In this paper‎, ‎a numerical scheme based on shifted Vieta-Lucas polynomials is utilised to solve mentioned equation‎. ‎The main characteristic of the presented method is to approximate Brownian motion with help of the Gauss-Legendre quadrature‎, ‎which makes calculations easier‎. ‎Another characteristic of this method are employed suitable collocation points to convert the stochastic equation under the study into a system of algebraic equations by using the operational matrices‎. ‎So that‎, ‎Newton's method is applied to solve them‎. The convergence analysis and error bound of the suggested method are well established‎. ‎Additionally‎, ‎the proofs related to the existence and uniqueness of the solutions for the equations under investigation have been provided‎. ‎In order to illustrate the effectiveness‎, ‎compatibility and plausibility of the proposed technique‎, ‎four numerical examples are presented.

In this paper, a type of time-Fractional Fokker-Planck equation (FPE) of the Ornstein-Uhlenbeck process is solved via Riemann-Liouville and Caputo derivatives. An analytical method based on symmetry operators is used for , nding reduced form and exact solutions of the equation. A numerical simulation based on the M, untz-Legendre polynomials is applied in order to , nd some approximated solutions of the equation.

The ultimate goal of this performance study is to provide a  proposed scheme for solving the time-Fractional stochastic advection-diffusion equation (TFSADE) of order $\alpha (0\le \alpha <1)$. In this proposed scheme, we utilize an approach based on cubic trigonometric B-spline collocation methods (CTBSCM).  In this study, we replace the existing Fractional derivative with the Fractional Caputo derivative for time discretization and then replace the first and second derivatives of the equation using cubic trigonometric B-spline functions for spatial discretization. Applying this proposed scheme to TFSADE causes the equation to reduce to the linear system. In the end, the examples show that the order of convergence of the proposed method is $O(\tau ^{2-\alpha}+h^2)$ where $h$ and $\tau$  are the spatial and time step lengths, respectively.

In this paper, we , nd an integral representation for the fundamental solution of the Fractional Ostrovsky equation in terms of the Airy and Bessel-Wright functions. The equation is studied in the sense of the Weyl Fractional derivative and the solu-tion is presented as the Airy transforms of Wright functions. Using the asymptotic expansion of Wright function the asymptotic behavior of solution is also discussed.

In this study, we construct the Fractional-order Bernstein wavelets for solving stochastic Fractional integro-differential equations. Fractional-order Bernstein wavelets and their properties are presented for the first time.  The Fractional integral operator of Fractional-order Bernstein wavelets together with the Gaussian integration method is applied to reduce stochastic Fractional integro-differential equations to the solution of algebraic equations which can be simply solved to obtain the solution of the problem.  Also, an error estimation for our approach is introduced.  The numerical results demonstrate that our scheme is simply applicable, efficient, powerful and very precise at the small number of basis functions.

In this article, we study a new nonlinear Langevin equation of two Fractional orders with Fractional boundary value conditions which is a generalization of previous Langevin equations. Based on Banach and Schauder fixed point theorems, the existence and uniqueness of solutions of this equation are investigated. Moreover, our hypotheses are simpler than similar works.

Recently Fractional cable equation has been investigated by many authors who have applied it in various areas. Here we introduce and investigate a generalized space-time Fractional cable equation associated with Riemann-Liouville and Hilfer Fractional derivatives. By mainly applying both Laplace and Fourier transforms, we express the solution of the proposed generalized Fractional cable equation as H-functions. The main results here are general enough to be specialized to yield many new and known results, only several of which are demonstrated in corollaries. Finally, we consider the moment of the Green function with its several asymptotic formulas.

In this paper, we intend to solve special kind of ordinary differential equations which is called Heun equations, by converting to a corresponding stochastic differential equation (S.D.E.). So, we construct a stochastic linear equation system from this equation which its solution is based on computing fundamental matrix of this system and then, this S.D.E. is solved by numerically methods. Moreover, its asymptotic stability and statistical concepts like expectation and variance of solutions are discussed. Finally, the attained solutions of these S.D.E.s compared with exact solution of corresponding differential equations.