Equations of dynamics. Three domains in series

In numerous cases, the determination of the output function with a given accuracy requires that the two-domain cascading system should be enlarged with an additional first-order inertial object (domain). 

A cascading system composed of three domains in series can be described by the following set of equations of dynamics  

As before, Eqs (4.49)-(4.51) are assumed to be general differential equations of dynamics describing the courses of the temperature changes in time in the domains considered. In Eq. (4.49)/(t) is the input function. It may be assigned an arbitrary course. Function T (t) is the domain 1 output function, which at the same time constitutes the input function for domain 2. The output function of domain 2 constitutes the input function of domain 3, whose output function is T-i(t). 

For the solution of the above system of differential equations, the Laplace transform method was used, yielding 

For a forcing function described by Eq. (4.45) and initial conditions described by Eq. (4.55), the temperature changes T-i(0 are expressed by the equation 

---