Application of the Exp−φξ-Expansion Method to Find the Soliton Solutions in Biomembranes and Nerves

Abstract

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(−φ(ξ)) −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.

Closed-Form Solutions in a Magneto-Electro-Elastic Circular Rod via Generalized Exp-Function Method

“…Step 4. Equation ( 32) is substituted into Equation (31), and then, Equation ( 33) is used, with all the coefficients of exp − A 1 φ (η) +A 2 A 3 φ (η) +A 4 i to zero, yielding an algebraic equation system for k 1 , A, B, k 2 and (i = 0, 1, 2, 3, . .…”

“…The homogeneous balance method [12], the expfunction method [13], the tanh and extended-function method [14,15], the extended direct algebraic method [16], the modified simple equation method [17], the (G'/G)-expansion method [18,19], the modified extended tanh function method [20], the iteration transform method [21,22], and homotopy perturbation method [23] are a few examples of important methods. Using these techniques, numerous researchers have discovered various kinds of exact and solitary wave solutions to various nonlinear evolution equations, for example, see references [24][25][26][27][28][29][30][31][32]. It is reminiscent of Lie algebraic methods, another class of strong algebraic techniques that are frequently employed to solve nonlinear ordinary differential equations [33][34][35].…”

Exact Solutions for the Generalized Atangana-Baleanu-Riemann Fractional (3 + 1)-Dimensional Kadomtsev–Petviashvili Equation

“…[8][9][10][11]. To date, many powerful methods have been proposed for this subject, such as the Bäcklund transformation method [12], Darboux transformation [13], Hirota bilinear method [14], improved F-expansion method [15], sine-Gordon method [16], projective Riccati equations method [17], G'/G-expansion method [18], (G'/G,1/G)-expansion method [19], improved (m + G'/G)-expansion method [20], improved G'/G 2 -expansion method [21], the first integral method [22], Generalized Exp-Function Method [23], Exp(−ϕ(ξ))-Expansion Method [24], and Lie symmetry method, which are connected to the wave transformations of equations that do not change the set of solutions [25], among others [26][27][28][29][30][31][32][33][34][35][36].…”