Abstract
In this research article, we present exact solutions with parameters for two nonlinear model partial differential equations(PDEs) describing microtubules, by implementing the exp(−Φ(ξ))-Expansion Method. The considered models, describing highly nonlinear dynamics of microtubules, can be reduced to nonlinear ordinary differential equations. While the first PDE describes the longitudinal model of nonlinear dynamics of microtubules, the second one describes the nonlinear model of dynamics of radial dislocations in microtubules. The acquired solutions are then graphically presented, and their distinct properties are enumerated in respect to the corresponding dynamic behavior of the microtubules they model. Various patterns, including but not limited to regular, singular kink-like, as well as periodicity exhibiting ones, are detected. Being the method of choice herein, the exp(−Φ(ξ))-Expansion Method not disappointing in the least, is found and declared highly efficient.
Highlights
MTs are cytoskeletal biopolymers shaped as nanotubes. They are hollow cylinders formed by Proto-Filaments (PFs) representing a series of proteins known as tubulin dimers
MTs dynamical behavior is modeled by nonlinear partial differential equations (NPDEs)
To date solving NPDEs exactly or approximately, a plethora of methods have been in use
Summary
MTs are cytoskeletal biopolymers shaped as nanotubes. They are hollow cylinders formed by Proto-Filaments (PFs) representing a series of proteins known as tubulin dimers. MTs dynamical behavior is modeled by nonlinear partial differential equations (NPDEs). These equations are mathematical models of physical circumstances that emerge in various fields of engineering, plasma physics, solid state physics, optical fibers, chemistry, hydrodynamics, biology, fluid mechanics and geochemistry. To date solving NPDEs exactly or approximately, a plethora of methods have been in use. These include, but are not limited to, (G1 /G)-expansion [1,2,3,4,5,6], Frobenius decomposition [7], local fractional variation iteration [8], local fractional series expansion [9], multiple exp-function algorithm [10,11], transformed rational function [12], exp-function method [13,14], trigonometric series function [15], inverse scattering [16], homogeneous balance [17,18], first integral [19,20,21,22], F-expansion [23,24,25], Jacobi function [26,27,28,29], Sumudu transform [30,31,32], solitary wave ansatz [33,34,35,36], novel (G1 /G) -expansion [37,38,39,40,41,42], modified direct algebraic method [43,44], and last but not least, the exppΦpξqq-Expansion Method [45,46,47,48,49,50]
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