New optical soliton solutions of space-time fractional nonlinear dynamics of microtubules via three integration schemes

Abstract

In this study, we implement three efficient integration algorithms to retrieve the solutions of optical soliton space-time fractional nonlinear equation for the dynamics of microtubules MTs, which considered as one of the most important part in cellular processes biology. In this work we used three integration methods, firstly, the method of exp(-Ω)-expansion equation function, secondly the Kudryashov equation method in the general case and the method of extended Bernoulli sub-equation function, with the help of the fractional complex transformation and conformable derivatives which including the solution of complex function, rational function, hyperbolic function and exponential function. Finally, our results give good solution and understanding of the properties of the non-linear waves in fractional medium.

Keywords

Generalized Kudryashov method Bernoulli sub-equation function method nonlinear dynamical equation of microtubules

1 Introduction

The recent years attained much interest and many applications of the fractional calculus, where is widely used in the modeling and simulation in several research fields, for example, physics, biology, biomedical engineering, signal processing, control theory, systems identification, marketing tools and finance, dynamical systems [1 –20] and so on. More precisely, the fractional differential equations provide a good models in the biological systems. The dynamical process occurs in the tissue is perfectly modeled by the fractional differential equations in [21]. The fractional differential equations FDE succeed to describe and model many physiological systems like (dynamics of the oxygen in the blood and filtration, action potential propagation, and controlling the process of insulin secretion) and these models introduce a good description to understand the normal physiological balancing, and the main changes in the dynamics of the stable system that is the reason of several disease [22, 23]. Also, some of the nonlocal epidemics diseases, like SARS, and avian flu is studied by the FDE [24]. Moreover, application found for the fractional differential equations (FDE) in the area biomedical signal processing, signal modeling and quantum area [25 –29].

Microtubules (MTs) plays an important role as a connection for the intracellular transport of motor proteins. The structure of Microtubules (MTs) is extensively studied by several groups [30 –39]. Microtubules (MTs) is considered as one of the most important biological systems simulated by the differential equations.

The following two Hamiltonian models Equation (1 & 2) can simulate and describe the dynamics of Microtubules (MTs) very well [3 , 40], $H_{n} = \sum_{n} [\frac{l}{2} {\dot{φ}}_{n}^{2} + \frac{k}{2} (φ_{n + 1} - φ_{n})^{2} - pEcos φ_{n}],$ (1)

$\begin{matrix} H_{n} & = & \sum_{n} [\frac{m}{2} {\dot{u}}_{n}^{2} + \frac{k}{2} (u_{n + 1} - u_{n})^{2} - \frac{1}{2} {Au}_{n}^{2} \\ + \frac{1}{4} {Bu}_{n}^{4} - {Cu}_{n}], C = qE \end{matrix}$ (2) where the parameters of the models can be defined as I and m are the dimer moment and mass, and k represents the inter-dimer bonding interaction. The dimer position is presented by the integer n, the exact value of the substantial electric field at the position n, is described by E > 0. We can express the charge excess inside the dipole by q > 0 and the single dimer electric dipole moment by p > 0 and A and B are constants. From the following equations (3 & 4) we can obtain the equation describing the space-time fractional partial differential equations by utilizing continuum approximation.

$\begin{matrix} m \frac{\partial^{2 α} z (x, t)}{\partial t^{2 α}} - {kl}^{2} \frac{\partial^{2 β} z (x, t)}{\partial x^{2 β}} - qE - Az (x, t) \\ + {Bz}^{3} (x, t) + γ \frac{\partial^{α} z (x, t)}{\partial t^{α}} = 0, \end{matrix}$ (3)

$\begin{matrix} l \frac{\partial^{2 α} z (x, t)}{\partial t^{2 α}} - {kl}^{2} \frac{\partial^{2 β} z (x, t)}{\partial x^{2 β}} + pEz (x, t) \\ - \frac{pE}{6} z^{3} (x, t) + Γ \frac{\partial^{α} z (x, t)}{\partial t^{α}} = 0 \end{matrix}$ (4)

The previous work motivated us to introduce new modified solutions for the governing equation of nonlinear dynamics of microtubules MTs. So in this regard we in this paper we presents three strategic integration algorithms to encore soliton solutions to the model. The method exp(-Ω)-expansion function and Kudryashov generalized method scheme and extended Bernoulli sub-equation function method are employed to find the microtubules MTs dynamical nonlinear equation solution.

More details about the organization of the paper are described in the next coupled sections. The definitions and properties of Conformable fractional derivatives of order α is introduced in section 2. A comprehensive description of the three integration schemes for solving FPDEs is introduced in section 3. We apply these three integrations to obtain the new exact solutions of the nonlinear dynamical equation of microtubules MTs as you can see in section 4. We conclude or findings in section 5.

2 The conformable derivatives method of order α

The definition of the conformable derivatives method of order α is given in several papers as [1, 2].

Definition : Let g : (0, ∞) ⟶ R, then some definition, and theorems for conformable derivatives and its properties are stated as follows:

$\begin{matrix} U_{α} (g) (t) & = & \lim_{ɛ ⟶ 0} \frac{g (t + ɛ t^{1 - α}) - g (t)}{ɛ}, \\ t > 0, 0 < α < 1, \end{matrix}$ (5) $\begin{array}{l} U_{α} (b g + c h) = B U_{α} (g) + C U_{α} (h), B, C \in R, \\ U_{α} t^{λ} = λ t^{λ - α}, λ \in R, \\ U_{α} (gh) = g U_{α} (h) + {hU}_{α} (g), \\ U_{α} (\frac{g}{h}) = \frac{h U_{α} (g) - g U_{α} (h)}{h^{2}} \end{array}$ (6)

If the parameter g differentiable, thereafter $U_{α} (g) (t) = t^{1 - α} \frac{dg}{dh}$

Theorem. Let g : (0, ∞) ⟶ R be differentiable, where α is the differentiable function, then

$U_{α} (g * h) = t^{1 - α} h' (t) g' (h (t))$ (7)

3 Succinct overview of integration strategies

3.1 The extended Bernoulli Sub-Equation Function Method (IBSEFM)

To obtain the IBSEFM [41 –46] several steps should be conducted as listed below:

Step 1. Suppose the general fractional evolution equation has the general structure as in Equation 8,

$\begin{matrix} H (u, D_{t}^{α} u, D_{x}^{β} u, D_{t}^{2 α} u, D_{t}^{2 α} u, D_{t}^{α} D_{t}^{β} u, . . . .) = 0, \\ 0 < α, β > 1 \end{matrix}$ (8)

Step 2. Fractional complex transformation is applied as in the next equation 9, $u (x, t) = U (ξ), ξ = \frac{c t^{α}}{Γ (1 + α)} + \frac{δ x^{β}}{Γ (1 + β)},$ (9)

The symbols δ and c in this equation are constants and we will use the chain rule for its calculation $D_{t}^{α} u = σ_{t}^{'} \frac{dU}{d ξ} D_{t}^{α} ξ, D_{x}^{α} u = σ_{x}^{'} \frac{dU}{d ξ} D_{x}^{α} ξ$ (10) where $σ_{x}^{'} = σ_{t}^{'} = l$ , and l is a constant. By inserting equation (10) in equation (9), we obtain the following ODE. $ψ (U (ξ), U^{'} (ξ), U^{″} (ξ), U^{‴} (ξ), ......) = 0$ (11)

Step 3. From the point of view of this method, the solution of Equation (11) is proposed as in the following equations, $U (ξ) = \frac{\sum_{i = 0}^{n} A_{i} F^{i} (ξ)}{\sum_{j = 0}^{m} B_{j} F^{j} (ξ)},$ (12) $F^{'} (ξ) = b F (ξ) + d F^{M} (ξ), b \neq 0, d \neq 0, M \in R$ (13) where in the previous equation (13) the F (ξ) is the polynomial of Bernoulli differential equation. With the aid of the above relations in Equation (13) admits to $Ω (F (ξ)) = ρ_{s} F (ξ)^{s} + . . . . . . + ρ_{1} F (ξ) + ρ_{0} = 0$ (14) The real values of n, m and M is obtained from the balance principle. The solutions of Bernoulli equation (14) in general form are given in Ref. [43].

Step 4. By substituting all Ω (F (ξ)) coefficients are equal to zero, this led to generate a system of an algebraic equations; $ρ_{i} = 0, i = 0, . . . . . . ., s$ (15)

If we could solve the above equation of the system, the values of the parameters a₀,...a_n,b₀,...b_n are obtained.

3.2 Modified exp(-Ω) -expansion Function Method (MEFM)

Next target is to explore solution of Equation (11), which will be used for constructing the exact solutions to Equation (12). The following expression in equation (16) represents Equation (11) solution; $U (ξ) = \frac{\sum_{i = 0}^{N} A_{i} {\exp (- Ω (ξ))}^{i}}{\sum_{j = 0}^{M} B_{j} {\exp (- Ω (ξ))}^{j}}$ (16) where the constants A_i, B_j, (0 < i < N, 0 < j < M) will be calculated later, also Ω = Ω (ξ) establish the ODE $Ω^{'} (ξ) = e x p (- Ω (ξ)) + μ e x p (Ω (ξ)) + λ,$ (17) where the constants λ and μ will be evaluated later.

The general auxiliary Equation (17) has solutions are given [43]. By inserting equation (6) in addition with equation (7) in equation (11), a polynomial of exp(-Ω (ξ)) is obtained, taking in your consideration that all power of exp(-Ω (ξ)) is equal to zero. This method produces a group of equations, can be evaluated to obtain the values of A_i, B_j, μ, λ, c, k. By using the obtained values in Equation (16), then the optical solitons solutions of Equation (8) are obtained.

3.3 Pen picture of the generalized Kudryashov scheme

This part is devoted to develop the optical solutions to Equation (11) by using Kudryashov method in the general form. The quick glance of this method suggests to possess the following solution structure for Equation (11) as $u (x, t) = U (ξ), ξ = \frac{c t^{α}}{Γ (1 + α)} + \frac{δ x^{β}}{Γ (1 + β)},$ (18) with $D_{t}^{α} u = σ_{t}^{'} \frac{dU}{d ξ} D_{t}^{α} ξ, D_{x}^{α} u = σ_{x}^{'} \frac{dU}{d ξ} D_{x}^{α} ξ$ (19) the constants δ, c can be calculated later,

Making use of Equation (8) with (19), then Equation (8) take the ODE form, $ϕ (U, U^{'}, U^{″}, U^{‴},) = 0$ (20)

In view of this method, the solution in equation (20) as in the following formula in (21), $U (ξ) = \frac{\sum_{i = 0}^{n} a_{i} Q^{i} (ξ)}{\sum_{j = 0}^{m} b_{j} Q^{j} (ξ)}$ (21) where Q = Q (ξ) satisfies $Q^{'} (ξ) = Q^{2} (ξ) - Q (ξ)$ (22)

By the solving of equation (22) we obtain the solution as follows: $Q (ξ) = \frac{1}{1 \pm Aexp (ξ)}$ (23)

By using of equation (21) in addition with equation (22) within equation (20), collecting all power of Qⁱ (ξ), and taking all equal to zero, a group of algebraic equations is obtained. By the solving the generated equations, the values of the parameters A_i, B_j, c, δ is quantified. The optical solutions of Equation (8) can be found from the solutions which represented by Equation (21) and using of Equation (22) solutions,

4 The implementations of three integration schemes

4.1 New optical solitons solutions via the IBSEFM method

For finding the solution of Equation (3) through the IBSEFM method for constructing the exact traveling wave solutions, the fractional complex transform is considered, $z (x, t) = U (ξ), ξ = \frac{c t^{α}}{Γ (1 + α)} + \frac{δ x^{β}}{Γ (1 + β)}$ (24)

Inserting Equation (24) into (3), we have $P U^{″} (ξ) - Q U^{'} (ξ) - U (ξ) + U^{3} (ξ) - R = 0,$ (25)

$\begin{matrix} P & = & \frac{{mc}^{2} - {kl}^{2} δ^{2}}{A}, \\ Q & = & \frac{γ c}{A}, \\ R & = & \frac{qE}{A \sqrt{A / B}}, \\ z (ξ) & = & \sqrt{A / B} u (ξ) \end{matrix}$ (26)

Balancing between U″ (ξ) and U³ (ξ), we have the relation between M, m and n $m + M = n + 1$

By choosing special values of M = 3, m = 1 and n = 3, we can obtain the following equation, $U (ξ) = \frac{A_{0} + A_{1} F (ξ) + A_{2} F^{2} (ξ) + A_{3} F^{3} (ξ)}{B_{0} + B_{1} F (ξ)}$ (27)

By substituting of Equation (27) into (25) with Equation (13), and collecting all power of F (ξ), group of algebraic equations can be obtained and solved to have the following equations of the system parameters,

$\begin{matrix} d = d, b = b, B_{1} = B_{1}, A_{3} = A_{3}, \\ α = \sqrt{- 3 A_{3}^{2} b^{2} + 12 d^{2} B_{1}^{2}}, \\ Q = \pm \frac{α A 3}{d^{2} B_{1}^{2}}, \\ A_{0} = \frac{(3 A_{3} b \pm α) B_{0}}{6 {dB}_{1}}, \\ R = \pm \frac{(A_{3}^{2} b^{2} - d^{2} B_{1}^{2}) α}{9 d^{3} B_{1}^{3}}, \end{matrix}$

$A_{1} = \frac{3 A_{3} b \pm α}{6 d}, A_{2} = \frac{A_{3} B_{0}}{B_{1}}$ (28)

In view of Equation (28) along with Equation (27) with the solution of Equation (13), admits to the new complex function solution for Equation (13) as $u (ξ) = \frac{\frac{(3 A_{3} b \pm α) B_{0}}{6 {dB}_{1}} + [\frac{3 A_{3} b \pm α}{6 d}] F (ξ) + [\frac{A_{3} B_{0}}{B_{1}}] F^{2} (ξ) + A_{3} F^{3} (ξ)}{B_{0} + B_{1} F (ξ)},$ (29) for b ≠ d $F (ξ) = [- \frac{d}{b} + \frac{c_{1}}{e^{2 b ξ}}]^{- 2},$ (30) and for b = d $F (ξ) = [\frac{(c_{1} - 1) + (c_{1} + 1) \tanh (- b ξ)}{1 - \tanh (- b ξ)}]^{- 2},$ (31) $ξ = \frac{c t^{α}}{γ (1 + α)} + \frac{δ x^{β}}{γ (1 + β)}, z (ξ) = \sqrt{A / B} u (ξ)$ (32)

Remak 1. It is to be noted that the optical solutions obtained Equation (30 - 32), these outcomes are new and have not been accounted for in previous work.

4.2 Optical solitons solutions using MEFM method

In this subsection, the solution of Equation (3) will be obtained via the MEFM method. By using Equation (24) into (3), we have

$P u^{″} (ξ) - Q u^{'} (ξ) - u (ξ) + u^{3} (ξ) - R = 0,$ (33)

$\begin{matrix} P & = & \frac{{mc}^{2} - {kl}^{2} δ^{2}}{A}, \\ Q & = & \frac{γ c}{A}, \\ R & = & \frac{qE}{A \sqrt{A / B}}, \\ z (ξ) & = & \sqrt{A / B} u (ξ) \end{matrix}$ (34)

The balance principle is considered in the treatment of U″ (ξ) and U³ (ξ), so the following equation is obtained, $N = M + 1$

If we choose a fixed values of the parameters M = 1 and N = 2, so the equation (16) is expressed as follows, $U (ξ) = \frac{A_{0} + A_{1} \exp (- Ω) + A_{2} \exp (- 2 Ω)}{B_{0} + B_{1} \exp (- Ω)}$ (35)

The same as in the previous subsection, by using Equation (35) into (33) with the help of Equation (17), also, with collecting the formula exp(-Ω) power. As result of the last point, a system of algebraic equations is obtained, and it is solution is given in the following equations,

$\begin{matrix} λ = λ, B_{0} = B_{0}, B_{1} = B_{1}, α = \sqrt{- 2 P}, \\ μ = \frac{6 P + 3 P^{2} λ^{2} + Q^{2}}{12 P^{2}}, A_{2} = \pm α B_{1}, \end{matrix}$

$\begin{matrix} R = \pm \frac{2 (2 Q^{2} + 9 P) Q}{27 PQ}, \\ A_{1} = \pm \frac{({QB}_{1} + 3 P λ B_{1} + 6 {PB}_{0})}{3 α}, \end{matrix}$ $A_{0} = \pm \frac{B_{0} (Q + 3 P λ)}{3 α}$ (36)

In view of Equation (36) with the general solutions of Equation (17) through Equation (35), a new solution of Equation (3) is obtained as in the next equation (37). $u (ξ) = \frac{\pm \frac{B_{0} (Q + 3 P λ)}{3 α} \pm \frac{({QB}_{1} + 3 P λ B_{1} + 6 {PB}_{0})}{3 α} \exp (- Ω) + \pm α B_{1} \exp (- 2 Ω)}{B_{0} + B_{1} \exp (- Ω)},$ (37)

Family 1. By taking μ ≠ 0, Δ = λ² - 4μ > 0, the hyperbolic function solution is given by, $Ω (ξ) = \ln [\frac{- Δ}{2 μ} \tanh (\frac{Δ}{2} (ξ + c_{1})) - \frac{λ}{2 μ}]$ (38)

Family 2. By taking μ ≠ 0, Δ = λ² - 4μ < 0, the trigonometric function solution is given by, $Ω (ξ) = \ln [\frac{Δ_{1}}{2 μ} \tan (\frac{Δ_{1}}{2} (ξ + c_{1})) - \frac{λ}{2 μ}],$ (39)

Δ₁ = - λ² + 4μ

Family 3. By taking μ = 0λ ≠ 0, Δ = λ² - 4μ > 0 $Ω (ξ) = - \ln (\frac{λ}{\exp (λ (ξ + c_{1}) - 1})$ (40)

Family 4. By taking μ ≠ 0, λ ≠ 0, Δ = λ² - 4μ = 0 $Ω (ξ) = \ln (\frac{2 λ (ξ + c_{1}) + 4}{λ^{2} (ξ + c_{1})})$ (41)

Family 5. By taking μ = 0, λ = 0, Δ = λ² - 4μ = 0, then the rational function solution is $Ω (ξ) = \ln (ξ + c_{1}),$ (42) where c₁ is constant, $ξ = \frac{c t^{α}}{γ (1 + α)} + \frac{δ x^{β}}{γ (1 + β)}, z (ξ) = \sqrt{A / B} u (ξ)$

Remak 2. It is worth noting that, in case of A₂ = 0, B₁ = 0, B₀ = 1, the new solutions obtained are exactly the same with that obtained in [29]. In the samr manner Equation (4) can be solved directly using the three schemes mentioned above. For simplicity should be omitted here.

4.3 Optical solitons solutions using generalized Kudryashov method

As before, to solve Equation (3) via generalized Kudryashov scheme, the following formate is considered [47, 48], $z (x, t) = U (ξ), ξ = \frac{c t^{α}}{Γ (1 + α)} + \frac{δ x^{β}}{Γ (1 + β)}$ (43)

Then Equation (3) yields $P U^{″} (ξ) - Q U^{'} (ξ) - U (ξ) + u^{3} (ξ) - R = 0,$ (44)

$\begin{matrix} P & = & \frac{{mc}^{2} - {kl}^{2} δ^{2}}{A}, \\ Q & = & \frac{γ c}{A}, \\ R & = & \frac{qE}{A \sqrt{A / B}}, \\ z (ξ) & = & \sqrt{A / B} u (ξ) \end{matrix}$ (45)

By balancing between U″ (ξ) and U³ (ξ), the following equation is obtained, $N = M + 1$

If we take M = 1 and N = 2, we can write Equation (21) as $U (ξ) = \frac{a_{0} + a_{1} Q (ξ) + a_{2} Q^{2} (ξ)}{b_{0} + b_{1} Q (ξ)}$ (46)

According to Equation (46) into (44), with the gathering of all power of Q (ξ), a group of algebraic equations is obtained and from the solution of these equations, we have

Case(1).

$\begin{matrix} a_{2} = a_{2}, b_{1} = b_{1}, b_{0} = - \frac{b_{1}}{2}, a_{1} = \pm \frac{γ α}{3} - a_{2}, \\ a_{0} = \frac{γ α}{6} + \frac{a_{2}}{2}, c = \pm \frac{α}{b_{1}^{2}}, \end{matrix}$

$\begin{matrix} δ = \pm \frac{ν a_{2}}{l b_{1}^{2}}, R = \pm \frac{2 γ (- b_{1}^{2} + 4 a_{2}^{2}) α}{9 b_{1}^{3}}, \\ α = \sqrt{- \frac{- 3 b_{1}^{2} + 3 a_{2}^{2}}{γ^{2}}}, \\ ν = \sqrt{- \frac{6 {mA}^{2} a_{2}^{2} - A γ^{2} b_{1}^{2} - 6 A^{2} {mb}_{1}^{2}}{2 k γ^{2}}} \end{matrix}$ (47)

Case(2).

$\begin{matrix} c = c, δ = δ, a_{0} = a_{0}, a_{2} = 0, b_{0} = b_{0}, b_{1} = b_{1}, \\ a_{1} = \frac{b_{1} a_{0}}{b_{0}}, R = \frac{a_{0} (- b_{0}^{2} + a_{2}^{2})}{b_{0}^{2}} \end{matrix}$ (48)

In case of Equation (47) and Equation (46)), led to obtain the exact solution of Equation (3) as $u_{1} (ξ) = \frac{[\frac{γ α}{6} + \frac{a_{2}}{2} \pm (\frac{γ α}{3} - a_{2}) Q (ξ) + a_{2} Q^{2} (ξ)]}{[- \frac{b_{1}}{2} + b_{1} Q (ξ)]}$ (49)

In view of Equation (48), we can obtain the solution of Equation (3) exactly as follows: $u_{2} (ξ) = \frac{a_{0} + \frac{b_{1} a_{0}}{b_{0}} Q (ξ)}{b_{0} + b_{1} Q (ξ)},$ (50) Then $Q (ξ) = \frac{1}{1 \pm Aexp (ξ)},$ (51) $ξ = \frac{c t^{α}}{Γ (1 + α)} + \frac{δ x^{β}}{Γ (1 + β)}$ and $z (ξ) = \sqrt{A / B} U (ξ)$ .

Remark 3. We can see that, the obtained exact solutions in this paper are completely new and can not be found in the previous studies. In the same manner Equation (4) can be solved directly via the proposed method. For simplicity should be omit here.

5 Conclusion

In this paper, two models of space-time nonlinear fractional order differential equations for describing nonlinear dynamics of microtubules MTs is introduced. Three integration schemes are suggested to structure the optical soliton solutions of fractional space-time of nonlinear dynamics of microtubules, namely, exp (- (Ω))-expansion equation method, modified Bernoulli sub-equation method and generalized Kudryashov method. More integration schemes will be later implement to study this equation such as Lie symmetry, semi inverse variational principle, F-expansion method, Kudryashov’s scheme, mapping method and several others. From the results, it can be noted that the fractional parameter have the main role in the obtained solutions, where if we take the parameters α = 1 and β = 1, the solutions give the obtained solutions in the case of normal derivative. Moreover, a lot of free parameters are admit with obtained solutions in this paper, so we can say that, these kinds of the dynamic solutions are more suitable in the solving of the initial and boundary values problems.

Footnotes

Acknowledgments

The authors extend their appreciation to the Deanship of Scientific Research at King Khalid University for funding this work through research groups program under grant number G. R.P-38-40.

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