A new function, namely the Gamma Interval Function is defined first. The function is used in modelling physically inspired problems of first and second order differential equations. The differential equations corresponding to different subdomains are solved and the solutions are combined to each other by imposing the physical constraints of the problem. It is shown that the solutions can be expressed as a single function with the aid of Gamma Interval Functions. Sample problems are selected in the area of applied mechanics such as dynamics and mechanical vibrations. This new approach presented here may inspire further work related to modelling of the mathematical physics problems.