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.

This paper presents the introduction of two novel equation types: the partial hesitant fuzzy equation and the half hesitant fuzzy equation‎. Additionally, ‎ an efficient method is proposed to solve these equations by defining four solution categories: Controllable‎, ‎Tolerable Solution Set (TSS)‎, Controllable ‎Solution Set (CSS)‎, ‎and Algebraic Solution Set (ASS)‎. ‎ Furthermore, ‎ the paper establishes eight theorems that explore different types of solutions and lay out the conditions for the existence and non-existence of hesitant fuzzy solutions‎. ‎ The practicality of the proposed method is demonstrated through numerical examples.

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.

The Zakharov equation is a nonlinear plasma fluid model, used for ion-acoustic waves in a magnetized plasma. In the present study, Langmuir waves of the dimensionless Zakharov equation are investigated by using the Sardar-subequation method. The obtained solutions lead to a variety of exact solutions in the form of dark, bright, periodic singular,  singular, and combined dark-bright type solutions. These acquired solutions are depicted graphically by the 2D, contour and 3D plots which show the physical behaviour of obtained solutions. All the graphs confirm the validity of the obtained solutions. These types of solutions have a large range of applications in mathematical and applied sciences.