Efficient simulation of plasma physics' time fractional modified Korteweg-de Vries equations

In many science and engineering fields, integer-order differential equations are unable to provide a satisfactory explanation for a wide range of phenomena when compared to fractional-order differential equations. The fractional modified Korteweg-de Vries (mKdV) equations are investigated in this work by employing effective analytical methods within the Caputo operator. The findings for the given problems are computed using the Elzaki transformation, the homotopy perturbation method, and the Adomian decomposition method. With these techniques, the problems were first made simpler utilizing the Elzaki transform, and the problems were then comprehensively solved by employing the decomposition and perturbation approaches. A few numerical cases with their approximate analytical solutions are considered to demonstrate the conclusions drawn from the findings. To verify these approaches, we examined two cases and compared them with the real outcomes. By using these methods, the solution to the suggested problem is represented by recurrence relations. The selected issues have series solutions that can be found and have features that more quickly approach the exact results. It is found that there is a strong correlation between the derived results and the real results of every problem when the number of terms increases in the series solution of the problem. The use of efficient techniques that provide higher levels of accuracy with less computation makes the current work innovative. To further benefit the scientific community, the proposed methods can also be used in the future to solve other fractional nonlinear problems.