The present article introduces an operational approach based on Modified hat functions to solve the space-time-fractional differential equations in the Caputo sense. In this method, the derivative of the unknown function is considered as a linear combination of Modified hat functions. We use the operational matrix of the Riemann–Liouville fractional integral of Modified hat functions to approximate the Caputo fractional derivative in order to reduce the problem to a system of Sylvester equations. The error of the mentioned method is of the order O(h3). In addition, we examine several  numerical examples to confirm the ability of the proposed approach.

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.

In this paper, we apply spectral method based on the Bernstein polynomials for solving a class of optimal control problems with Jumarie’s Modified Riemann-Liouville fractional derivative. In the first step, we introduce the dual basis and operational matrix of product based on the Bernstein basis. Then, we get the Bernstein operational matrix for the Jumarie’s Modified Riemann-Liouville fractional derivative, which has not been undertaken before. By using the function approximations based on the Bernstein basis and mentioned operational matrices, the optimal control problems with Jumarie’s Modified Riemann-Liouville fractional derivative is reduced to a system of algebraic equations that easily solvable by Newton’s iteration method. We apply the proposed method for solving two examples. The numerical results show that present method is simple in implementation and the approximate solutions are in high accuracy. Some comparisons with other method guarantee that the results are reasonable. Also, the obtained solutions approach to classical solutions as the order of the fractional derivatives approach to 1, as expected.

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.