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Abstract
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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
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