On the “Tantawy technique” and other methods for analyzing fractional Fokker–Plank type equations

The family of Fokker–Planck (FP) equations has been widely used in various physical applications, especially optical physics. It has proven effective in understanding numerous nonlinear phenomena observed in optical fiber. Accordingly, we analyze the time fractional FP (FFP) equations to understand the underlying mechanics of the phenomena described by the suggested models, control the generation and propagation of these phenomena, or prevent their occurrence altogether to achieve the desired applications. Thus, in this investigation, some techniques, such as a novel technique, which is called the “Tantawy technique” and the optimal auxiliary function method (OAFM) are implemented to analyze different types of FFP equations in the framework of the Caputo operator. Using fractional calculus, we develop a complete framework for effectively managing complex diffusion processes and their corresponding probability density functions. We apply the OAFM and the “Tantawy technique” to three different models of the FFP. Some highly accurate approximations for the two models are derived and discussed numerically. The accuracy of the obtained approximations is checked by analyzing these approximations numerically and graphically using some suitable values for the related parameters. In addition, the absolute error for the obtained approximations is estimated compared to the exact solutions for the integer cases to confirm the validity and accuracy of these approximations. Moreover, the novel “Tantawy technique,” developed for the first time, is highly accurate and efficient in analyzing the most complicated and nonlinear problems. It is also far more precise than the optimal auxiliary function method. Moreover, it is characterized by extreme simplicity, high accuracy, and stability throughout the study domain, which is not achieved in many other methods. The derived approximations using the “Tantawy technique” are compared to the Laplace homotopy perturbation method.