On the “Tantawy Technique” and other methods for analyzing the family of fractional Burgers’ equations: Applications to plasma physics

This paper explores some innovative and modern techniques for analyzing the family of fractional Burgers-type equations, which are extensively utilized in studying shock waves in plasma physics and marine environments. The most notable novel and effective technique employed in this study for the first time is the “ Tantawy Technique,” named after its creator, Prof. Samir El-Tantawy. Two other effective methods, known as, the Aboodh residual power series method (ARPSM) and the Aboodh new iteration method (ANIM), are also applied to analyze the suggested problems and compare them with the Tantawy Technique results. We employ these methods to address the complexities associated with fractional partial differential equations (PDEs), specifically using the Caputo operator for fractional derivatives. The study provides a comprehensive analysis of the convergence and accuracy of the proposed methods through various numerical examples. All three proposed methods can effectively yield precise and rapid solutions for fractional Burgers’ type equations. We assess the precision of all derived approximations by comparing them with the exact solutions for the integer case and calculating the absolute errors for these approximations. This work contributes significantly to fractional calculus and nonlinear differential equations, offering practical tools for scientists and engineers to solve complex fractional PDEs with enhanced accuracy. The results will improve our understanding and investigation of several complex phenomena that occur in nature and different nonlinear media, such as optics fiber, fluid mechanics, and plasma physics. As real-world applications for this study, we will investigate various evolutionary equations, such as the Burgers and KdV–Burgers equations used in plasma physics, to explain the properties of fractional shock waves while considering viscosity.