Extraction of soliton solutions for the time–space fractional order nonclassical Sobolev-type equation with unique physical problems

Abstract

In this paper, we investigate the soliton solutions of the time–space fractional order nonclassical Sobolev-type (TSFNST) equation. Sobolev equations are used in thermodynamics, the flow of fluid through fractured rock, and other fields. There are certain significant partial differential equations having a third-order mixed derivative with regard to time and space that are now being used in real-world applications. The study aims to analyze the TSFNST equation analytically by employing the ϕ6-model expansion and modified G′/G2 expansion techniques respectively. The solution is obtained successfully in the Jacobi elliptic function solutions that will prove to us the hyperbolic, trigonometric, and rational solutions. Additionally, we select some unique physical problems of the solution. Lastly, the 3-dimensional and their corresponding contour plots are drawn by choosing the different values of constants