In this research work, we established exact solution for the conformable space–time fractional (3 + 1) dimensional modified Korteweg de Vries equations (mKdV). A Sine- Cosine method is used for obtaining travelling wave solutions for these models with minimal algebra. We can conclude that the proposed scheme is reliable and efficient as its required minimal algebra without using sophisticated Mathematical tools (maple, Mathematica and others). The goal has been achieved with minimal computational cost and the present solutions obtained will serve as new solutions to the modified Korteweg–de-Vries (mKdV) equations.

In this paper, we propose a Dai–Liao (DL) conjugate gradient method for solving large-scale system of nonlinear equations. The method incorporates an extended secant equation developed from modified secant equations proposed by Zhang et al. (J Optim Theory Appl 102(1):147–157, 1999 ) and Wei et al. (Appl Math Comput 175(2):1156–1188, 2006 ) in the DL approach. It is shown that the proposed scheme satisfies the sufficient descent condition. The global convergence of the method is established under mild conditions, and computational experiments on some benchmark test problems show that the method is efficient and robust.

In this paper, we present an innovative approach to acquire the exact solutions of the Shynaray-IIA equations (S-IIAE), by using the improved modified Sardar sub-equation method (IMSSEM). The S-IIAE are nonlinear and coupled partial differential equations that arise in various fields of physics and engineering such as optical fibers and ferromagnetic materials. The IMSSEM is applied to S-IIAE; we successfully derived exact solutions that accurately described the wave propagation behavior of the system under consideration. The obtained solutions include rational, trigonometric, and trigonometric hyperbolic function solutions. The obtained solutions are concise and offer a deeper insight into the dynamics and characteristics of the S-IIAE. Moreover, some of the new solutions to S-IIAE are plotted in different dimensions through which bright, anti-kink and bright solitary wave structures are established. The results of the study also indicated that the proposed method is a valuable approach for achieving analytical solutions to a wide range of nonlinear partial differential equations.

In this work, we investigate soliton solutions for the conformable nonlinear differential equation governing wave-propagation in low-pass electrical transmission lines. Adopting two integration techniques, we construct dark and bright solitary waves, jacobian elliptic function solutions and trigonometric solutions. The obtained results are relevant and will probably help to carry data and codify them in telecommunication.