Elzaki Homotopy perturbation method for modeling fractional ion-acoustic solitary and shock waves in a non-Maxwellian plasma

Weaam Alhejaili¹Adnan Khan²Amnah S. Al-Johani³Samir A El-Tantawy^4,5*

• ¹Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia
• ²Abdul Wali Khan University Mardan, Mardan, Khyber Pakhtunkhwa, Pakistan
• ³University of Tabuk, Tabuk, Tabuk, Saudi Arabia
• ⁴Port Said University, Port Said, Egypt
• ⁵Al Baha University, Al Bahah, Saudi Arabia

It is known that the family of nonlinear Korteweg-de Vries-type (KdV) equations is widely used in modeling many realistic phenomena that occur in nature, such as the propagation of solitons, shock waves, multiple solitons, cnoidal waves, and periodic waves in seas and oceans, plasma physics, fluid mechanics, and electronic circuits. Motivated by these applications, we proceed to analyze the time-fractional forms of this family, including the planar quadratic nonlinear fractional KdV (FKdV) and planar cubic nonlinear fractional modified KdV (FmKdV) using Elzaki Homotopy perturbation method (HPTM). By implementing this method, we can derive some highly accurate approximations to both FKdV and FmKdV equations. Using the suggested method, the nonlinear planar FKdV equation is solved and analytical FKdV-soliton approximation is obtained. For the nonlinear planar FmKdV equation, two general formulas are derived depending on the polarity of the cubic nonlinearity coefficient "C". At C > 0, the mKdV-soliton is used as an initial solution, and an analytical FmKdV-soliton approximation is generated. On the contrary, for C < 0, the nonlinear planar FmKdV equation does not support solitons but instead supports shock waves.Using the suggested approach, a general formula for the FmKdV-shock wave approximation is derived. As a practical application of the derived approximations, the fluid-governed equations of a collisionless and unmagnetized plasma composed of inertial cold ions and inertialess Cairns-Tsallis distributed electrons are reduced to both the FKdV and the FmKdV equations to study the properties of fractional ion-acoustic waves and gain a deeper understanding of their dynamical behavior. The derived approximations for both nonlinear planar FKdV and FmKdV equations are not limited to plasma physics and its applications but extend to the simulation of many nonlinear phenomena described by these equations, as the derived approximations are general.