A Study of Deviating Arguments in Existence Differential equations of Solutions

Derivatives and integrals are used in mathematical equations to describe the most fascinating natural phenomenon. If the equations are integral or differential, they are referred to as such. There are many distinct types of differential equations that arise in numerous fields, for as in the fields of biology and engineering. When we find a differential equation while analyzing a given system, it is referred to as modeling the system using differential equations. Ordinary differential equations are a common technique in modeling. ODE models presume that systems are independent of their previous states and that their future states are only dependent on the current state. In ODE models, the assumption is made that the system and its subsystems interact instantly and without any delay. Realistic models, on the other hand, incorporate a small bit of lag. As a result, the past, present, and sometimes the derivative of the past states must be taken into account while determining the future state. The functional differential equations (FDEs) represent these models (FDEs). Many models are better characterized by FDEs rather than ODEs because we presume that the system's history has an impact on its future state. The simplest form of FDEs are known as delay differential equations (DDEs). Because the unknown function's derivative depends on the past, they are known as differential equations with retarded arguments. It is natural to extend the DDEs to include NDDEs, in which the unknown function is derived at the delayed argument. Some special types of differential equations have intriguing features, and one of these types is differential equations with piecewise constant or continuous arguments. (EPCA). There have been several studies on EPCA, which are a hybrid of difference equations and differential equations, that is, a system that can have both discrete and continuous delays. [40]

Using Newton's second law of motion, an initial value problem for a second order ordinary differential equation can be stated for a basic problem of identifying the location of a damped harmonic oscillator in relation to time under the t −→ time variable, t ≥ 0. x(t) −→ position or displacement of the oscillator, a bounded and infinitely differentiable function of t. x’(t) −→ velocity of the oscillator. X’’(t) −→ acceleration of the oscillator. m −→ mass of the oscillator. −2mx −→ restoring force acting on the oscillator. −2mx’(t) −→ damped force acting on the oscillator. me−t −→ external force acting on the oscillator. x(0) = 1 −→ initial displacement. x0(0) = 0 −→ initial velocity. Then the initial value problem is By Picard's theorem, the initial value issue has a single solution because the coefficients of x'(t), x(t), and et are all continuous functions. First, the problem can be reduced to a simpler problem without the damping term as follows::

Let us apply Adomian decomposition series [1]-[3] on both sides of with the initial condition to get

Let us apply Laplace transform on both sides of Hence we get the desired exact solution

Let us apply decomposition series technique in the Laplace transform method, we take straightaway the equation The Laplace decomposition series [5] can be taken as We can obtain an iteration to compute L {yn(t)} as follows :

Hence by using Laplace decomposition series for L {y(t)}, we arrive at and hence the desired solution is .

Differential equations with diverging arguments provide the finest description of realistic models. There are two fundamental issues in studying mathematical models: the existence of solutions and their uniqueness. Deviating arguments and unique solutions are two of the main topics covered in this chapter, so we'll focus on those in this section. Illustrations include the following examples. We consider the following NDDE: condition on f is relaxed, and the existence of a solution is proven. LEMMA 1.x(t) is a solution of (3.1), (3.2) on J if and only if x(t)is a solution of Note: In what follows ||.|| means Eucledian norm unless otherwise specified. THEOREM 3.2.2. Suppose that f satisfies the following: Series (3.8) likewise converges (uniformly) on [0, β] according to the comparison test, which shows that the series to the right of (3.10) also converges. Let x be the sum of the (3.8) series (t).

It is shown in this section that the solution x(t) of (3.1) is continually dependent on the beginning conditions under which we get sufficient conditions. We also consider the following equation in addition to (3.1): Applying Gronwall’s inequality, it follows that Choosing suitable δ, it is easy to see that ||x(t) − y(t)|| < e for t∈ [0, β1]. This completes the proof of the theorem. THEOREM 3.3.2. Consider (3.1) where f(t,y(t),y([t]),y 0 ([t])) satisfies the assumptions of Theorem 3.2.2. Let x(t) be solution of (3.1) on [0, β] with initial condition x(0) = x0 for t = 0 and y(t) be solution of (3.1) on [0, β ∗ ] with initial condition y(0) = y0 for t = 0. Moreover, if β1 = min{β, β ∗}, then for e > 0, there exists a δ(e) > 0 such that ||x0 − y0|| < δ implies The proof is similar to Theorem 3.3.1 with function g = 0

Mathematical equations use derivatives and integrals to describe some of nature's most amazing phenomena. These equations can be categorized as either differential or integral equations, depending on the case. Differential equations, both linear and nonlinear, appear often in the physical, biological, social, and engineering sciences. When we discover a differential equation when researching a system, we are referring to the system's differential equation modeling. Odd-order delay differential equations have been given more attention than even-order ones in the study of oscillatory behavior, contrary to popular opinion. This study focuses on odd-order equations with

If you're trying to deal with FDEs, you have to consider the history. First research into FDEs was done during Euler's investigation on the general shape of curves that are identical to their own evolutes. FDEs were studied in the eighteenth and nineteenth century by Bernoulli, Laplace, Poisson, and Condorcet. You should know all of these names. Schmidt studied differential-difference equations in 1911, when they were first introduced. He made a significant contribution to the study of interactions between species by studying changes and oscillations in animal populations in 1928, and later wrote a book about the importance of history in 1931. For the first time in 1977, a comprehensive theory of differential equations with trailing arguments that are piecewise continuos or constant was proposed by A. D Mykhus, arguing that a comprehensive theory was needed. The first piecewise constant argument mathematical model was devised by Busenberg and Cooke in 1982 when they were investigating the biomedical problem of vertically transmitted illnesses. Differential equations with piecewise constant arguments have grown tremendously since then, with various researchers and references referenced]. Neuronal networks, lossless transmission, nuclear reactors and population dynamics are just a few of the fields in which it is widely employed.

[1] Aftabizadeh, A. R., & Wiener, J. (1988). Oscillatory and periodic solutions for systems of two first order linear differential equations with piecewise constant argument. Applicable Analysis, 26(4), 327-333. [2] Agarwal, R. P., Bohner, M., Li, T., & Zhang, C. (2013). A new approach in the study of oscillatory behavior of even-order neutral delay differential equations. Applied Mathematics and Computation, 225, 787-794. [3] Ahmad, S., &Rao, M. R. M. (1999). Theory of Ordinary Differential Equations: With Applications of Biology and Engineering. Affiliated East-West Private Lmt.. [4] Ahmed, F. N., Ahmad, R. R., Din, U. K. S., &Noorani, M. S. M. (2013). Oscillation criteria of first order neutral delay differential equations with variable coefficients. In Abstract and Applied Analysis (Vol. 2013).Hindawi. [5] Ahmed, F. N., Ahmad, R. R., Din, U. K., &Noorani, M. S. (2014). Oscillations for neutral functional differential equations.In [6] Bainov, D. D., &Mishev, D. P. (1991). Oscillation theory for neutral differential equations with delay.CRC Press. [7] Bellman, R. E., &Kalaba, R. E. (1965). Quasilinearization and nonlinear boundary-value problems.American Elsevier, New York. [8] Busenberg, S., & Cooke, K. L. (1982). Models of vertically transmitted diseases with sequentialcontinuous dynamics.In Nonlinear phenomena in mathematical sciences (pp. 179-187). [9] Candan, T. (2016). Existence of non-oscillatory solutions to first-order neutral differential equations.Electron. J. Diff. Equ, 39, 1-11. [10] Chaplygin, S. A. (1954). Collected papers on mechanics and mathematics.Moscow, 1, 954. [11] Chang, M. H., Pang, T., &Pemy, M. (2010). An approximation scheme for Black-Scholes equations with delays. Journal of Systems Science and Complexity, 23(3), 438-455.

Research Scholar, Sunrise University, Alwar, Rajasthan