A novel approach to $ q $-fractional partial differential equations: Unraveling solutions through semi-analytical methods - Khalid K. Ali; Mohamed S. Mohamed; M. Maneea (AIMS Mathematics)
This paper presents an innovative approach to solve $ q $-fractional partial differential equations through a combination of two semi-analytical techniques: The Residual Power Series Method (RPSM) and the Homotopy Analysis Method (HAM). Both methods are extended to obtain approximations for $ q $-fractional partial differential equations ($ q $-FPDEs). These equations are significant in $ q $-calculus, which has gained attention due to its relevance in engineering applications, particularly in quantum mechanics. In this study, we solve linear and nonlinear $ q $-FPDEs and obtain the closed-form solutions, which confirm the validity of the utilized methods. The results are further illustrated through two-dimensional and three-dimensional graphs, thus highlighting the interaction between parameters, particularly the fractional parameter, the $ q $-calculus parameter, and time.