Novel analysis of linear and nonlinear fractional KdV and Kolmogorov equations

In this paper, the Tantawy Least Square Method (TLSM) is presented and used to achieve approximate solutions to fractional-order partial differential equations (FPDEs) with the Caputo derivative. The suggested method is a combination of the least squares approximation and Tantawy decomposition framework, which ensures minimal residual error and maximum accuracy. To show its efficiency, TLSM is implemented on a set of test problems, among them the fractional Kolmogorov equation, the linearized Korteweg-de Vries (KdV) equation, and the nonlinear KdV equation. The results obtained are compared to the known techniques including the fractional reduced differential transform method (FRDTM) and the differential transform method (DTM). Experimental results indicate that TLSM produces solutions with minimal absolute errors and hence is more precise and stable than available methods. Moreover, the fractional order is also found to be important as the analysis demonstrates that the order parameter has an influential effect on the diffusion rates, memory effects, and nonlinear wave structures. These results make TLSM a dependable and effective tool to tackle linear and nonlinear FPDEs, and may be extended to multidimensional and coupled systems to applied sciences and engineering.