‎    This paper introduces a modified version of the Variational Iteration Method, incorporating $\mathbb{P}$-transformation. We propose a novel semi-analytical technique named the modified variational iteration method   for addressing fractional differential equations featuring Tempered Liouville-Caputo derivatives. The modified variational iteration method emerges as a highly efficient and powerful mathematical tool, offering exact or approximate solutions for a diverse range of real-world problems in engineering and the natural sciences, specifically those expressed through differential equations. To assess its effectiveness and accuracy, we scrutinize the modified variational iteration method by applying it to three problems related to the heat-like multidimensional diffusion equation with a fractional time derivative in a Tempered Liouville-Caputo form.

In this paper, we investigate the solutions of a class of-Hilfer fractional differential equations with the initial values in the sense of-fractional integral by using the successive approximation techniques. Next, the continuous dependence of a solution for the given Cauchy-type problem is presented.

In this study, beta-derived Sturm–Liouville problems are discussed. First, the existence and uniqueness problem for such equations is discussed. Then, self-adjointness is obtained with the help of boundary conditions. Eigenfunction expansion was obtained with the help of characteristic determinants and Green’s function. Finally, an example is given showing the theoretical results obtained.

Integral and differential of fractional order are important notion that is often used in dealing with Frechet geometry. Based on pseudo-operations given by monotone and continuous function g, we study pseudo-derivative and pseudo-integral of fractional order through this paper.

In this paper, we study linear fractional fuzzy differential equations involving the Caputo generalized Hukuhara derivative. Using the fuzzy Laplace transform, we present the general form of solutions in terms of Mittag-Leffler functions. Finally, some examples are provided to illustrate our results.

The existing Muskingum models vary only in the form of the storage equation. So far, for accounting the nonlinearity characteristic of flood wave, the form of the storage equation has been modified based on the experience and in a trial and error manner in order to capture the overall flood propagating characteristics more accurately. One of the theoretical based method for considering the nonlinear characteristics in Muskingum model is to use the fractional derivative in the continuity differential equation. This study presents a new Muskingum model in which the linear storage equation and the continuity equation with fractional derivative are used. As shown in this study, the new model can simulate flood waves with both linear and nonlinear behaviors. The proposed Muskingum model with fractional derivative order was implemented and tested on three different sets of flood data. The results of this study indicate that the proposed Muskingum model improves the results and estimates the flood wave characteristics more accurately than the traditional linear Muskingum models.

In the paper, an oscillatory system with liquid dampers is considered, when the mass of the head is large enough. By means of expedient transformations, the equation of motion with fractional derivatives is reduced to an equation of fractional order containing a small parameter. The corresponding nonlocal boundary value problem is solved and the zero and first approximations of solutions of the relative small parameter are constructed. The results are illustrated on the concrete example, where the solution differs from the analytical solution by 10−, 2 order.

In this paper, we present a BOX mathematical model for the release of COVID-19. We intend to generalize the model to fractional order derivative in Atangana-Baleanu sense and to show the existence of solution for the fractional model using Schaefer's , xed point theorem and for the uniqueness of solution we make use of Banach , xed point theorem. By using Shehu transform and Picard successive iterative procedure, we explore the iterative solutions and its stability for the considered fractional model. Given the beginning of a new wave of COVID-19 spread in Indonesia, we present a numerical simulation to study and predict the spread of the disease in this country.