Discrete and Continuous Multiplicative Differential Equations and Applications in Solving Non−Linear Difference and Differential Equations

Boundary and initial value problems, including nonlinear difference equations and nonlinear differential equations, are the mathematical models of many physics and engineering problems and natural phenomena. Usually, due to the lack of a solid theory for solving these types of equations, these equations are solved by using numerical and approximate methods. In this paper, first some elementary and basic definitions and concepts of discrete and continuous multiplicative calculus are given. Next we apply some ideas and methods to obtain invariant functions with respect to their associated derivative. These invariant functions are used to solve several types of nonlinear difference and differential equations that have appeared in natural sciences and physical problems. After that, these methods are expanded for solving nonlinear difference and differential equations through discrete and continuous multiplicative differential equations. Finally, some applications of multiplicative forms of differential equations are given which simplify numerical methods for solving nonlinear biological problems and exponential approximations for nonlinear functions.