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.

We present a class of higher-dimensional Hilbert-type inequalities on a fractal set $(\mathbb{R}_+^{\alpha n})^{k}$. The crucial step in establishing our results are higher-dimensional spherical coordinates on a fractal space. Further, we impose the corresponding conditions under which the constants appearing in the established Hilbert-type inequalities are the best possible. As an application, our results are compared with the previous results known from the literature.

We are exploring solitons and other solutions describe new interaction between two solitons besides, new three soliton solutions are generated. we examine the commutative product between multi unknown Lie infinitesimals for the (2+1) dimensional variable-coefficients Bogoyavlensky Konopelchenko (VCBK) equation and this study result some new Lie vectors. The commutative product generates a system of nonlinear ODEs which had been solved manually. Through two stages of Lie symmetry reduction, (VCBK) equation is reduced to non-solvable nonlinear ODEs using various combinations of optimal Lie vectors. Using the Integration method, we investigate new analytical solutions for these ODEs. Back substituting to the original variables generates new solitons and other solutions for (VCBK). Some selected solutions illustrated through three-dimensional plots.

In this paper we apply the fractional integrals with arbitrary order depending on the fractional operators of Riemann type (ABR) and Caputo type (ABC) with kernels of Mittag Lefller in three parameters $E_{\alpha,\mu}^\gamma(\lambda,t)$ for solving the time fractional parabolic nonlinear equation. We utilize these operators with homotopy analysis method (HAM) for constructing the new scheme for generating the successive approximations. This procedure are used successfully on two examples for finding the solutions. The effectiveness and accuracy are verified by clarifying the convergence region in the $h$-curves as well as by calculating the residual error and the results were accurate. Depending on this results, this treatment can be used to find the approximate solutions to many fractional differential equations.

Aim of the presented study is to investigate the numerical solution of fifth-order nonlinear Lane-Emden (LE) based singular system at the origin with different shape factors developed on the analogous pattern of standard second order LE equations. The stochastic neuro-evolution computing is exploited for numerical outcomes by using the artificial neural networks (ANNs) for applicable mapping and learning of decision variables with integrated meta-heuristics of genetic algorithms (GAs) for global search aided with the rapid local search of active-set (AS) i.e., ANN-GA-AS algorithm. The designed numerical computing approach ANN-GA-AS implemented effectively for solving fifth-order nonlinear LE singular system and results of statistical assessments further authenticate the accuracy, convergence, and stability.

This paper aims to construct new mixed-type periodic and lump-type solutions via the dependent variable transformation and the Hirota’s bilinear form (general bilinear techniques). This study will be investigated by considering the (3+1)-dimensional generalized B-type Kadomtsev-Petviashvili equation which describes the weakly dispersive waves in a homogenous medium in fluid dynamics. The obtained solutions contain abundant physical structure. Consequently, the dynamical behaviors of these solutions are graphically discussed for different choices of the free parameters through 3D- and contour plots.

In this article, we use a fuzzy number in its parametric form and transform a non-linear fuzzy integral equations to its parametric form of the second kind as in the crisp case. The main focus is to solve the fuzzy non-linear integral equations for semi analytical solutions. The suggested treatment are presented for the solution of respective fuzzy non-linear integral equations including fuzzy non-linear Fredholm integral equation, fuzzy non-linear Volterra integral equations and fuzzy non-linear singular integral equation of Able's type kernel via an hybrid method of integral transform and decomposition technique. The proposed method is illustrated in details by solving few examples.

In this paper, we have considered two hybrid systems of FDEs for the existence results and stability criteria. The problems are more general than considered in the literature and to the best of our study in the field, no one considered singular hybrid-FDE with p-Laplacian operator for the existence and stability results. We utilize the well known Guo-Krasnoselskii's fixed point theorem.

In this paper a new so called Iterative laplace transform method is implemented to investigate the solution of certain important population models of non-integer order. The iterative procedure is combined effectively with Laplace transformation to develop the suggested methodology. The Caputo operator is applied to express non-integer derivative of fractional-order. the series form solution is obtained having components of convergent behavior toward the exact solution. For justification and verification of the present method some illustrative examples are discussed. The closed contact is observed between the obtained and exact solutions. Moreover, the suggested method has small volume of calculations and therefore it can be applied to handle the solutions of various problems with fractional-order derivatives.