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.
Citation: Khalid K. Ali, Mohamed S. Mohamed, M. Maneea. A novel approach to $ \mathit{q} $-fractional partial differential equations: Unraveling solutions through semi-analytical methods[J]. AIMS Mathematics, 2024, 9(12): 33442-33466. doi: 10.3934/math.20241596
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.
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