In this paper, we use the modified exp$\left(-\psi \left(\theta \right)\right)$-function method to observe some of the solitary wave solutions for the microtubules (MTs). By treating the issues as nonlinear model partial differential equations describing microtubules, we were able to solve the problem. We then found specific solutions to the nonlinear evolution equation (NLEE) covering various parameters that are particularly significant in biophysics and nanobiosciences. In addition to the soliton-like pulse solutions, we also find the rational, trigonometric, hyperbolic, and exponential function characteristic solutions for this equation. The validity of the method we developed and the fact that it provides more solutions are demonstrated by comparison to other methods. We next use the software Mathematica 10 to generate 2D, 3D, and contour plots of the precise findings we observed using the suggested technique and the proper parameter values.

Using the new generalized exp-function method, we were able to derive significant novel closed form solutions to the nonlinear dispersive modified Benjamin–Bona–Mahony (DMBBM) equation. The general framework of the new generalized exp-function method has been given. Many novel closed form solutions have been obtained in the form of hyperbolic, trigonometric, and rational function solutions. Using the computer application Wolfram Mathematica 10, we plotted 2D, 3D, and contour surfaces of closed form solutions found in this work. In the form of a table, the acquired results are compared to the known solutions in the existing literature.

Heimburg and Jackson devised a mathematical model known as the Heimburg model to describe the transmission of electromechanical pulses in nerves, which is a significant step forward. The major objective of this paper was to examine the dynamics of the Heimburg model by extracting closed-form wave solutions. The proposed model was not studied by using analytical techniques. For the first time, innovative analytical solutions were investigated using the exp$\left(-\phi \left(\xi \right)\right)$-expansion method to illustrate the dynamic behavior of the electromechanical pulse in a nerve. This approach generates a wide range of general and broad-spectral solutions with unknown parameters. For the definitive value of these constraints, the well-known periodic- and kink-shaped solitons were recovered. By giving different values to the parameters, the 3D, 2D, and contour forms that constantly modulate in the form of an electromechanical pulse traveling through the axon in the nerve were created. The discovered solutions are innovative, distinct, and useful and might be crucial in medicine and biosciences.

A number of solitary wave solutions for microtubules (MTs) are observed in this article by using the modified exp-function approach. We tackle the problem by treating the results as nonlinear RLC transmission lines, and then finding exact solutions to Nonlinear Evolution Equation (NLEE) containing parameters of particular importance in biophysics and nanobiosciences. For this equation, we find trigonometric, hyperbolic, rational, and exponential function solutions, as well as soliton-like pulse solutions. A comparison with other approach indicates the legitimacy of the approach we devised as well as the fact that our method offers extra solutions. Finally, we plot 2D, 3D and contour visualizations of the exact results that we observed using our approach using appropriate parameter values with the help of software Mathematica 10.

In this article, a new method for obtaining closed-form solutions of the simplified modified Camassa-Holm (MCH) equation, a nonlinear fractional partial differential equation, is suggested. The modified Riemann-Liouville fractional derivative and the wave transformation are used to convert the fractional order partial differential equation into an integer order ordinary differential equation. Using the novel (G′/G²)-expansion method, several exact solutions with extra free parameters are found in the form of hyperbolic, trigonometric, and rational function solutions. When parameters are given appropriate values along with distinct values of fractional order α travelling wave solutions such as singular periodic waves, singular kink wave soliton solutions are formed which are forms of soliton solutions. Also, the solutions obtained by the proposed method depend on the value of the arbitrary parameters H. Previous results are re-derived when parameters are given special values. Furthermore, for numerical presentations in the form of 3D and 2D graphics, the commercial software Mathematica 10 is incorporated. The method is accurately depicted, and it provides extra general exact solutions.

In the areas of tidal and tsunami waves in oceans, rivers, ion and magneto-sound waves in plasmas, electromagnetic waves in transmission lines, homogeneous and stationary media, etc., the Riemann wave equations are attractive nonlinear equations. The modified exp$\left(-\Phi \left(\eta \right)\right)$-function method is used in this article to show how well it can be applied to extract travelling and solitary wave solutions from higher-order nonlinear evolution equations (NLEEs) using the equations mentioned above. Trigonometric, hyperbolic, and exponential functions solitary wave solutions can be extracted using the above-mentioned technique. By changing specific values of the embedded parameters, we can obtain bell-form soliton, consolidated bell-shape soliton, compacton, singular kink soliton, flat kink shape soliton, smooth singular soliton, and other sorts of soliton solutions. The solutions are graphically illustrated in 3D and 2D for the accuracy of the outcome by using the Wolfram Mathematica 10. The verification of numerical solvers on the stability analysis of the solution is substantially aided by the analytic solutions.