This paper proves that the monotone iterative method is an effective method to find the approximate solution of fractional nonlinear Langevin equation involving $\varphi$-Hilfer fractional derivative with multi-point boundary conditions. First, we apply a approach based on the properties of the Mittag-Leffler function to derive the formula of explicit solutions for the proposed problem. Next, by using the fixed point technique and some properties of Mittag-Leffler functions, we establish the sufficient conditions of existence of a unique solution for the considered problem. Moreover, we discuss the lower and upper explicit monotone iterative sequences that converge to the extremal solution by using the monotone iterative method. Finally, we construct a pertinent example that includes some graphics to show the applicability of our results.

In this study, we presented the conformable Laplace transform iterative method to find the approximate solution of the systems of nonlinear temporal-fractional differential equations in the sense of the conformable derivative. The advantage of the suggested approach was to compute the solution without discretization and restrictive assumptions. Three distinct examples were provided to show the applicability and efficacy of the proposed approach. To examine the exact and approximate solutions, we utilized the 2D and 3D graphics. Furthermore, the outcomes produced in this study were consistent with the exact solutions; hence, this strategy efficiently and effectively determined exact and approximate solutions to nonlinear temporal-fractional differential equations.