Applications of Laplace Transform in science and technology

In the present paper, we discussed applications of Laplace Transform to solve the equations that occur in mathematical modeling of various engineering problems like simple electric circuit, in analysis and modeling of Mechanical system, in population growth, in conduction of heat equation (with example), in Bessel’s function and in economics. We interpreted a relation between beta and gamma function and applications to simultaneous differential equations with suitable example using Laplace Transform Method.


Introduction
The important field of Mathematical Analysis is Laplace transformation is referred as integral transforms with wide applications in various fields like engineering technology, basic sciences, and mathematics and in economics. It is also used to find the solution of differential equations at boundary value. Mathematical formulations of most of the engineering problems are in the form of differential equations.
In day to day life we commonly use mathematical models and its applications. Population growth model plays a very important role (see, [3]). Daci [4] used Laplace transform in Mathematical model on population projection in Albania and he experimentally verified how Laplace transform is used in population growth. The simple applications of Laplace transform in engineering fields related to transfer function of mechanical system, nuclear as physics are discussed by Sawant [9]. In [1], Patil explained how present discounted value in finance related to Laplace transforms and the application of time derivative property using Laplace transforms. The Laplace transform theory violates a very fundamental requirement of all engineering systems that is in control system (see. Das [2]).

Relation between Gamma and beta function
Since beta and gamma functions are definite integrals called Euler's integral of first and second kind respectively. They are used to solve definite integrals which we cannot solve or reduce to standard form easily. Relation between beta and gamma function is also helps us to solve definite integrals. The relation between beta and gamma function is Let f(t) = t m−1 and g(t) = t n−1 then the convolution of ( ) and ( ) is given by Put tr = u, tdr = du, we have

Application in solving simultaneous Differential equations
In engineering, the mathematical formulations of some of the models / problems are not only in terms of single differential equations but also in terms of simultaneous linear differential equations.  Taking its inverse Laplace transform, we obtain Hence the desired particle moves through the path 22

Application of Laplace transform in Heat conduction equation
To determine the flow of heat in semi-infinite solid > 0, when initially the solid is at zero temperature and at = 0 the boundary 0 x  is raised to a temperature and integrating with respect to t from 0 to  and using given initial conditions we get, To find a solution which remains finite, as x  , we must take . Multiply given partial differential equation by st e  and integrating with respect to t from 0 to ∞, we get y xs is bounded, A must be zero and Using the inversion formula we get

Laplace Transform in the analysis of Electric Circuits
Consider R, L and C are connected in series with electromotive power of voltage E, where R is resistance, L is an inductance and C is capacitance. A switch is connected in the circuit. By Kirchhoff's law, we get Example: If R is 16 ohms, I is 3-henry, and C is 0.02 F are joined in series having an supply of 300v (See. figure1). At time t is equal to zero, the charge on the capacitor and current flowing in the closed circuit is zero. To  These are required expressions for charge q and current respectively at any time > 0.

1) y (t), the displacement which is measured in m and
2) F (t), the force which is measured in N.   Laplace transform is not limited to particular branch of Engineering it is used in civil Engineering and also in Economics problems.

Laplace Transform and Population Growth
We use Laplace transform for calculating growth of population. A group of the same species of crops, animals or other organisms which live together and reproduces is termed as Population. The logistic equation is in variable separable form, by applying transformation of Laplace and its properties to establish the exponential expression. In population growth rate of birth, death rate as well as rate of migration and immigration rate plays vital role. We know that as birth rate and death rate varies inversely i.e one increases other decreases. The above mentioned mechanism mathematically can be formulated as ∆ = − + − , where P, B, D, M and I are rate of change population with respect to time t, rate of birth, death, migration and immigration respectively. If we assume polulation is 'closed', that is migration and immigration are equal then Considering the density-independent pollution the simple model is that polulation size combined with freedom in density does not affect birth and death rates. Therefore, number of people are proportional to birth and death. (3.6.1) Now using equation (3.6.1) we can determine r and the population can be evaluated for different years. We will Apply the equation (3.6.1) for Population Albanian. Table 1 Applying the data for the year 2010 with t = 0, we have P (0) = 2918674, we can evaluate r using the fact that P = 2819697when which is the year 2000. By using (3.6.1) = −0.345 × 10 −2 . The general solution is ( ) = 291674 −0.00345 Now we calculate the population in later years and compare it with actual data. In between 2010 and 2019, we have a good agreement in between predicted and actual data. To predict our future it also helps us. Therefore we can apply this model to estimate the millions of population that year.

Representation of present value in Laplace transform
In the investment project, for various alternatives one wishes to determine the present value of series of cash receipts and transactions. The Present value of a sequence of payments given by, Where at time t, the present discounted value is [PV], the flow of cash is C(t), the rate of discount is s and t be the time period. Let us assume the current value with constant compounding which is the current value of cash flow. In other words, this is the amount which we need to pay today in order to get a cash flow or a series of them in the future. Now with the help of exponential series we derive equation (3.9.1) can be written as Which is same result as above. The insurance company has launched a protection that will pay Rs.1000 indefinitely, starting its first payment from next year. If the right product is 20% how much this security worth today? Using time line we get, PV = k s = 1000 0.2 = 5000

Conclusion
Throughout the paper, we have discussed some applications of Laplace Transform in various fields of Engineering.