**Stochastic Differential Equations**

A diffusion are often thought of as a powerful Markoff process (in ℝn) with continuous ways. Before the event of Itô’s theory of random integration for pedesis, the first methodology of finding out diffusions was to check their transition semigroups. This was adore finding out the small generators of their semigroups, that square measure partial differential operators. so Feller’s investigations of diffusions (for example) were really investigations of partial differential equations, galvanized by diffusions.** [1]**

**User’s guide to viscosity solutions of second order partial differential equations**

The notion of body solutions of scalar totally nonlinear partial differential equations of second order provides a framework within which surprising comparison and individuality theorems, existence theorems, and theorems regarding continuous dependence could currently be tried by terribly economical and putting arguments. The vary of vital applications of those results is big. this text may be a self-contained exposition of the essential theory of body solutions. **[2]**

**The Painlevé property for partial differential equations**

In this paper we have a tendency to outline the Painlevé property for partial differential equations and show however it determines, in an exceedingly remarkably straightforward manner, the integrability, the Bäcklund transforms, the linearizing transforms, and also the Lax pairs of 3 well‐known partial differential equations (Burgers’ equation, KdV equation, and also the changed KdV equation). this means that the Painlevé property could offer a unified description of integrable behavior in resurgent systems (ordinary and partial differential equations), while, at an equivalent time, providing AN economical methodology for determinant the integrability of explicit systems. **[3]**

**Analog simulator of integro-differential equations with classical memristors**

An analogue computer makes use of ceaselessly changeable quantities of a system, like its electrical, mechanical, or hydraulic properties, to resolve a given drawback. whereas these devices are sometimes computationally a lot of powerful than their digital counterparts, they suffer from analog noise that doesn’t give error management. we’ll specialize in analog computers supported active electrical networks comprised of resistors, capacitors, and operational amplifiers that are capable of simulating any linear standard equation. However, the category of nonlinear dynamics they will solve is proscribed.** [4]**

**A Substitution Method for Partial Differential Equations Using Ramadan Group Integral Transform**

In this paper we tend to introduce the conception of Ramadan cluster integral remodel substitution (RGTS) technique to resolve some sorts of Partial differential equations. This new technique may be a convenient thanks to notice precise resolution with less process price as compared with technique of separation of variables (MSV) and variation iteration technique (VIM). The planned technique solves linear partial differential equations involving mixed partial derivatives. **[5]**

**Reference**

**[1]** Protter, P.E., 2005. Stochastic differential equations. In Stochastic integration and differential equations (pp. 249-361). Springer, Berlin, Heidelberg. (Web Link)

**[2]** Crandall, M.G., Ishii, H. and Lions, P.L., 1992. User’s guide to viscosity solutions of second order partial differential equations. Bulletin of the American mathematical society, 27(1), (Web Link)

**[3]** Weiss, J., Tabor, M. and Carnevale, G., 1983. The Painlevé property for partial differential equations. Journal of Mathematical Physics, 24(3), (Web Link)

**[4]** Analog simulator of integro-differential equations with classical memristors

G. Alvarado Barrios, J. C. Retamal, E. Solano & M. Sanz

Scientific Reportsvolume 9, Article number: 12928 (2019) (Web Link)

**[5]** Ramadan, M. A., Raslan, K. R., Hadhoud, A. R. and Mesrega, A. K. (2017) “A Substitution Method for Partial Differential Equations Using Ramadan Group Integral Transform”, Asian Research Journal of Mathematics, 7(4), (Web Link)