CHAPTER ONE
INTRODUCTION
1.1 Background of the Study In this particular area of research, I wish to study transformation which plays an important role in pure and applied mathematics. This class of transformation is an integral transformation called the Laplace transform. It is very effective in the study of initial value problem involving linear differential equation with constant coefficient. Laplace transform was first introduced by a French mathematician called Pierre Simon Marquis de Laplace about 1780’s. This method associated with the isolation of the original problem that is function ƒ(t) of a real variable and some function ƒ(s) of a complex variable so that the ordinary differential equation for the function ƒ(t) is transformed into an algebraic equation for ƒ(s) which in most cases can readily be solved. The solution of the original differential equation can be arrived at by obtaining the inverse transformation. The transformation and its inverse can be derived by consulting already prepared table of transform. This method is particularly useful in the solution of differential equation and has more application in various fields of technology e.g. electrical network, mechanical vibrations, structural problems, control systems. Meanwhile in this research work, I shall look into the Laplace transform, the properties of the Laplace transform and the use of this technique in solving delay differential equation will be looked into.
1.2 Statement of the Problem There are so many engineering and other related problems that can be expressed in the form of ordinary differential equations. But such problems cannot easily be solved using the elementary method of solution. In such cases, the Laplace transform becomes a more successful and useful tool in solving these problems. Therefore, I intend to carry out this research called Laplace transform and its application in solving delay differential equations.
1.3 Justification of the Study As a tool for solving delay differential equations (DDEs), Laplace transform has some essential features that makes it more desirable method compared to others. (a) Laplace transform reduces a differential equation to an algebraic which makes the problem very easy to handle. (b) The solution satisfying the given initial condition is automatically found of which is justified. (c) Laplace transforms handles both homogenous and non-homogenous equation at the same time, no need of solving the homogenous part first before non-homogenous on the process of finding the solution of the given problem.
1.4 Scope and Limitations The work will cover Laplace transform, its properties and the applications of Laplace transform in solving Delay Differential Equations (DDEs).
1.5 Aim and Objectives The aim of the project is to know the application of laplace transform in solving delay differential equation The following are the aim and objectives of the study; (i) To have more insight on the Laplace transform. (ii) To be conversant with the application of Laplace transform in solving delay differential equation and know its area of application in real life situation. (iii) To determine the inverse transformation to obtain a particular solution.
1.6 Definition of Basic Terms
1.6.1 Differential Equations Let x be an independent variable and y be the dependent variable, then the differential equation is a relationship between an independent variable x and a dependent variable y and one or more derivatives of y with respect to x.
i. dydx=x+6

ii xdydx-y2=0

iii d2ydx2=4dydx+10y

1.6.2 Order of a Differential Equation The order of a differential equation is given by the highest derivative involved in the differential equation.
Example:
i xdydx=y2 +4 is of first order

ii d2ydx2+8dydx+2=0 is of second order

1.6.3 Degree of a Differential Equation The degree of a differential equation is the power of the highest derivative involved in the differential equation.
Example:
i xdydx +y2 =0 is of degree one

ii (d2y)3dx2+(d2y)3dx2 is of degree two iii d2ydx2=4dydx+10y

1.6.4 Delay Differential Equations (DDEs)
The delay differential equations (DDEs) is the equation in the form y't=f(t,yt, yt-T1t,yt, yt-T2t,yt.
This was originally motivated mainly by problems in feedback control theory. The delays Ti = 1, 2, … are measurable physical quantities and may be constant, a function of t (the variable or time dependent case). Examples of delays include; (i) The time taken for a signal to travel to the controlled object. (ii) The driver reaction time. (iii) The time for the body to produce red blood cells and cell division time in the dynamics of viral exhaustion or persistence.
Time delays are natural component of the dynamic process of biology, ecology, physiology, economics, epidemiology and mechanics and to ignore them is to ignore reality.
A delay differential equation (DDE) is a functional differential equation where the highest order derivative only occurs with one value argument and this argument is not less than the arguments of the unknown function and its lower order derivatives appearing in the equation. Such equations are called retarded functional differential equations (RFDE) or differential equations with retarded arguments.
Example:
i x'(t)=x(t-π)

ii x'(t)=x(t)-xt2

1.6.5 Application of Delay Differential Equation Delay differential equations arise when modeling certain physical problems as the following applications show;
Application 1: Balancing a pencil on your finger tip Let m > 0 be the mass of the pencil and l > 0 be its length. Model the pencil as a damped inverted pendulum. If there is no applied force, the model will be mlθ''+γlθ'-mgsinθ=0, Where γ≥0the damping coefficient and g is is the acceleration due to gravity. When balancing, the finger applies a force which depends on the deviation of the pencil from the vertical position, Ɵ and the angular velocity of the pencil, θ' mlθ''t+γlθ't-mgsinθt=f(θt-τ, θ'(t-τ))
Here τ > 0 represents your reaction time i.e. the time between when you observe the position and velocity of the pencil and the time when you actually apply the force. This is an example of a system with delayed feedback control. Such systems can be schematically represented as in figure 1.2. One of the main ideas of control theory is to design feedback laws to force a system to exhibit a certain desired behaviour.
Application 2: Population Models
The standard predator-prey model is of the form x't=r1xt1-xtL-b1xtyt y't=-r2yt+b2xtyt

Where: x(t) is the size of the prey population at time t, y(t) is the size of the predator population at time t, r1> 0, r2 > 0 are the per capita growth / death rates b1 > 0, b2 > 0 are the predation coefficients
A more realistic model allows for a time delay, τ > 0 between when the prey is eaten and when this food is converted into new biomass through births of predators. In this case the model becomes x't=r1xt1-xtL-b1xtyt y't=-r2yt+b2xt-τyt-τ

And even more realistic model uses a distribution of delays to take into account the fact that the gestation period is not exactly the same for each individual. In this case, the model becomes. x't=r1xt1-xtL-b1x(t)y(t) y't=-r2yt+b20∞Gsxt-syt-sds,
0∞G(s)ds=1

θ θ Fig. 1.1: Geometry for Pencil Balancing

1.6.6 A Functional Differential Equation (FDE) A functional differential equation is an equation for an unknown function which involves derivatives of the functions and in which the function and possibly its derivatives, occur with various different arguments.
Example:

i x't=t2xt-x'(t-1) ii x'(t)=x(t)x(t-1)+xt+2

1.6.7 Laplace Transform
Suppose f(t) represents some expression t defined for t≥0, the Laplace transform of f(t) denoted by Ltis defined to be
Lft=0∞e-stf(t)dt
Where
L: the Laplace operator f(t): a general function of time
S: a complex independent variable
F(s) : the symbol for the Laplace transform f(t)
This is Fs=0∞e-stf(t)dt
The function F(s) is called the Laplace transform of the given function F(t) and it is denoted by Lft such that f(s)=Lft=0∞e-stf(t)dt 1.7.0 Transformation of some Standard Function
1.7.1 Transformation of a Constant
Suppose f(t) is a constant
If f(t)=a then
Lft=La=0∞ae-stdt
=a[e-st]o∞=as
∴Lft=as provided s>0
1.7.2 Transformation of an Exponential Function If ƒ(t) is an exponential function say eat where a is a constant, then the Laplace transform of eat is given by
L{eat}=0∞e-st.eatdt
=0∞e-(s-a)tdt
Leat=-1s-ae-s-at0∞=-10-1s-a =1s-a
Leat=1s-a provided s-a>0

1.7.3 Transformation of a Periodic Function
In the periodic function, if we want to transform sin at or cos at, we write it in the form of exponential function, thus cos at +i sinat =eiat
Here sin at is the imaginary part if eiat that is L=sinat=imaginary of
0∞eiat.e-stdt
=Im 0∞e-(s-ia)tdt Now using the exponential form we have that 0∞e-(s-ia)tdt =1(s-ia) provided s-ia>0 Lsinat=In 1s-ia By rationalizing the denomination we have
1(s-ia)∙s+ias+ia =(s+ia)(s2+a2)=ia(s2+a2)
L=sin at=a(s2+a2) provided s>0

1.8 Existence of Laplace Transform
The existence of the Laplace transform of a function ƒ(t) depends on the convergence of an improper integral
0∞e-stf(t)dt
The theorem of the existence of Laplace transform states that “if ƒ(t) is piecewise continuous and of exponential order, then the Laplace transform of ƒ defined by
Lft0∞e-stftdt exist for s sufficiently large Therefore, with the existence of Laplace transform, I shall go on and make further research on this topic “Laplace Transform and its Application in solving Delay Differential Equations (DDEs).

CHAPTER TWO
LITERATURE REVIEW This chapter consists of previous research work made by some science authors for the development of this great topic “Laplace transform”. Strond (2001) defined Laplace transform as follows; If ƒ(x) represents some expression in x, defined for x ≥ 0, the Laplace transform of ƒ(x) denoted by L{ ƒ(x)} is defined to be
Lƒx=0∞e-sxf(x)dx

Where s is a variable whose values are chosen so as to ensure that the semi-infinite integrals converges.
He further stated that “The Laplace Transform is an expression in the variable s which is denoted by ƒ(s)”. It is said that ƒ(x) and ƒ(s) = L { ƒ(x)} from a transform pair. This means that if ƒ(s) is the Laplace transform of ƒ(x), then ƒ(x) is the inverse Laplace transform of ƒ(s), we write it as; ƒ(x) = L' { ƒ(s)}
Murray (1965) provides the linearity property of the Laplace transform that if ƒ1(t) and ƒ2(t) are two functions whose Laplace exist for s > a, and s > a2, c1 and c2 be real constants.
Then for a greater than, the maximum of a1 and a2.
L{c1f1t+c2f2(t)}=c1L{c1f1t+c2f2(t)}
Kreyzig (2002) developed the first translation property of Laplace transform that if ƒ(t) has transform ƒ(s) where s > K, then the linear Laplace transform that is L{eat at} = ƒ(s - a) and the inverse is given as ƒ(t) = L-1 { ƒ(s – a)} where a is any real number.
Gupta (1978) developed the Laplace transform of integrals as follows;
Suppose the Laplace transform of ƒ(t) is ƒ(s) that is L{ ƒ(t)} = ƒ(s)
Then
L0tf(u)du=f(s)s Stroud (2003) defined the Laplace transform of an expression ƒ(t) denoted by L{ ƒ(t)} as the semi-infinite integral.
L[f(t)]=0∞f(t)e-stdt

Stroud went further to develop the transformation of derivatives as follows; Suppose f'(t) denotes the first derivatives of ƒ(t) with respect to t1 and f'(t) denote the second derivative of ƒ(t) with respect to t e.t.c. then;
Lft=0∞fte-stdt by defination
Integrating by part we have
L[f(t)]=[e-stf(t)]∞0-0∞f(t)(-se-st)dt
When t→∞ e-stft→0
Since s is positive and large enough to ensure that e-st decays faster than any possible growth of ƒ(t).
L[f'(t)]=f0+sL[f(t)]
Murray (1965) developed the relationship which exist between the production of two function say ƒ and g and the Laplace transform of the individual function is called convolution of ƒ and g defined by f×g=0tf(t-u)g(u)du He states that if L{ƒ(t)} = ƒ(s) and L{g(t)} = G(s) that is if the Laplace transform of ƒ(t) is ƒ(s) and g(t) = f(s)G(s) then, L{ ƒ(s) x g(t)} = ƒ(s) G(S) or
L-1[f(s)g(s)]=f×g0tf(t-u)g(u)d(u)
Note that we use this in solving or finding the inverse transformation of the product of two functions.
Stroud (2003) in addition to the above review provision, went further to develop the theorem of multiplying by t and n tn as follows; If L{ƒ(t)} = ƒ(s) then L {t ƒ(t)} =f' (s) since L ƒt=fs=0∞e-stftdt as stated earlier, then

L ƒt=fs=0∞e-stftdt=-f's

L ƒt=-f's

CHAPTER THREE
METHODOLOGY
This chapter shows the techniques employed in solving delayed differential equation by the method of Laplace transform. The methods to be used in carrying out this research include; (i) Taking the transform of the derivatives involved in the problem. (ii) Solving equation by the method of rational approximations.
3.1 Methods of Transforms of a Derivative
Suppose f'(t) denotes the first derivatives of ƒ(t) with respect to t and f''t, denotes the second derivatives of ƒ(t) with respect to t, e.t.c. then,
Lf't=0∞e-stf'tdt by defination 3.2.1
Integrating by part we have Lf't =[e-stft]0∞-0∞ft-se-stdt 3.2.2
When t→∞, e-stf(t)→0

Since s is positive and large enough to ensure that e-st decay faster than any possible growth of ƒ(t).
Lf't=-f(0)+sL{f(t)}
Replacing ft by f't
Lf''t=-f(0)+sL{f(t)}
Lf''t=-f(0)+sL{f'(t)}
Lf''t=-f'(0)+s-f0+sLft
Note that Lft=F(s)
Lf't=SF(s)-F(0)
Lf''t=s2F(s)-sF(0)-f'(0)
In general, the Laplace transform of derivative can be written as
Lfnt=snFs-sn-1f0-sn-2f'0…………………. -sfn-20-fn-10 3.2.3
Therefore we denote the Laplace transform of Y, i.e Y=L{y}=L{f(t)}=f(s) And by so, doing, we can have the transform of the following derivative
L(Y)=Y
L(Y')=SY-Y0
L(Y'')=s2Y-sY0-sY1
3.2 Method of Rational Approximations
Laplace transform technique are used in the solution of delay differential equations (DDEs). By the method of rational approximation these are explain below. i. Taking the laplace transforms of the given differential equation ii. Using the given initial condition (if specified) iii. Rearranging the equation algebraically to give the transform of the solution iv. Determining the inverse transform to obtain the particular solution

Table 1: Table of the standard Laplace Transform Function f(t) | L[f(t)]=f(s) | a | as, s>0 | t | 1s2,s>o | eat | 1(s-a),s>0 | sinat | as2+a2,s>0 | cosat | as2+a2,s>0 | sinhat | as2-a2,s>⃓a⃓ | coshat | as2-a2,s>⃓a⃓ | tn | n!sn+1,s>0 | eatsinbt | b(s-a)2+b2,s>a | tneat | n!(s-a)n+1,s>a | e-atf(t) | F(s+a) | f(t)t | s∞F(u)du | f''(t) | s''Fs-sn-1f0……..fn-1(0) |

CHAPTER FOUR
RESULTS
4.1 Application of Laplace Transform in Solving Delay Differential Equation
This chapter is based on the solution of delay differential equation (DDE) by laplace transform.
In solving delay differential equation, laplace transform technique are used in the solution of DDE’s arising in control theory by rational approximation method. Are illustrate the solution of DDE by laplace transform method in the following example. 1. Solve the equation. x't=2x(t-1) … 4.1
Solution
Taking the laplace transform leads to
Lx't=2L[x(t-1)]

=1∞x'te-stdt=21∞xt-1e-stdt
Taking the L.H.S
1∞x'te-stdt
By applying integration by part dV=x't, V=dV=x(t) u=e-st, dudt=se-st uv-1∞vdudt =e-st.xt-1∞xt-se-st
=x(t)e-st/1∞- 1∞xt.(-se-st)

Taking the R.H.S
21∞xt-1e-stdt
Let u=t-1 dudt=1 du=dt u=t-1 t=u+1
Recall
21∞xt-1e-stdt
Since t-1=u then
21∞xt-1e-stdt
Recall from equation (4.1) above

1∞x'te-stdt=21∞xt-1e-stdt x(t)e-st/1∞-1∞x(t)-se-stdt=21∞xue-s(u+1)dt Assuming that xte-st→0 as t→∞
This leads to x1e-st+s1∞e-st.x(t)dt=2e-s01xue-sudu+2e-s1∞x(u)e-sudu Hence assuming that s-2e-s≠0 we obtain
1∞xte-stdt=x1e-s+2e-s+2e-s01x(u)e-sudus-2e-s
Hence assuming the inversion formular can be applied, we obtained. xte-st=cx1e-s+2e-s01x(u)e-sudu.estds Divide both side by e-st xt=cx1e-s+2e-s01x(u)e-sudu s-2e-s .estds
Example 2.
Solve the equation f''t+4ft=sin2t …4.2
Taking the laplace transform of both sides L[f''t]+L[4ft]=sin2t s2fs-sf0-f'0+4fs=ks2+k2 s2fs-sf0-f'0+4fs=2s2+4
With the initial condition f0=0 and f'0=0 s2Fs-s0-0+4Fs=2s2+4 s2Fs+4F(s)=2s2+4
Fss2+4=2s2+4
Divide both side by s2+4

Fss2+4s2+4=2s2+4s2+4
Fs=2s2+4÷s2+41=2s2+4×1s2+4
Fs=2(s2+4)2
Now apply the laplace inverse transform to get ft=L-1F(s) ft=L-12s2+42 ft=18sin2t-t4cos2t. Example 3.
Solve the equation y''-3y'+2y=4e2t …4.3
Where Y0=-3, Y'0=5
Taking the laplace transform.
L[y'']-3[L]y'+2L[y]=4[L]e2t

s2y-sy0-y'0-3sy-y0+2y=4. 1s-k s2y-sy0-y'0-3sy+3y0+2y=4s-2 With initial condition y0=-3, y' 0=5 s2y-s-3-5-3sy+3-3+2y=4s-2 s2y-3s-3sy-5+2y-9=4s-2 s2-3s+2y+3s-14=4s-2 Collect like terms s2-3s+2y=4s-2-3s-141 Divide the both side by s2-3s+2 y=4s-2s2-3s+2=3s-141s2-3s+2 y=4(s2-2s+2)(s-2)+14-3s(s2-3s+2)
Separate y into its partial fraction y=-3s2+20s-24(s-1)(s-2)2 Separate y into partial fraction. y=-7s-1+4s-2+4(s-2)2 Taking the inverse transform we get. y=L-1[y] y=L-1-7s-1+L-14s-2+L-14s-22 y=-7L-11s-1+4L-11s-2+4L-11s-22 y=-7et+4e2t+4te2t
The laplace transform method can be considerably shatter than the classical methods which require determination of the complimentary function, determination of a particular intergral, and the general solution. With all these we see that the method of laplace transform is more preferable.

CHAPTER FIVE SUMMARY, CONCLUSION AND RECOMMENDATIONS
5.1. SUMMARY
This project reviewed the basic theory of laplace transform given by a function f(t). We examine the property of laplace transform and the method itself in solving delay differential equations (DDES).
Moreso, taking the transform of the deferential equations and rearranging the equations algebraically gives the transform of the solution. Also we use the given initial condition (if at all specified in the problem) and finally determine the inverse transform to obtain the particular solution.
5.2 Conclusion
Above all, laplace transform has been found to have many advantages over other methods of solving deferential equation which include the possession of the useful property that many relationships and operated over the original f(t) corresponds to simplex relationship and operation over its imagine F(s). laplace transform is crucial for the study of control system, hence they are used for analysis of HVAC (Heating, Ventilation and Air Conditioning) control systems, which are useful in all modern building and construction with the above mentioned advantages we can observed that the laplace transform method is the most widely acceptable method, the method also has the ready-made table of laplace transform which makes the work easier in solving problems. 5.3 Recommendation. The laplace transform is a widely used integral transform in mathematics with many applications in physic and engineering. the laplace transform has the following advantages over other method of solving differential equation. -The laplace transform has the useful property that many relationships and operations over the original f(t) corresponds to simpler relationships and operations over its imagine F(s)
The laplace transform is used for solving differential and integral equations. In physics and engineering, it is used for analysis of linear time-invariant. * The laplace transform provide an alternative functional discretions that often simplifies the process of analyzing the behavior of the system or synthesizing a new system based on a set specifications
-Finally, it is equal in value to the initial function.

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