. Eq. 54 is the first integral to 52 and 53. Due to the Division Theorem, there exists a polynomial Here, we take two different cases, corresponding to when m = 1 and m = 2 in Eq. 56.
Case 1. Suppose that m = 1, by equating the coefficients of Y i (i = 2, 1, 0) on both sides of Eq. 56, we havė Since a i (X) (i = 0, 1, 2, . . . , m) are polynomials, then from Eq. 57 we deduce that a 1 (X) is a constant and h(X) = 0. For simplicity, take a 1 (X) = 1, then Eq. 58 becomesȧ 0 (X) = g(X) (60) Balancing the degrees of g(X) and a 0 (X) in Eq. 61, we conclude that deg(g(X)) = 1 only. Suppose that g(X) = B 0 X + B 1 , and B 0 = 0, then we get a 0 (X) as Substituting a 0 (X), a 1 (X) and g(X) in Eq. 61 and setting all the coefficients of X i (i = 0, 1, 2, . . .) to zero, we obtain a system of non-linear algebraic equations and by solving it, we obtain Using Eq. 63 in 54, we obtain the first integral to Eq. 52 and 53 as Solving Eq. 64, we get the exact solutions to the space and time fractional DNA model as Case 2. Suppose that m = 2, by equating the coefficients of Y i (i = 3, 2, 1, 0) on both sides of Eq. 56, we haveȧ 2 (X) = h(X)a 2 (X) (67) a 1 (X) = g(X)a 2 (X) + h(X)a 1 (X) (68) Since a i (X) (i = 0, 1, 2) are polynomials, then from Eq. 67, we deduce that a 2 (X) is a constant and h(X) = 0. For simplicity, we take a 2 (X) = 1, then Eqs. 67 -70 becomesȧ Balancing the degrees of g(X) and a 0 (X) in Eq. 73, we conclude that deg(g(X)) = 1 only. Suppose that g(X) = B 0 X +B 1 , and B 0 = 0, then we obtain a 0 (X) and a 1 (X) as Substituting a 0 (X), a 1 (X) and g(X) in Eq. 73 and setting all the coefficients of X i (i = 0, 1, 2, ) to zero, we get a system of non-linear algebraic equations and by solving it, we obtain Using Eq. 76 in 54, we obtain the first integral to Eq. 52 and 53 as Solving Eq. 77, we get the exact solutions to the space and time fractional DNA model as  Figure 1 is generally flat characterized with P (z, t) = 2.5 and a sharp drop in P (z, t) near z, t = 0. The effect of the different σ and γ values is a change in the nature of the drop in P (z, t) near z, t = 0 with σ = 1 and γ = 1 corresponding to the integer order case. In Figure 2, for V = −

Conclusion.
In this study, we have showed the efficacy of the first integral method for generating exact soliton and periodic solutions to fractional non-linear evolution equations. The time and space fractional models for non-linear blood flow in large vessels and Deoxyribonucleic acid (DNA) molecule dynamics were defined in the sense of Jumarie's modified Riemann-Liouville derivative via a fractional complex transformation. The validity of the solutions to the fractional biological models obtained with first integral method was achieved by putting them back into the models with the aid of Mathematica. Thus, we conclude that the first integral method can be extended to solve other non-linear biological problems.