A novel approach to q-fractional partial differential equations: Unraveling solutions through semi-analytical methods

Abstract

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.

Keywords

• $ \mathit{q} $-calculus
• fractional calculus
• $ \mathit{q} $-fractional partial differential equations
• homotopy analysis method
• residual power series method
• semi analytical techniques