Differential equations including a quantity and its derivatives probably are the fundamental tool of modeling in all fields of engineering, mathematics, physics, biology, economics, etc. For instance, the well-known Newton’s second law of motion relates the force to the acceleration that is the second derivative of the position. Differential equations have been classified into different types as follows. Partial differential equations (PDEs) involve rates of change concerning continuous parameters for describing the behavior of dynamical systems that take place infinite-dimensional configuration space. Fractional-order differential equations (FDEs) are a type of differential equations whose orders are relaxed to accommodate fractional numbers. This alteration results in the more realistic modeling of natural dynamical processes with memory or hereditary properties. Besides, delay differential equations (DDEs) occupy a place of central importance in all areas of science with a multitude of practical applications. They are used in time-delayed systems such as high-speed machining, communication, power systems, and control systems. Recent studies show that enhancing differential equations with nonlocal operators, such as fractional-order operators, delays, and partial derivatives, results in possessing unique mathematical tools that model natural phenomena especially with infinite dimensions in the highest level of satisfaction. One can remark the importance of this field by observing the exponential increase in the number of research articles on this subject, as well as the expansion of new areas for nonlocal operator applications in the last few years. PDEs often describe Infinite-dimensional dynamical systems with an infinite number of roots in their characteristic equations. A meaningful use of the nonlocal operators, i.e., fractional-order operators and delays, can be in stabilizing infinite-dimensional dynamical systems governed by PDEs. They can inject an infinite number of roots to the characteristic equation of infinite-dimensional dynamical systems, and potentially relocate their unstable poles and stabilize them.

Although enhancing PDEs with nonlocal operators is phenomenal to model natural phenomena especially with infinite dimensions, the initialization, realization, and integration of them are still considered as open problems. Furthermore, two main burdens of employing these remarkable tools in practical problems are (1) missing computational methods that handle a large amount of data and (2) the lack of well-established control theories for dynamical systems with nonlocal operators. Moreover, integration is the cornerstone of solving differential equations used in almost all the fields that remain as the core function in all dynamical systems’ simulations. They ultimately deliver all the computational algorithms in a way to accomplish proper results. However, PDEs with nonlocal operators require a large physical memory to retain all the states, which is the main challenge in integration. Besides, the state-space realization is the cornerstone of many approaches in control theory. The state-space realization of ordinary differential equations is trivial due to the unique size of their solution space, and hence the size of their minimal state-space realization is equal to their order. However, the previous statement is not exact for differential equations with nonlocal operators since the size of their state-space is not finite. Besides, the initialization of differential equations including a combination of different type of nonlocal operators is one of the most challenging problems in control theory. The construction of any form of state-space forms in a finite dimensional space results in losing the connection between the solution of the original equation with infinite dimension and that of the new representation, which is due to its undetermined initial conditions. This critical issue has been addressed in the literature for decays without proposing any promising solution.