A novel approach to $\mathit{q}$-fractional partial differential equations: Unraveling solutions through semi-analytical methods

This paper presents an innovative approach to solve $\mathit{q}$-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 $\mathit{q}$-fractional partial differential equations ( $\mathit{q}$-FPDEs). These equations are significant in $\mathit{q}$-calculus, which has gained attention due to its relevance in engineering applications, particularly in quantum mechanics. In this study, we solve linear and nonlinear $\mathit{q}$-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 $\mathit{q}$-calculus parameter, and time.