Analyzing multiplicative noise effects on stochastic dynamical ϕ4 equation using the new extended direct algebraic method

The stochastic dynamical ϕ4 equation is obtained by adding a multiplicative noise term to the classical ϕ4 equation. The noise term represents the random fluctuations that are present in the system and is modeled by a Wiener process. The stochastic dynamical ϕ4 equation is a powerful tool for modeling the behavior of complex systems that exhibit randomness and nonlinearity. It has a wide range of applications in physics, chemistry, biology, and finance. Our goal of this paper is to use the new extended direct algebraic method to find the stochastic traveling wave solutions of the dynamical ϕ4 equation. We explore the new trigonometric, hyperbolic, and rational functions using the new extended direct algebraic method. Furthermore, we use Matlab to plot 3D surfaces of exact solutions to show how multiplicative noise affects the solutions to the stochastic dynamical ϕ4 equation.