In this paper, we consider the Space-Time Fractional Advection-Dispersion equation on a finite domain with variable coefficients. Fractional Advection- Dispersion equation as a model for transporting heterogeneous subsurface media as one approach to the modeling of the generally non-Fickian behavior of transport. We use a semi-analytical method as Reproducing kernel Method to solve the Space-Time Fractional Advection-Dispersion equation so that we can get better approximate solutions than the methods with which this problem has been solved. The main obstacle to solve this problem is the existence of a Gram-Schmidt orthogonalization process in the general form of the reproducing kernel method, which is very time-consuming. So, we introduce the Improved Reproducing Kernel Method, which is a different implementation for the general form of the reproducing kernel method. In this method, the Gram-Schmidt orthogonalization process is eliminated to significantly reduce the CPU-time. Also, the present method increases the accuracy of approximate solutions.

This paper aims to solve the celebrated Fuzzy Fractional Differential Equations (FFDE) using an Artificial Neural Network (ANN) technique. Compared to the integer order differential equation, the proposed FFDE can better describe several real application problems of various physical systems. To accomplish the aforementioned aim, the error back propagation algorithm and a multi-layer feed forward neural architecture are utilized using the unsupervised learning in order to minimize the error function as well as the modification of the parameters such as weights and biases. By combining the initial conditions with the ANN, output provides an appropriate approximate solution of the proposed FFDE. Then, two illustrative examples are solved to confirm the applicability of the concept as well as to demonstrate both the precision and effectiveness of the developed method. By comparing with some traditional methods, the obtained results reveals a close match that confirms both accuracy and correctness of the proposed method.

In many mathematical types of research, in order to solve the fuzzy fractional differential equations, we should transform these problems into crisp corresponding problems and by solving them the approximate solution can be obtained. The aim of this paper is to present a new direct method to solve the fuzzy fractional differential equations without this transformation. In this work, the fuzzy generalized Taylor expansion by using the sense of fuzzy Caputo fractional derivative for fuzzy-valued functions is presented. For solving fuzzy fractional differential equations, the fuzzy generalized Euler's method is applied. In order to show the accuracy and efficiency of the presented method, the local and global truncation errors are determined. Moreover, the consistency, the convergence and the stability of the generalized Euler's method are proved in detail. Eventually, the numerical examples, especially in the switching point case, show the flexibility and the capability of the presented method.