Analytical Solutions for Advanced Functional Differential Equations with Discontinuous Forcing Terms and Studying Their Dynamical Properties

Abstract: This paper aims to find new analytical closed-forms to the solutions of the nonhomogeneous functional differential equations of the n order with finite and constants delays and various initial delay conditions in terms of elementary functions using Laplace transform method. As well as, the definition of dynamical systems for ordinary differential equations is used to introduce the definition of dynamical systems for delay differential equations which contain multiple delays with a discussion of their dynamical properties: The exponential stability and strong stability.


Introduction:
In electrical engineering, in which electronic components are often controlled by on-off switches, discontinuous forcing functions are the norm. Also, in physics, forces often change suddenly and are best described by discontinuous functions. A useful way for representing discontinuous functions is in terms of the unit step function (1). It is well known that the step function is discontinuous at the origin, but that is not necessary in the signal theory. The step function is an important tool for testing and introducing other signals such as multiplying shifted step functions by other different shifted step functions (2).
A bat hitting a ball and two billiard balls colliding are impulsive forces occur at nearly an instant of time. To deal with these types of forces mathematically, the impulse function (Dirac delta function or unit impulse) is defined (1). In engineering, the idea of an action occurring at a point is dealt with. Whether it is a force at a point in space or a signal at a point in time, it becomes a useful way to develop a quantitative definition for this phenomena. This leads us to the notion of a unit impulse, probably the second most essential function, in systems and signals (2). The delay differential equations (DDEs) are usually used in many mathematical, physical and engineering models. Several methods have been used to solve some of them by using numerical methods, others by using analytical methods. Many researchers used the Lambert W function to obtain solutions of the DDEs (3)(4)(5)(6). Recently, Abdullah and et al. found analytical solutions of retarded dynamical systems of the third order and of the n th order by using Lambert W function and a discussion of their stability in their two papers (7)(8) respectively. Abdullah and et al. found approximate characteristic roots for DDEs with multiple delays via the method of spectral tau (9).
A few researchers have worked on finding analytical solutions without using Lambert W function such as Pospíšil and Jaroš who used the unilateral Laplace transform to introduce a closedform formula for a solution of a system of nonhomogeneous linear delay differential equations with a finite number of constant delays (10). In this paper, many rules for finding analytical solutions of nonhomogeneous DDEs are obtained using Laplace transform without using the nonelementary Lambert W function. In addition, some of dynamical systems are constructed with a discussion of their stability.
Basic Concepts Definition 1 (11) A dynamical system is a map X X G :   t  satisfies: (1) X X : 0   is the identity; that is (2) The composition In this work In the case R G  the dynamical system is called flow and in the case Using the assumption with the initial delay condition:  is the Laplace transform of () gt , ( ) 0 hs  . Then, the solution of the equation (1) is: . ) (

If
 , the solution of the delay differential equation (1) is:  , the solution of the delay differential equation (1) is:   The proofs of 1, 2 and 4 are similar to the proof of 3, so they are omitted.

Analytical Solutions of New Forms of Advanced Differential Equations With Multiple Delays
In this section, analytical solutions for many new forms of nonhomogeneous DDEs have been found with Heaviside functions and Dirac delta functions and multiple delays and is differentiable on with the initial delay condition: where b and c are constants.

2.
The solution of the DDE: with the initial delay condition: 3. The solution of the DDE: with the initial delay condition:

4.
The solution of the DDE: with the initial delay condition: with the initial delay condition: with the initial delay condition: By equation (3) . Since the DDE (14) is in the form of (8), the solution of the DDE (14) is given as follows: In a similar manner, many DDEs and their analytical solutions can be introduced as shown in the following Table 1: Table 1. DDEs and the corresponding closed-forms formulas. 1 ),  In this section, the previous results are used to construct dynamical systems resulting from nonhomogeneous DDEs with multiple delays and discussing some of their properties (exponentially stable and strongly stable) with the following initial delay conditions: Let us formulate our problem by considering the first order DDE: where , 0  j T with the initial delay condition: