Dynamical wave structures for time-fractional (3+1)-dimensional p-type model via two improved techniques

Abstract

In this work, we investigates the conformable time-fractional (3+1)-dimensional p-type model for the analytical solutions. The underlying model is explained the material characteristics and spontaneous processes in solid-state physics, such as magnetism and conventional particle physics. To obtain the analytical solutions, we used the novel Kumar–Malik method and the new extended direct algebraic method. We derived the analytical solutions through the application of the conformal fractional derivative and the fractional wave transformation. We successfully obtain several solutions in the form of rational, hyperbolic, mixed trigonometric, mixed hyperbolic, exponential, Jacobi elliptic, and trigonometric functions by using these methods. The found solutions include various solitary wave solutions as well as bright, dark, and w-shaped soliton solutions. With the use of Mathematica 13.0, the analytical soliton solutions are further shown in 3D, contour and 2D representations, assisting in the understanding of these complex wave phenomena. © 2025 The Authors