New Exact Solutions of (4+1)-dimensional Space-Time Fractional-order Fokas Partial differential Equation arising in Mathematical Physics

In the last few decades, the study of fractional calculus has been extended with great significance due to its variety of applications in physics, mathematical physics, fluid flow, viscoelasticity, control theory, wave theory, chemical physics, electrical networks, etc. Unlike the integer-order derivative, the fractional derivative operators are nonlocal and have a weak singularity, so theoretical analysis and analytical studies are more complicated. Moreover, it is not an easy job to find the analytical/exact solutions of higher-dimensional fractional-order problems. As we know, there are no classical methods to handle the non-linear fractional partial differential equations (FPDEs) and provide the explicit solution due to the complexities of fractional calculus. For this reason, new techniques/methods are always welcome do find exact solutions to such problems. This project is devoted to formulating the appropriate method such as the F-expansion method and auxiliary equation method to find the exact solution of (4+1)-dimensional fractional-order Fokas partial differential equation. We will also construct the different types of wave structures (e.g. solitary wave, a periodic wave, and other structures) based on the obtained exact solutions under different parametric conditions. These new structures will help to understand the physical phenomena and interpretations of these fractional-order nonlinear models in real-world problems such as water wave theory, ocean dynamics, and many others. The obtained results and computational work will also show the power, simplicity, and effectiveness of the proposed methods. Many other such types of high dimensional nonlinear fractional-orders models can also be solved by these methods.