Narrow Results

Nonlinear fractional-order partial differential equations play an important role in science and engineering by illustrating a variety of nonlinear processes. The nonlinear space-time fractional combined Korteweg-de Vries and modified Korteweg-de Vries equation is a very significant fractional partial differential equation and is used to simulate shallow water surface wave phenomena, pulse waves in large arteries, ion acoustic waves in plasmas, and atmospheric dust-acoustic solitary waves. The improved Bernoulli sub-equation function method and the new generalized $(G^{\prime }/G)$-expansion method are two noteworthy approaches that have been used to analyze and extract solutions to the above-mentioned equation that include various types of traveling waves as well as soliton solutions via beta-derivative. Through the utilization of a wave transformation, the fractional-order equation is transformed into a nonlinear ordinary differential equation. The exponential function, trigonometric function, rational function, and hyperbolic trigonometric function solutions with arbitrary constants have been used to articulate the obtained solutions. By utilizing the aforementioned approaches, several standard waveforms have been recognized including multiple periodic types, kink shapes, bell-shaped, single solitons, and other types of solitons. Mathematica software has been used to illustrate the wave profiles through 3D and contour plots, providing a clearer physical sketch based on diverse values of free parameters. The suggested methods have demonstrated their reliability in establishing more generalized wave solutions and exhibit computational efficiency, making them suitable for soliton solutions.

In this paper, we present a work on dynamic equation of Deoxyribonucleic acid (DNA) derived from the Peyrard–Bishop (PB) model oscillator chain for various dynamical solitary wave solutions. In order to construct novel soliton solutions in the DNA dynamic PB model with beta-derivative, the efficiency of the newly developed algorithms is being investigated, which could include a new auxiliary equation strategy (NAES). Some precise soliton solutions comprising dark, light and other forms of multi-wave soliton solutions are achieved via the proposed methods. Furthermore, mathematical models demonstrate the singularity of our work in comparison to current literary materials and even describe some results using the classic Peyrard–Bishop model. All the established results contribute to the possibility of extending the approach to solve other nonlinear equations of fractional space–time derivatives in nonlinear sciences. The strategy that has been proposed recently is specific and is being employed to produce novel closed-form solutions for all many other FNLEEs.

For the ($2+1$)-dimensional time conformable Maccari system with beta-derivative, exact solutions are obtained in this paper using the improved sub-equation method. This technique has been used to find rational solutions, generalized hyperbolic function solutions and generalized trigonometric function solutions. The new structures of the solutions are shown after the proper parameter values have been given. It is easier to understand the intricate physical processes of these dynamical models by viewing the physical surfaces of some obtained solutions in various forms. The outcomes demonstrate the efficiency of the applied strategy, which may be used to solve a number of nonlinear physical problems. ($2+1$)-dimensional time conformable Maccari system is considered and the graphs of solutions are sketched using MAPLE software after its exact solutions which are acquired using the improved sub-equation method.

In this work, a method called the $G^{\prime }/G^2$-extension method which is effective and yields soliton solutions is utilized to scrutinize the nonlinear Burger-like equation with beta derivative and the nonlinear Boussinesq–Burger equation with beta derivative. The method is applied in a reliable and efficient style using a symbolic computing program called Maple. In addition, some graphs of the solutions are presented. When these equations are solved using the suggested method, several novel exact solutions have been achieved that differ from those found by utilizing them earlier. The findings of this work add to the body of literature by offering insightful definitions of different nonlinear systems. The outcomes showed that the proposed technique is a useful mathematical tool and the symbolic computation program makes them easier, more reliable and faster.

The deoxyribonucleic acid (DNA) dynamical equation, which emerges from the oscillator chain known as the Peyrard–Bishop (PB) model for abundant optical soliton solutions, is presented, along with a novel fractional derivative operator. The Kudryashov expansion method and the extended hyperbolic function (HF) method are used to construct novel abundant exact soliton solutions, including light, dark, and other special solutions that can be directly evaluated. These newly formed soliton solutions acquired here lead one to ask whether the analytical approach could be extended to deal with other nonlinear evolution equations with fractional space–time derivatives arising in engineering physics and nonlinear sciences. It is noted that the newly proposed methods’ performance is most reliable and efficient, and they will be used to construct new generalized expressions of exact closed-form solutions for any other NPDEs of fractional order.

This study purposes to extract some fractional analytical solutions of the Peyrard-Bishop-Deoxyribo-Nucleic-Acid ($\beta$-PBDNA) dynamic model with the beta-derivative by the unified Riccati equation expansion method (UREEM). Furthermore, we examine the role of various parameters of the fractional model on the soliton dynamic. The research focuses on computational biophysics and materials science, examining the impact of various parameters on the fractional model. This paper contributes to understanding soliton solutions and the $\beta$-PBDNA dynamic model, demonstrating the applicability of the UREEM method to various fractional models. Some soliton solutions of the model are successfully generated by applying the UREEM. Implementing the UREEM, we take a fractional wave transformation to convert the model into a nonlinear ordinary differential equation. So, a linear equation system is generated. After the system is solved, the soliton solutions are gained for the appropriate solution sets. Finally, 3D, 2D and contour graphs of diverse solutions are depicted at suitable values of parameters. In addition, this paper presents 3D, 2D and contour graphs of various solutions with suitable parameter values. The results are beneficial for interpreting the model in future work and confirm that UREEM is effectively applicable to diverse fractional models, coupled with a comprehensive graphical analysis of how different parameters influence these solutions.

This work suggests a fractional modification of the KdV–Kadomtsev–Petviashvili model with the beta-derivative to consider unsmooth boundary. Some new interesting solitary waves are found for the first time ever by the fractional sine–cosine method and the fractional ansatz method. These dynamical characteristics of new solitary waves are discussed by some three-dimensional (3D) figures, and the effect of the fractal parameters on the solitary waves traveling is also discussed and explained.