Fractional soliton dynamics of electrical microtubule transmission line model with local -derivative
本站小编 Free考研考试/2022-01-02
Nauman Raza1, Saima Arshed1, Kashif Ali Khan2, Mustafa Inc,3,4,5,∗1Department of Mathematics, University of the Punjab, Quaid-e-Azam Campus, Lahore, Pakistan 2Department of Mathematics, University of Engineering and Technology, Lahore, Pakistan 3Department of Computer Engineering, Biruni University, Istanbul, Turkey 4Department of Mathematics, Science Faculty, Firat University, Elazig, Turkey 5Department of Medical Research, Medical University Hospital, China Medical University, Taichung, Taiwan, China
First author contact:∗Author to whom any correspondence should be addressed. Received:2021-02-10Revised:2021-06-9Accepted:2021-06-11Online:2021-07-16
Abstract In this paper, two integrating strategies namely $\exp [-\phi (\chi )]$ and $\tfrac{{G}^{{\prime} }}{{G}^{2}}$-expansion methods together with the attributes of local-M derivatives have been acknowledged on the electrical microtubule (MT) model to retrieve soliton solutions. The said model performs a significant role in illustrating the waves propagation in nonlinear systems. MTs are also highly productive in signaling, cell motility, and intracellular transport. The proposed algorithms yielded solutions of bright, dark, singular, and combo fractional soliton type. The significance of the fractional parameters of the fetched results is explained and presented vividly. Keywords：solitons solution;microtubule;nonlinear transmission line;$\left(\tfrac{{G}^{{\prime} }}{{G}^{2}}\right)$-expansion method
PDF (575KB)MetadataMetricsRelated articlesExportEndNote|Ris|BibtexFavorite Cite this article Nauman Raza, Saima Arshed, Kashif Ali Khan, Mustafa Inc. Fractional soliton dynamics of electrical microtubule transmission line model with local M-derivative. Communications in Theoretical Physics, 2021, 73(9): 095002- doi:10.1088/1572-9494/ac0a67
1. Introduction
The investigation of a nonlinear routine is mandatory to understand the principles of the universe. Because of the wide significance of physical systems, the dissemination of modulated matter-waves like the solitary waves is the significance parameter as it is persisting for several years. To expedite the solitons experiment, Russel [1] in 1834 started work on it, and Zabusky et al [2] performed computer simulation regarding these soliton waves. The said domain of research in the systems of nonlinear theory has flourished as a rich field in the last three decades. No doubt, representation of chaotic electrical transmission (ET) lines are managed by laboratory experiments but these are very narrow in numbers. Further, the influence of network properties along the nonlinear ET lines was done by Pelap et al [3] where the soliton’s energy has been talked over with several frequency modes and inductances. Apart from that, some of the modern development of mathematical physics animated the probe of solutions [4–6] for nonlinear equations which boost-up in the myriad fields of physics. These unfolding are imperative in observing the principles of physical systems. In the ongoing past, there are multiple but distinctive ongoing schemes for integrability that choose several traveling wave solutions of diverse nonlinear evolutionary equations. These schemes incorporate the technique of Jacobi elliptic, extended trial equation, F-expansion, the generalized differential transform, modified simple equation, Adomian decomposition, the $\tfrac{G^{\prime} }{G}$-expansion, the first integral, sine-cosine algorithm, the single wave ansatz strategy, and a series of other schemes mentioned in [7–9].
Microtubules (MTs) are one of the three principal components of cytoskeletal polymers involved in a certain figure of particular cellular actions incorporating the organelle’s traffic using motor proteins, cellular engineering, cell division, motility, and a doable functioning in information processing within neuronal tasking. MT are hollow cylinders with a lumen, a polymer of alpha-tubulin and beta-tubulin. As they join/fuse, they make a dimer. The dimer has two identical or similar tubulin, named as a heterodimer. These dimers arrange themselves in head the tail manner and forming protofilaments (PFs). Thirteen of these PFs make a long chain of polymers and these polymers will put together in the form of a sheet and this sheet is rolled or converted into a tube. That is called a MT whose diameter is 25 nm thick. Its applications are in cardiac growth like the cardiomyocyte MT network (which is implemented in hypertrophy). According to what MTs work and exercise the electrical signals/electrical information was still untold in a larger sense. Sataric et al [10] launched the new model of MT as a chaotic ET line converted it into some rings. And developed a stochastic mathematical relation for the local electrostatic potential integrated with the ionic wave disseminating along an MT. Further, he [11] presents a nonlinear dimer's dynamics structure within MTs. And with the help of numerical and analytic techniques, the kink soliton solution is formed, which is superintended for the energy shifting along the MTs. The dynamical behavior of MTs can be modified into the nonlinear equation(s). These equations portrayed the natural experiences into the mathematical prototypes that crop up in the several fields of engineering, optical fibers, human biology, fluid mechanics, geochemistry, etc. In the current article, the electrical MT model and impact of capacitance nonlinearity on the engendering of ionic heartbeats are diagnostically considered. Quickened movement of the MT offers to ascend to electrodynamic wonders which assumes a crucial job in intracellular flagging and data handling [12–24]. Nur et al [25] present a methodology to elucidate the solutions of nonlinear equations depicting the MT dynamics by the aid of $\exp [-\phi (\chi )]$-expansion method. The majority of these electrodynamic frameworks happen as nonlinear halfway differential conditions. These conditions have moved toward becoming a consideration in pretty much every field of life, for example, hypothetical material science, material science, liquid elements, acoustics, cosmology, oceanography, astronomy, atomic physics and plasma material science, shallow water wave stream, nonlinear optics [26, 27], and some with biophysical implications can be seen in [28, 29].
Fractional calculus (FC) is known as the extension of integer-order calculus [30]. And most of the investigation of derivatives and anti-derivatives of non-integer (arbitrary, real, or complex) order is one of the emerging branches of (FC), as FC has a lot of significance in the numerous fields of science and engineering: at present its applications can be seen in the economy, fluid mechanics, image processing, biological models, and the most impressive in weather-controlling. Usage of special functions also provides insights into the importance of FC in the other branches of mathematics. Some of the certain attributes of FC involve rules of linearity, product, quotient rule and, chain rule, also include the definition of function composition in the calculus of integer-order. Especially, the attributes of FC are connected with the ordinary calculus of order one, as we set the order of the FC equal to one. In 2014, Khalil is the person who prompted the ‘local derivative’, that is persistent with the classical description of derivatives and holds all the patent obligations of the said rate of change, but note that, it does not meet the criteria of some famous results like theorem of mean value, extended mean value, and Roll’s theorem [31]. To remove that glitch, in 2018, the local M-derivative have been introduced by Sousa et al, where this new definition endorsed the idea prompted by Khalil et al and fulfill all of the above attributes [32].
In fractional derivative, several useful techniques have been introduced in multiple investigations to cite the nonlinear fractional evolution equations for exploring the traveling wave solutions [33–35]. Currently, Gao et al [36] suggested an easy method, the modified expansion function method plus the sinh-Gordon technique to expedite some new solitons solutions with the M-fractional paraxial wave equation. And under the influence of several nonlinear forms, Ghafri et al [37] explored the symmetry version in the fractional Ginzburg–Landau equation. Further, Fujioka et al [38] discussed the fractional dimensions by creating the fractional optical solutions in a generalized nonlinear Schrödinger equation.
No doubt, the mathematical processes managed with derivatives and integrals of arbitrary and complex orders are named FC. So, it opens a smart window to describe and understand the behavior of complex systems in an upgraded way. To apprehend the dynamics of biological systems, mathematical prototypes employing ordinary differential equations with integer-order, have been significantly utilized. But the attributes of most biological models have memory or aftereffects. These effects have a significant role in the disease spread. The disease spread in the future will be influenced by the involvement of memory effects on past events. The distance of memory effect reveals the history of disease dissemination. Thus, memory effects on the escalation of an infectious malady can be probed with the aid of fractional derivatives. And the development of such models by fractional differential equation has more privileges than classical integer-order mathematical modeling, in which such actions are abandoned. In some cases, the models of FODEs look more well established with the real happening than the integer-order models. The reason behind that FD activates the description of the memory and hereditary attributes inherent in numerous materials and mechanisms. So, there is an exponential need to study and use the differential of fractional-order as proposed by Rihan and Utoyo [39, 40]. Some of the current examinations in biological science proposed that MT dynamics in the adult brain function is the crucial process of learning and memory and it may be agreed in degenerative maladies, such as Alzheimer's disease. This enhances the likelihood of targeting MT dynamics in the making of current therapeutic agents as can be noticed in [40, 41]. FC is also appropriate for designing the electrical attributes of biological models. As Zoran et al [42] found a modern family of generalized prototypes for electrical impedance and invoked them to human skin by laboratory data fitting. Baleanu with other authors [43] present research by adopting some productive integration schemes to achieve the outcomes of optical soliton space–time-fractional nonlinear equation for the MTs dynamics, which examined as one of the most significant parts in cellular processes of biology. They use the attributes of the fractional complex transformation and conformable derivatives which involve the results of hyperbolic, rational, complex, and exponential functions. The outcomes of this research in fractional medium provide significant upshots and grabbing the attributes of the nonlinear waves.
The current work is carried out to explore the effects of time-fractional evolution on the MT transmission lines model by employing the properties of local M-derivative. For its implementation, we are providing the general results or solutions of the proposed model by adopting $\exp [-\phi (\chi )]$ and $\tfrac{{G}^{{\prime} }}{{G}^{2}}$-expansion methods together with graphical illustrations. The next part of the article is designed as follows: the elementary portion of the local M-derivative along with certain attributes is explained in the 2nd section. The concise overview of the adopted technique is insight in section 3. The detail of the suggested model is presented in section 4 and the quest of soliton results via suggested techniques can be seen in section 5. Results of the obtained solutions have been discussed in the same section. Lastly, section 6 provides the conclusion corner.
2. Interpretation of local M-derivative
This component proposed the fundamental attributes and definition of the local M-derivative [32]. If $h:{R}^{+}\cup \{0\}\,\to {\mathfrak{R}}$, $0\lt \lambda \lt 1$, and $t\gt 0$ then the function h, which is defined as$\begin{eqnarray}{D}_{M}^{\lambda ,\delta }\{h(t)\}=\mathop{\mathrm{lim}}\limits_{\omega \to 0}\displaystyle \frac{h\left({{tE}}_{\delta }(\omega {t}^{-\lambda })\right)-h(t)}{\omega },\,\,\forall \,t\gt 0,\end{eqnarray}$is called the local M-derivative of order λ, where ${E}_{\delta }(\cdot ),\forall \ \omega \gt 0$, is referred as the Mittag–Leffler function containing one parameter [44]. In the specified cut $(0,b)$ and $b\gt 0$, if h(t) is differentiable, and as one get the existence of ${\mathrm{lim}}_{t\to {0}^{+}}{D}_{M}^{\lambda ,\delta }\{h(t)\}$,$\begin{eqnarray}{D}_{M}^{\lambda ,\delta }\{h(0)\}=\mathop{\mathrm{lim}}\limits_{t\to {0}^{+}}{D}_{M}^{\lambda ,\delta }\{h(t)\},\end{eqnarray}$and$\begin{eqnarray}{D}_{M}^{\lambda ,\delta }\{g(t)\}=\displaystyle \frac{{t}^{1-\delta }}{{\rm{\Gamma }}(\delta +1)}g^{\prime} (t),\end{eqnarray}$where ${}^{{\prime} }=\tfrac{{\rm{d}}}{{\rm{d}}{t}}$, and then$\begin{eqnarray}{D}_{M}^{\lambda ,\delta }\left(\displaystyle \frac{{\rm{\Gamma }}(\delta +1){t}^{\lambda }}{\lambda }\right)=1.\end{eqnarray}$Apply the chain rule, one get:$\begin{eqnarray}{D}_{M}^{\lambda ,\delta }(g\cdot h)(b)={g}^{{\prime} }(h(b)){D}_{M}^{\lambda ,\delta }h(b),\end{eqnarray}$therefore, the connection from equations (4) and (5) can be determined mentioned below:$\begin{eqnarray}\begin{array}{l}{D}_{M}^{\lambda ,\delta }F\left[{\rm{\Gamma }}(\delta +1){\lambda }^{-1}{t}^{\lambda }\right]={F}^{{\prime} }\left[{\rm{\Gamma }}(\delta +1){\lambda }^{-1}{t}^{\lambda }\right]{D}_{M}^{\lambda ,\delta }\\ \left[{\rm{\Gamma }}(\delta +1){\lambda }^{-1}{t}^{\lambda }\right]={F}^{{\prime} }\left[{\rm{\Gamma }}(\delta +1){\lambda }^{-1}{t}^{\lambda }\right],\end{array}\end{eqnarray}$together with$\begin{eqnarray}\eta ={\lambda }^{-1}l{\rm{\Gamma }}\left(\delta +1\right){t}^{\lambda },\end{eqnarray}$and eventually we get$\begin{eqnarray}{D}_{M}^{\lambda ,\delta }\{F(\eta )\}={{lF}}^{{\prime} }(\eta ).\end{eqnarray}$where l represent some constant.
3. Suggested analytic techniques
This component provides the delineation of two suggested techniques like the $\exp [-\phi (\chi )]$ and $\tfrac{{G}^{{\prime} }}{{G}^{2}}$-expansion method. Consider the following general nonlinear PDE$\begin{eqnarray}\pi \left(p,{D}_{\eta }p,{D}_{\zeta }p,{D}_{\eta }^{2}p,{D}_{\zeta \eta }p,{D}_{\zeta }^{2}p,\ldots \right)=0,\end{eqnarray}$where $p=p(\zeta ,\eta )$ is function to be determined. Invoking the wave transformation$\begin{eqnarray}p(\zeta ,\eta )=\gamma (\chi ),\quad \quad \chi =\zeta -c\eta ,\end{eqnarray}$the above PDE is reduced to nonlinear ODE as$\begin{eqnarray}S(\gamma ,{\gamma }^{{\prime} },{\gamma }^{{\prime\prime} },{\gamma }^{\prime\prime\prime },\ldots )=0,\end{eqnarray}$where ${}^{{\prime} }=\tfrac{{\rm{d}}}{{\rm{d}}\chi }$.
3.1. Method of $\exp [-\phi (\chi )]$-expansion
In the current method, the truncated series i.e. the wave solution of equation (9) is of the type$\begin{eqnarray}\gamma (\chi )=\sum _{i=0}^{\kappa }{p}_{i}{\left(\exp [-\phi (\chi )]\right)}^{i},\end{eqnarray}$where pi are the values to be evaluated and $\phi (\chi )$ is the result or outcome of the below mentioned equation$\begin{eqnarray}{\phi }^{{\prime} }(\chi )=\alpha +\exp [-\phi (\chi )]+\beta \exp [-\phi (\chi )].\end{eqnarray}$Some of the cases related to the general solutions of equation (13) are mentioned below.
Case 4When $\beta \ne 0$, $\alpha \ne 0$, and ${\alpha }^{2}-4\beta =0$, then$\begin{eqnarray*}{\phi }_{4}(\chi )=\mathrm{ln}\left(-\displaystyle \frac{2(\alpha (\chi +W))+2}{{\alpha }^{2}(\chi +W)}\right).\end{eqnarray*}$
Case 5When $\beta =0$, $\alpha =0$, and ${\alpha }^{2}-4\beta =0$, then$\begin{eqnarray*}{\phi }_{5}(\chi )=\mathrm{ln}\left(\chi +W\right),\end{eqnarray*}$where W being the integration constant. The value of κ is originated from equation (11), by acknowledging a balance between greatest order derivative with greatest order nonlinearity.
Inserting the value of $\gamma ,{\gamma }^{{\prime} }$ in equation (11), we get$\begin{eqnarray*}Q(\exp (-\phi (\chi )))=0.\end{eqnarray*}$As each powers of ${\left(\exp [-\phi (\chi )]\right)}^{j}$ identified to zero, one can obtained a nonlinear network of algebraic equations in pi. After putting the values of pi in equation (12), the exact solutions are obtained for equation (9).
In this technique, the truncated series i.e. the traveling wave solution of equation (9) is of the type$\begin{eqnarray}\gamma (\chi )={a}_{0}+\sum _{i=1}^{\kappa }\left({\alpha }_{i}{\left(\displaystyle \frac{{G}^{{\prime} }}{{G}^{2}}\right)}^{i}+{\beta }_{i}{\left(\displaystyle \frac{{G}^{{\prime} }}{{G}^{2}}\right)}^{-i}\right),\end{eqnarray}$Note it that, $G=G(\chi )$ is the result of an ODE.$\begin{eqnarray}{\left(\displaystyle \frac{{G}^{{\prime} }}{{G}^{2}}\right)}^{{\prime} }=\beta +\alpha {\left(\displaystyle \frac{{G}^{{\prime} }}{{G}^{2}}\right)}^{2},\end{eqnarray}$with the constraint that $\alpha \ne 0$, $\beta \ne 1$ are integers. a0, ${\alpha }_{i}$, ${\beta }_{i}$ are constants to be found.
Below mentioned are some cases related to the general solutions of equation (15).
If $\alpha \ne 0,\beta =0$,$\begin{eqnarray*}\displaystyle \frac{G^{\prime} }{{G}^{2}}=-\displaystyle \frac{T}{\alpha \left(T\chi +U\right)},\end{eqnarray*}$where T and U are nonzero constants. On the same way, the value of κ is originated from equation (11), by acknowledging a balance between greatest order derivative with greatest order nonlinearity.
Inserting the value of $\gamma ,{\gamma }^{{\prime} }$ in equation (11), we get$\begin{eqnarray*}{\rm{\Omega }}\left(\displaystyle \frac{{G}^{{\prime} }}{{G}^{2}}\right)=0.\end{eqnarray*}$As each powers of ${\left(\tfrac{{G}^{{\prime} }}{{G}^{2}}\right)}^{j}$ identified to zero, one can obtained a network of nonlinear algebraic equations. After putting the figures of ${a}_{0},{\alpha }_{i},{\beta }_{i}$ in equation (14), the exact solutions are obtained for equation (9).
4. Governing model
Contemplate$\begin{eqnarray}{R}_{2}{C}_{0}{l}^{2}{u}_{{xxt}}+{l}^{2}{u}_{{xx}}+2{R}_{1}{C}_{0}\delta {{uu}}_{t}-{R}_{1}{u}_{t}=0.\end{eqnarray}$as the structure of MTs like (nonlinear) RLC transmission line, where ${R}_{1}={10}^{9}{\rm{\Omega }}X$, ${R}_{2}=7\times {10}^{-9}$ represent the longitudinal and transversal parcel of the resistance of an rudimentary ring, while $l=8\times {10}^{-9}$ and ${C}_{0}=1.8\times {10}^{-15}\,{\rm{F}}$ represent the length and entire maximal capacitance of the elementary rings (ER). δ $(\delta \lt 1)$ is the tool to describe the nonlinear features of ER in MT. Note that, with an uprising concentration of counter-ions, the capacitance C0 should vary because of the $\beta -{TTs}$ variability. The connection between the elementary capacitor with voltage is chaotic, as like the charge-voltage connection for a reverse-biased diode junction. Now we consider the$\begin{eqnarray}{Q}_{n}={C}_{0}{V}_{n}-{C}_{0}{{bV}}_{n}^{2}.\end{eqnarray}$the value of b is anticipated to be infinitesimal, and then from the figure 1 and after applying the local M-derivative , we can write down
Now from the nth MT ring, its section BC after applying the local M-derivative wave transformation and usage of section 2, we get:$\begin{eqnarray}{v}_{n}-{v}_{n+1}={{LD}}_{M,t}^{\lambda ;\delta }{I}_{n}+{I}_{n}{R}_{1},\end{eqnarray}$where the voltages across the transmission lines AE and CF are vn and ${v}_{n+1}$. Likewise, if ${V}_{n}+{V}_{0}$ is the voltage over the variable type capacitor ${C}_{0}{V}_{n}-{C}_{0}{{bV}}_{n}^{2}$, then we get$\begin{eqnarray}{v}_{n}={V}_{0}+{V}_{n}+({I}_{n-1}+{I}_{n}){R}_{2}.\end{eqnarray}$Note it that, the bias-voltage of the capacitor is represented by V0. Further$\begin{eqnarray}{I}_{n-1}-{I}_{n}={D}_{M,t}^{\lambda ;\delta }{Q}_{n}\end{eqnarray}$indicates the current through the line BD. From equation (18), we get$\begin{eqnarray*}{{LD}}_{M,t}^{\lambda ;\delta }{I}_{n-1}={v}_{n-1}-{v}_{n}-{I}_{n-1}{R}_{1}\end{eqnarray*}$and$\begin{eqnarray*}{{LD}}_{M,t}^{\lambda ;\delta }{I}_{n}={v}_{n}-{v}_{n+1}-{I}_{n}{R}_{1}.\end{eqnarray*}$Use equation (20), we will get$\begin{eqnarray}{{LD}}_{M,t}^{2;\delta }{Q}_{n}={v}_{n+1}+{v}_{n-1}-2{v}_{n}+({I}_{n}-{I}_{n-1}){R}_{1}.\end{eqnarray}$Further, it implies$\begin{eqnarray}\begin{array}{l}{{LD}}_{M,t}^{2,\delta }({C}_{0}{V}_{n}-{C}_{0}{{bV}}_{n}^{2})={v}_{n+1}+{v}_{n-1}-2{v}_{n}\\ -\,{R}_{1}{C}_{0}{D}_{M,t}^{1,\delta }({V}_{n}-{{bV}}_{n}^{2})-{R}_{2}{C}_{0}\{2{D}_{M,t}^{1,\delta }({V}_{n}-{{bV}}_{n}^{2})\\ -\,{D}_{M,t}^{1,\delta }({V}_{n+1}-{{bV}}_{n+1}^{2})-{D}_{M,t}^{1,\delta }({V}_{n-1}-{{bV}}_{n-1}^{2})\}.\end{array}\end{eqnarray}$Remember that the voltage Vn switches moderately from the ring n to its vicinity, and with the help of Taylor’s expansion, we can carefully unfold Vn in a continuum estimation in the form of a small spatial parameter l,$\begin{eqnarray}\begin{array}{l}{V}_{n\pm 1}=V\pm {{lD}}_{M,t}^{\lambda ;\delta }{V}_{x}+\displaystyle \frac{{l}^{2}}{2!}{D}_{M,t}^{\lambda ;\delta }{V}_{{xx}}\\ \pm \,\displaystyle \frac{{l}^{3}}{3!}{D}_{M,t}^{\lambda ;\delta }{V}_{{xxx}}+\displaystyle \frac{{l}^{4}}{4!}{D}_{M,t}^{\lambda ;\delta }{V}_{{xxxx}}\pm ....\end{array}\end{eqnarray}$and ignoring some higher order terms of l, we can express the Taylor expansion as$\begin{eqnarray}{V}_{n+1}-2{V}_{n}+{V}_{n-1}={l}^{2}{D}_{M,t}^{\lambda ;\delta }{V}_{{xx}}+\displaystyle \frac{{l}^{4}}{12}{D}_{M,t}^{\lambda ;\delta }{V}_{{xxxx}}.\end{eqnarray}$Then equation (22) with the use of equations (23), (24) becomes$\begin{eqnarray}\begin{array}{l}{{LC}}_{0}{D}_{M,t}^{2;\delta }(V-{{bV}}^{2})={l}^{2}{D}_{M,t}^{\lambda ;\delta }{V}_{{xx}}\\ +\,\frac{{l}^{4}}{12}{D}_{M,t}^{\lambda ;\delta }{V}_{{xxxx}}-{R}_{1}{C}_{0}{D}_{M,t}^{1;\delta }(V-{{bV}}^{2})\\ +\,{R}_{2}{C}_{0}{D}_{M,t}^{1;\delta }\left\{{l}^{2}{D}_{M,t}^{\lambda ;\delta }{V}_{{xx}}+\frac{{l}^{4}}{12}{D}_{M,t}^{\lambda ;\delta }{V}_{{xxxx}}\right\}\\ -\,{R}_{2}{C}_{0}{{bD}}_{M,t}^{1;\delta }\{2{l}^{2}{D}_{M,t}^{\lambda ;\delta }{V}_{{xx}}+2{l}^{2}{\left({D}_{M,t}^{\lambda ;\delta }{V}_{x}\right)}^{2}\\ +\,\frac{{l}^{4}}{6}V({D}_{M,t}^{\lambda ;\delta }{V}_{{xxxx}})\\ +\,\frac{2{l}^{4}}{3}({D}_{M,t}^{\lambda ;\delta }{V}_{x})({D}_{M,t}^{\lambda ;\delta }{V}_{{xxx}})+\frac{{l}^{4}}{2}{\left({D}_{M,t}^{\lambda ;\delta }{V}_{{xx}}\right)}^{4}\}.\end{array}\end{eqnarray}$Since the time differences of the local voltage V are inappreciable as compared to the fixed background voltage V0, one can guess that the time-dependent derivative is of the order of the infinitesimal parameter ε, as well as the chaotic behavior, terms bV2 of voltage are of the order ${\epsilon }^{2}$. So equation (25) can be obtained by choosing only the principal terms:$\begin{eqnarray}\begin{array}{l}{{LC}}_{0}{D}_{M,t}^{2;\delta }V={R}_{2}{C}_{0}{l}^{2}{D}_{M,t}^{\lambda ;\delta }{uxx}+{l}^{2}{uxx}\\ \ \ -\ {R}_{1}{C}_{0}{D}_{M,t}^{1;\delta }u+2{{bC}}_{0}{R}_{1}\delta {{uD}}_{M,t}^{1;\delta }u.\end{array}\end{eqnarray}$Since LC0 is very small, so the term occurring on the left-hand side will be ignored and, we get; so the final fractional form of equation (16) with the help of above-mentioned work can be written as:$\begin{eqnarray}\begin{array}{l}{R}_{2}{C}_{0}{l}^{2}{D}_{M,t}^{\lambda ;\delta }{u}_{{xx}}+{l}^{2}{u}_{{xx}}\\ +2{R}_{1}{C}_{0}\delta {{uD}}_{M,t}^{\lambda ;\delta }u-{R}_{1}{D}_{M,t}^{\lambda ;\delta }u=0.\end{array}\end{eqnarray}$where the local M-derivative of order λ of a rule $u(x,t)$, w.r.t t is identified as ${D}_{M,t}^{\lambda ;\delta }u$. Important to note that if $\delta =1=\lambda $, equation (27) can be modified into the original equation.
5. Solitary wave solutions via proposed techniques
The theme of this work is to seek solutions like soliton, shock wave and singular type for the above said MT equation. Applying the above transformation, equation (27) takes the following form after integrating once$\begin{eqnarray}{\gamma }^{{\prime\prime} }(\chi )-\displaystyle \frac{a}{v}{\gamma }^{{\prime} }(\chi )-\displaystyle \frac{b}{2}{\gamma }^{2}(\chi )-r\gamma (\chi )-X=0,\end{eqnarray}$where the dimensionless parameters are defined as$\begin{eqnarray*}a=\displaystyle \frac{\tau }{{A}_{2}C},\quad b=\displaystyle \frac{-2{A}_{1}d}{{A}_{2}},\quad r=\displaystyle \frac{{A}_{1}}{{A}_{2}}.\end{eqnarray*}$Employing the homogenous balance principle in equation (19) impart $\kappa =2$. Then equation (12) becomes$\begin{eqnarray}\gamma (\chi )={p}_{0}+{p}_{1}\exp [-\phi (\chi )]+{p}_{2}{\left(\exp [-\phi (\chi )]\right)}^{2},\end{eqnarray}$where the factors ${p}_{0},{p}_{1},{p}_{2}$ needed to be calculated. As section 3 enforced to adopt the proposed method of solution technique, the values of unknown constants ${p}_{0},{p}_{1},{p}_{2}$ have been found. The emerging solutions are elaborated as as follows$\begin{eqnarray*}\begin{array}{rcl}{p}_{0} & = & \displaystyle \frac{-3{a}^{2}+\sqrt{36{a}^{2}+1250{v}^{4}{Xb}}+30{va}\alpha +75{v}^{2}{\alpha }^{2}}{25{v}^{2}b},\\ {p}_{1} & = & \displaystyle \frac{12(a+5v\alpha )}{5{vb}},\quad {p}_{2}=\displaystyle \frac{12}{b},\\ \beta & = & -\displaystyle \frac{{a}^{2}}{100{v}^{2}}+\displaystyle \frac{{\alpha }^{2}}{4},\quad r=\displaystyle \frac{\sqrt{2}\sqrt{18{a}^{4}+625{v}^{4}b}}{25{v}^{2}}.\end{array}\end{eqnarray*}$Following are the solutions obtained by $\exp [-\phi (\chi )]$-expansion method.
When ${\alpha }^{2}-4\beta \gt 0$ and $\beta \ne 0$, then hyperbolic solution has been gained as$\begin{eqnarray}\begin{array}{l}u(x,t)=\displaystyle \frac{{\rm{sech}} {\left[\tfrac{a(W+\chi )}{10v}\right]}^{2}}{50{v}^{2}b{\left(5v\alpha +a\tanh \left[\tfrac{a(W+\chi )}{10v}\right]\right)}^{2}}\\ \left(-(-6{a}^{2}+\sqrt{36{a}^{4}+1250{v}^{4}{Xb}})({a}^{2-25{v}^{2}{\alpha }^{2}})\right.\\ +\left({a}^{2}+\sqrt{36{a}^{4}+1250{v}^{4}{Xb}}+60{{va}}^{3}\alpha \right.\\ \left.+25{v}^{2}\sqrt{36{a}^{4}+1250{v}^{4}{Xb}}{\alpha }^{2}\right)\cosh \left[\displaystyle \frac{a(W+\chi )}{5v}\right]\\ +2a\left(3{a}^{3}+5v\sqrt{36{a}^{4}+1250{v}^{4}{Xb}}\alpha +75{v}^{2}a{\alpha }^{2}\right)\\ \left.\sinh \left[\displaystyle \frac{a(W+\chi )}{5v}\right]\right).\end{array}\end{eqnarray}$Figure 2 shows the graphical representation of equation (21) of the wave solutions $u(x,t)$ for different chooses of fractional values in (a) $\lambda =0.65$, $\delta =0.8$ and in (b) $\lambda =1$, $\delta =1$. Further, figure 2(c) explains the 2D plot of equation (21) versus time t at x=2 for several fractional values.
When ${\alpha }^{2}-4\beta \gt 0$ and $\beta =0$ and $\alpha \ne 0$ then singular soliton solution has been obtained as$\begin{eqnarray}\begin{array}{l}u(x,t)=\\ \displaystyle \frac{\sqrt{36{a}^{4}+1250{v}^{4}{Xb}}+3{a}^{2}\mathrm{csch}{\left[\tfrac{a(W+\chi )}{10v}\right]}^{2}\left(1+\sinh [\tfrac{a(W+\chi )}{5c}]\right)}{25{v}^{2}b}.\end{array}\end{eqnarray}$According to $\tfrac{{G}^{{\prime} }}{{G}^{2}}$—expansion method, equation (14) can be expressed as$\begin{eqnarray}\begin{array}{rcl}\gamma (\chi ) & = & {p}_{0}+{p}_{1}\left(\displaystyle \frac{{G}^{{\prime} }}{{G}^{2}}\right)+{q}_{1}{\left(\displaystyle \frac{{G}^{{\prime} }}{{G}^{2}}\right)}^{-1}\\ & & +\ {p}_{2}{\left(\displaystyle \frac{{G}^{{\prime} }}{{G}^{2}}\right)}^{2}+{q}_{2}{\left(\displaystyle \frac{{G}^{{\prime} }}{{G}^{2}}\right)}^{-2},\end{array}\end{eqnarray}$where ${p}_{0},{p}_{1},{p}_{2},{q}_{1},{q}_{2}$ are constants to be calculated. Adopt the same proposed technique as mentioned in section 3, the values of unknown constants have been obtained. The emerging results are compiled as follows.$\begin{eqnarray*}\begin{array}{rcl}{p}_{0} & = & -\displaystyle \frac{{vr}+2a\alpha }{{vb}},\quad {p}_{1}=\displaystyle \frac{12{\alpha }^{2}}{b},\\ {p}_{2} & = & {q}_{2}=0,\quad {q}_{1}=0,\\ \beta & = & \displaystyle \frac{{v}^{2}(2{Xb}-{r}^{2})+4{a}^{2}{\alpha }^{2}}{48{v}^{2}{\alpha }^{3}}.\end{array}\end{eqnarray*}$The following soliton solutions are obtained like
If $\alpha \beta \gt 0$, the said periodic but singular type is fetched as$\begin{eqnarray}\begin{array}{l}u(x,t)=-\displaystyle \frac{{vr}+2a\alpha }{{vb}}\\ +\displaystyle \frac{12{\alpha }^{2}\sqrt{\tfrac{\beta }{\alpha }}\left(\cos [\sqrt{\alpha \beta }\chi ]+\sin [\sqrt{\alpha \beta }\chi ]\right)}{b\left(\cos [\sqrt{\alpha \beta }\chi ]-\sin [\sqrt{\alpha \beta }\chi ]\right)}.\end{array}\end{eqnarray}$The said singular type solution for $\alpha \beta \lt 0$, is fetched as$\begin{eqnarray}u(x,t)=-\displaystyle \frac{r+\tfrac{2a\alpha }{v}+12\alpha \sqrt{| \alpha \beta | }\coth [\sqrt{| \alpha \beta | }\chi ]}{b}.\end{eqnarray}$The wave profiles $u(x,t)$ of equation (25) are shown vividly in figure 3 against different chooses of fractional values $\lambda =0.65$, $\delta =0.8$, in (a) $\lambda =0.85$, $\delta =0.9$ and in (b) $\lambda =1$, $\delta =1$ in (c). Figure 3(d) explains the 2D plot of equation (21) versus time t at x=2 for several fractional values.
If $\beta =0,\alpha \ne 0$, the wave solution of plane type is elucidated as$\begin{eqnarray}u(x,t)=-\displaystyle \frac{{vr}+2a\alpha }{{vb}}+\displaystyle \frac{12T\alpha }{T\chi +U}.\end{eqnarray}$
6. Conclusion
In this paper, $\exp [-\phi (\chi )]$ and $\tfrac{{G}^{{\prime} }}{{G}^{2}}$-expansion techniques have been invoked successfully on the electric model of ${\rm{MT}}$ to yield the fractional dark, singular, bright, and combo soliton results by using the attributes of local M-derivative. The 3D-graphs are depicted to fetch the fractional solutions, also the influence of fractional parameter λ is shown in 2D-graphs against the numerous values of λ. Such sorts of the solution are useful for the inquisition of some real-time physical processes. Although the above-mentioned techniques have some deficiencies i.e. it contains lengthy calculations but due to the availability of symbolic calculators such as Maple etc, one can get the controlled results efficiently. Also, the selection of factors produces several facts of these results and their graphical faces to be type illustrative without rendering the article so massive, should more realizations be expected. Further, these techniques can also be enforced adroitly for a wide range of NLEEs.
RussellJ S1844Report on Waves, Report of the 14th Meeting of the British Association for the Advancement of Science London John Murray 311390Plates XLVII-LVII [Cited within: 1]
ZabuskyN JKruskalM D1965 Interaction of solitons in a collisionless plasma and the recurrence of initial states Phys. Rev. Lett. 15 240243 DOI:10.1103/PhysRevLett.15.240 [Cited within: 1]
SirisubtaweeSKoonprasertS2018 Exact traveling wave solutions of certain nonlinear partial differential equations usnig the $\tfrac{{G}^{{\prime} }}{{G}^{2}}$-expansion methods Adv. Math. Phys. 2018 7628651 DOI:10.1155/2018/7628651 [Cited within: 1]
AlamM NAkbarM AMohyud-DinS T2014 General traveling wave solutions of the strain wave equation in microstructured solids via the new approach of generalized $\tfrac{{G}^{{\prime} }}{G}$-expansion method Alex. Eng. J. 53 233241 DOI:10.1016/j.aej.2014.01.002
RazaNJhangeerARezazadehHBekirA2019 Explicit solutions of the (2+1)-dimensional Hirota–Maccari system arising in nonlinear optics Int. J. Mod. Phys. B 33 1950360 DOI:10.1142/S0217979219503600 [Cited within: 1]
IncMAtesETchierF2016 Optical solitons of the coupled nonlinear Schrödinger’s equation with spatiotemporal dispersion Nonlinear Dyn. 85 13191329 DOI:10.1007/s11071-016-2762-9 [Cited within: 1]
RazaNAfzalJBekirARezazadehH2020 Improved $\tan \tfrac{\phi }{2}$-expansion approach for Burgers equation in nonlinear dynamical model of ion acoustic waves Braz. J. Phys. 50 254262 DOI:10.1007/s13538-020-00743-0
RazaNZubairA2018 Bright, dark and dark-singular soliton solutions of nonlinear Schrödinger’s equation with spation-temporal dispersion J. Mod. Opt. 65 19751982 DOI:10.1080/09500340.2018.1480066 [Cited within: 1]
ZimmermanJParameswaranRTianB2014 Nanoscale semiconductor devices as new biomaterials Biomater. Sci. 2 619626 DOI:10.1039/C3BM60280J [Cited within: 1]
HavelkaDCifraMKuceraO2014 Multi-mode electro-mechanical vibrations of a microtubule: in silico demonstration of electric pulse moving along a microtubule Appl. Phys. Lett. 104 243702 DOI:10.1063/1.4884118
FreedmanHRezaniaVPrielACarpenterENoskovS YTuszynskiJ A2010 Model of ionic currents through microtubule nanopores and the lumen Phys. Rev. E 81 051912 DOI:10.1103/PhysRevE.81.051912
RazaNMurtazaI GSialSYounisM2018 On soliton: the bimolecular nonlinear transmission line models with constant and time variable coefficients Waves Random Complex Media 28 553569 DOI:10.1080/17455030.2017.1368734
BakerN ASeptDJosephSHolstM JMcCammonJ A2001 Electrostatics of nanosystems: application to microtubules and the ribosome Proc. Natl Acad. Sci. 98 1003710041 DOI:10.1073/pnas.181342398
SataricM VSekulicD LZivanovM2010 Solitonic ionic currents along microtubules J. Comput. Theor. Nanosci. 7 22812290 DOI:10.1166/jctn.2010.1609
SekulicD LSataricB MTuszynskiJ ASataricM V2011 Nonlinear ionic pulses along microtubules Eur. Phys. J. E 34 111 DOI:10.1140/epje/i2011-11049-0
SekulicD LSataricM V2012 Microtubule as nanobioelectronic nonlinear circuit Serb. J. Electr. Eng. 9 107119 DOI:10.2298/SJEE1201107S
SekulicD LSataricM V2015 An improved nanoscale transmission line model of microtubule: the effect of nonlinearity on the propagation of electrical signals Facta Univ. Ser.: Electron. Energ. 28 133142 DOI:10.2298/FUEE1501133S
NdzanaFIIMohamadouA2019 Exact solitary wavelike solutions in a nonlinear microtubule RLC transmission line Chaos 29 013116 DOI:10.1063/1.5044722
EkossoM CFotueA JKenfackS CFotsinHFaiL C2020 The electrical analogue computer of microtubules protofilament Hindawi Disc. Dyn. Nat. Soc. 2020 4916202 DOI:10.1155/2020/4916202
AlamNBin BelgacemF M2016 Microtubules nonlinear Models dynamics investigations through the exp-expansion method implementation Mathematics 4 6 DOI:10.3390/math4010006 [Cited within: 1]
ZdravkovicSSataricM VMaluckovABalazA2014 A nonlinear model of the dynamics of radial dislocations in microtubules Appl. Math. Comput. 237 227237 DOI:10.1016/j.amc.2014.03.113 [Cited within: 1]
ZdravkovicSMaluckovADekicMKuzmanovicSSataricM V2014 Are microtubules discrete or continuum systems? Appl. Math. Comput. 242 353360 DOI:10.1016/j.amc.2014.05.068 [Cited within: 1]
OsmanM SRezazadehHEslamiM2019 Traveling wave solutions for (3+1)- dimensional conformable fractional Zakharov–Kuznetsov equation with power law nonlinearity Nonlinear Eng. 8 559567 DOI:10.1515/nleng-2018-0163
RazaN2018 Exact periodic and explicit solutions of the conformable time fractional Ginzburg Landau equation Opt. Qunatum Electron. 50 154 DOI:10.1007/s11082-018-1420-5 [Cited within: 1]
GaoWIsmaelH FMohammedS ABaskonusH MBulutH2019 Complex and real optical soliton properties of the paraxial nonlinear Schrödinger equation in kerr media with M-fractional Front. Phys. 7 197 DOI:10.3389/fphy.2019.00197 [Cited within: 1]
Al-GhafriK S2020 Soliton behaviours for the conformable space–time fractional complex Ginzburg–Landau equation in optical fibers Symmetry (MDPI) 12 219 DOI:10.3390/sym12020219 [Cited within: 1]
FujiokaJVelascoMRamirezA2016 Fractional optical solitons and fractional Noether’s theorem with ortigueira’s centered derivatives Appl. Math. 7 13401352 DOI:10.4236/am.2016.712118 [Cited within: 1]
UtoyoM IWindartoSa’adahA2018 Analysis of fractional order mathematical model of hematopoietic stem cell gene-based therapy Int. J. Math. Math. Sci. 2018 6180892 DOI:10.1155/2018/6180892 [Cited within: 2]
DentE W2016 Of microtubules and memory: implications for microtubule dynamics in dendrites and spines Mol. Biol. Cell 28 16 [Cited within: 1]
VosikaZ BLazovicG MMisevicG NSimic-KrsticJ B2013 Fractional calculus model of electrical impedance applied to human skin PLoS One 8 e59483 DOI:10.1371/journal.pone.0059483 [Cited within: 1]
OwyedSAbdouM AAbdel-AtyA HIbraheemA ANekhiliRBaleanuD2020 New optical soliton solutions of space–time fractional nonlinear dynamics of microtubules via three integration schemes J. Intell. Fuzzy Syst. 38 28592866 DOI:10.3233/JIFS-179571 [Cited within: 1]
GorenfloRKilbasA AMainardiFRogosinS V2014Mittag–Leffler Functions, Related Topics and Applications Berlin Springer [Cited within: 1]
,3,4,5,∗1Department of Mathematics, 
New window|Download| PPT slide
New window|Download| PPT slide
New window|Download| PPT slide
