The series solutions of fractional foam drainage and fractional modified regularized long wave problems

In general, integer order differential equations fall short of adequately describing a range of phenomena in many disciplines of science and engineering when compared to fractional order differential equations. In the context of the Caputo fractional derivative, this study examines the nonlinear fractional order foam drainage problem as well as the fractional order modified regularized long wave equation. In this work, we offer two novel techniques by combining the Elzaki transform with the Adomian decomposition method (ADM) and the homotopy perturbation method (HPM). We first translate the problem into its differential partner using the Elzaki transform, and then we use He's polynomials and the Adomian polynomials that are derived from HPM and ADM, respectively. These are two very powerful supports for nonlinear problems. The efficacy and precision of the recommended techniques are illustrated through the use of numerical and graphical findings. The numerical simulations of the derived solutions make use of a number of relevant values of the order of fractional parameter lambda. It is important to keep in mind that the recommended techniques can achieve higher overall performance since they need less computational effort than alternative strategies while yet retaining a high degree of numerical precision. The solutions acquired through the suggested techniques have been compared with the classical solutions and the solution obtained by ADM, new iterative method (NIM). The considerable results show that these approaches minimize the heavy calculation without restricting variables while requiring no assumptions. The present study can confirm the applicability of the new generalized Caputo type fractional operator to mathematical real-world problems. This approach shows to be one of the most efficient methods to solve fractional order nonlinear differential equations that arise in related fields of engineering and science. These methods also make clear how to use fractal calculus in real life. Furthermore, the results of this study support the value and significance of fractional operators in real-world applications.