Oscillatory Behavior of Euler Cauchy Equation with Heaviside Step Function of a Bulge Function Over Various Time Domain

Corresponding Author: Jitendra Kumar Pati Department of Math, C.V. Raman Global University, Bhubaneswar, Odisha, India Email: jkpati2015@gmail.com Abstract: In this study authors discuss oscillatory property of Euler Cauchy Equation with heaviside step function of a bulge function in respect of homogenous and non homogenous output over different time intervals through general solution using Transform method.


Introduction
As we know, Laplace transform is devised to solve linear ordinary differential equations with constant coefficient as well as variable coefficients. Here we apply the Laplace transform method to solve nonhomogeneous third order differential equation with right side as a step function called as bulge function.
The Heaviside step function, or the unit step function, usually denoted by H is a step function, named after Oliver Heaviside , the value of which is zero for negative arguments and one for positive arguments.
The function defined as follows known as Heaviside step function of a bulge function: The step function of a bulge function is usually known as Heaviside step function of a Bulge function.
Using unit step function, it is defined as: where, f(t) is defined in (1.1).

Preliminary Notes
Here we introduce some lemmas without proof whose results are useful in further section.

Lemma 2.1
The Laplace transform of the bulge function

Lemma 2.3
The Laplace transform of function f(t) as in ( 1.1)

Main Result
Here the Non Homogenous third order Euler Cauchy equation with Heaviside step function of a Bulge function is solved.      2 Similarly:

M y s A s B s D s aD s bs C s D s E s F s
So, using inverse Laplace transform we obtain: In simplifying we have: So the final solution is:

Comparative Study of Oscillatory Property of Euler Cauchy Equation
In this section, the authors desire to have a comparative study of oscillatory property of Euler Cauchy equation with homogenous and Non homogenous output. Take the homogenous equation as: The graph of the solution is in Fig. 1.
The equation seems to be oscillatory within the interval and converging towards oscillation. Going within the range [0,200] in Fig. 2, we find the equation is 37 oscillatory and seems to be converging towards oscillation within the t value from 50 to 100.
In discussion for a non homogenous equation, we have the Euler Cauchy equation with bulge function as: 1 32 2 , 0 1, 0 2, 0 3 t t y t y ty y e y y y The graph of the solution is in Fig. 3. It seems to be oscillatory within t value from 20 to 40 (refer Fig. 3).
Similarly within the range [0,200], the graph of solution is in Fig. 4. It seems to be oscillatory within t value from 50 to 100.
The graph of the solution for l = 1, ξ = 1 within the range [0,100] is in Fig. 5.
Within the range [0,200] the graph of solution is in Fig. 6. It seems to be oscillatory within t value from 50 to 100. Within the range [-5000,5000] the graph of solution is in Fig. 7.
It also seems to be oscillatory. If the range is extended to [-10,000,10,000], we have the graph in Fig. 8.
Graph of the equation is in Fig. 9. It is strongly oscillatory within the range [4,6].
Similarly within the range [-100,100], the graph of equation is in Fig. 10.
It is also strongly oscillatory within the range [0,100] and varies constantly towards the oscillation. Similarly within the range [-200,200], the graph of equation is in Fig. 11.
It is also oscillatory within the range [0,150] and at the point t = 100, the graph varies constantly towards the oscillation.
Similarly within the range [-500,500], the graph of equation is in Fig. 12.
Here also we conclude that it is oscillatory.
Since the Equation (4.1) is oscillatory and the Euler Cauchy equation with heaviside step function of a bulge function is also oscillatory as per the Fig. 5 to 8, so the solution (3.7) satisfies as per the condition of oscillation.
Here the author wants to point out that the oscillatory behavior of Euler Cauchy equation is really dirac(x) +…+ heaviside(x-1) sin(x-1) exp(x-1) 42 good by taking the non homogenous output as heaviside step function of a bulge function, since it includes discontinuous functions like Dirac delta and unit step function. The presence of these kind of functions helps to measure the oscillatory behavior up to a period of time.

Conclusion
In this article, authors discuss about solution of nonhomogeneous Euler Cauchy equation with heaviside step function of a bulge function in different time domain and at last there is a comparative study of oscillatory behavior of the concerned equation with homogenous and non homogenous output.
The non homogenous equation with right hand side as piecewise continuous function like heaviside step function, causes strongly oscillatory behavior in compare to homogenous output.
This process of comparative study may be applicable to check the oscillatory and non oscillatory behavior of equation concerned to mass spring system with damping and without damping. In the near future, we intend to conduct more research as a continuation of this work.