Description

Nonlinear time-fractional partial differential equations, especially nonlinear time-fractional Gas dynamic equations, can be resolved by applying the Optimal Homotopy Asymptotic Method (OHAM) and the Least Square Residual Power Series Method (LSRPSM). A Fractional-order derivative that has numerical values in the closed interval [0, 1] is being employed in the Caputo meaning. These approaches are compared based on their computing complexity, convergence rate, and approximation error. The present study demonstrates that when these techniques are assigned to nonlinear differential equations of fractional order, they exhibit differing convergence rates and approximation errors. Using the Matlab software, perform numerical computations and graphics for fractional gas differential equations. The results of this comparison are compared to the exact solution to demonstrate how much more efficient and precise our methods are at solving nonlinear differential equations. In comparison to (OHAM), the results demonstrate the validity and efficiency of the series solution utilizing (LSRPSM), showing the importance of these methods in the study of fractional differential equations.

URL

DOI

DOI:10.21608/FSRT.2023.233651.1104

Brief

Time-fractional Gas-Dynamic equations based on Caputo fractional derivation were approximated using the least Square Residual Power Series approach and the Optimal Homotopy Asymptotic method. At the same value of t and with the same number of term approximations, the acquired solutions in the Least Square Residual Power Series Method (LSRPSM) see (Table 2) and fig(4to6) converge to the exact solutions more rapidly than the Optimal Homotopy Asymptotic Method (OHAM) in table(1)and fig from 1to 3. This implies that the approximate solutions in (LSRPSM) technique are significantly closer to the exact solutions than the solutions in (OHAM) approach. According to the numerical results, the current approaches are simple, efficient, and provide extremely high precision for getting approximate solutions to various nonlinear fractional physical differential equations.

KeyWords

Least Square Residual Power Series Method, Optimal Homotopy Asymptotic Method, Fractional nonlinear gas dynamics equation, Caputo’s fractional derivative.