Welcoming you all, on the course of Transform Calculus and Its Applications In Differential

Equation. As you know the transform calculus is basically a technique by which we try to

solve a we try to solve a problem from we transform the problem from its original domain

to some other domain. The advantage is that in the original domain

if I try to solve the original equation it may become difficult for us to find the optimum

solution. Therefore, using this transform techniques we convert this original problem

into some other problem in some other domain, where it becomes easier for us to find the

solution of a particular equation. Now, in this particular topics at first initially

we will cover integral transform. And in the integral transform first let us see the definition

of integral transform say K s, t be a function of two variables s and t. Where s is a parameter,

please note this it may be s may be real or s may be complex also and s is independent

of t . So, we are taking a function K s t which is a function of two variables s and

t, where s is a parameter which may be real or complex and independent of t.

The function f s can be defined by the integral like this way, it is f of s . I will show

it afterwards also f of s this is equals to minus infinity to infinity K of s t f t d

t k of s t into f t d t. Suppose we have defined like this, this minus infinity to infinity

K of s t f t d t, this sometimes we may write as this also we may write as t 1 to t 2 also.

That means, on some finite domain t 1 to t 2 or it may be infinite domain minus t 1 to

minus infinity to infinity. So, if a function f s is defined like this

minus infinity to infinity K s t f t d t, this is called the integral transform of the

function f t and we denote it by T of F t, we denote this thing by T of F t . So, suppose

I am removing this and if I come to this again, here you see the thing which I told it is

now like this. The function f s has been defined like this way f s equals minus infinity to

infinity K s t F t d t is known as the integral transform of the function F t and is denoted

by T of F t, the function K s t is called basically the kernel of the transformation

and also sometimes we call it as integral kernel or nucleus.

So, effectively if you change the value of K s t then we can find out various type of

transforms like integral Laplace transform, Fourier transform like this way . Now, see

the input of the transform of a function and the output is another function is small f

. So, note this one the input of a transform is a function F t just here if you see, capital

F t is the input and your small f is the output of this .

Next is, a integral transform is a particular type of mathematical operator we can say.

Depending upon the specific choice of the kernel function K there can be numerous useful

integral transform as I was telling earlier. If I change the value of the kernel function

K I can obtain various kind of useful transforms like Laplace transform, Fourier transform,

Hankel transform like this . Some associated inverse which we call as K inverse u t which

we call this one as K inverse u t. And this K inverse u t is defined as F t this

is equals u 1 to u 2 I will come to this, u 1 to u 2 K inverse s t f s d s . So, which

will be a function of s and t and ultimately you are integrating over s so that it will

be a function of t in the earlier one it was f s that was a k was a function of s t and

you are integrating over a function of t. So, therefore, this we are telling as the

inverse kernel K inverse we are defining and from there also we can find out this F t . So,

if I go back to my original one . And so the next is F t equals u 1 to u 2 K

inverse s t f s d s which we call as the inverse kernel and the last one is a symmetric kernel

one that is unchanged when two variables are permuted. A symmetric kernel one please note

this one is that one where the values will remain unchanged, if we interchange or permute

the variables s and t and if the values remains same then also we can do the same thing.

Now, next come to the Laplace transform, basically if you see your Laplace transform is a mathematical

tool we say. It is a simply a mathematical tool which is used to solve various kind of

engineering and mathematical problems and which is used by mostly by the science departments,

engineering departments, various departments are using this particular Laplace transform

to solve a particular type of problems this one .

This Laplace transform was first introduced by French mathematician Laplace . Laplace

transform was developed by French mathematician Laplace in 1790's, please note this one. This

was defined in derived by Laplace French mathematician Laplace in 1790.

Now, this Laplace transform is used to solve ordinary differential equation. In general

whenever we try to find the solution of ordinary differential solution what we do? We try to

find out the general solution and from the general solution using the i c that is initial

conditions we try to find out the values of arbitrary constants. Whereas, if I use the

Laplace transform in that case what happens simply Laplace transform converts the ordinary

differential equation into an algebraic equation .

So, in turn to find the solution we have to solve only one algebraic equation which is

basically very easy to find the solution. Similarly if we try to find out the solution

of a partial differential equation using Laplace transform, it reduces the number of independent

variables by 1. Since it reduces the number of independent variables by 1 the solution

process becomes easier. So, if we go back to this one again, for Laplace

transform then . How to define the Laplace transform? If the kernel K s t is defined

as K s t equals 0 whenever t is less than 0 e power minus s t whenever t is greater

than equals 0. If the function is defined like this then the Laplace transform of this

capital F t is defined as f s equals 0 to infinity e power minus s t into f t d t . So,

you have a function F t the Laplace transform of the function capital F t is denoted by

small f s and is defined as 0 to infinity e power minus s t F t d t.

So, please note this one, whenever we try to use the Laplace transform the value of

the independent variable t always will be lying 0 to infinity or in the positive side

of this one . So, the function f s defined by the equation 1 is called the Laplace transform

of the function and we call it also denoted by Laplace transform of f t or small f s or

the capital F s . Laplace transform as I told is a mathematical

tool which can be used to solve several problems in science and engineering. It was first introduced

by Laplace as I told you earlier a French mathematician in the year 1790. The method

reduces the solution of an ordinary differential equation to the solution of an algebraic equation.

So, basically if I try to solve an ODE if I applied Laplace transform then the ODE will

be transformed or converted into the other domain and it will create one algebraic equation.

And to find the solution of the original problem instead of solving the ODE we have to simply

solve the algebraic equation. When Laplace transform technique is applied to an ordinary

differential equation sorry a one partial differential equation, this will be partial

differential equation it reduces the number of dependent variables by one .

Next is linear transformation, a transformation is said to be linear please note this one

this definition we are giving for in general. A transformation any transformation T you

take is said to be linear if for every pair of function F 1 and F 2 t and for every pair

of constants a 1 and a 2, we have this one .

So, linear transformation if we try to take this will be equals to T of a 1 F 1 t plus

a 2 F 2 t. This is equals a 1 transform of F 1 t plus a 2 into transform of F 2 t . So,

if we write down this thing . So, you have a transform T for this transform

T if we have two functions F 1 t and F 2 t. If you have these two functions F 1 t and

F 2 t and if for any constants a 1 and a 2 transformation of a 1 F 1 t plus a 2 F 2 t

equals a 1 transform of F 1 to F 1 t plus a 2 into transform of a 2 t then we say that

this is a linear transformation . So, if I have to go back to this we are writing T of

a 1 F 1 t plus a 2 F 2 t this is equals to a 1 into T of F 1 t plus a 2 into t of F 2

t. Now, come to the linearity property of Laplace

transform the Laplace transform of a linear transformation, that is Laplace transform

of a 1 F 1 t plus a 2 F 2 t this is equals a 1 Laplace transform of F 1 t plus a 2 Laplace

transform of F 2 t where a 2 and a 1 are constants . So, just the linearity property of general

transform what we told just before this slide same way we can tell that the linearity property

of Laplace transform holds if this Laplace transform of a 1 F 1 t plus a 2 F 2 t equals

a 1 Laplace transform of F 1 t plus a 2 Laplace transform of F 2 t where a 1 and a 2 are constants.

Now, the proof is very simple here. The proof here is simply we know that Laplace transform

of F t this is equals 0 to infinity e power minus s t into f t d t we know this thing.

Now if we take Laplace transform of a 1 F 1 t a a 1 F 1 t plus a 2 into F 2 t . So,

when you are writing it this equals using the definition of Laplace transform of F t

we can write it 0 to infinity e power minus s t into this whole term that is a 1 F 1 t

plus a 2 F 2 t a 2 into F 2 t d t. So, now I can break this into two parts two

integrals one I can write down 0 to infinity e power minus s t a 1 into F 1 t d t that

is a 1 into e power minus s t F 1 t d t plus 0 to infinity a 2 into e power minus s t f

t d t sorry this will be F 2 t into d t . So, this one is F 2 t into d t and this is nothing,

but a 1 into 0 to infinity e power minus s t F 1 t d t is nothing, but Laplace transform

of F 1 t plus a 2 into 0 to infinity e power minus s 2 minus s t this is basically your

F 2 t. So, if F 2 into the power minus s t F 2 t

d t this I can write down Laplace transform of this is F 2 t . So, this completes the

proof that Laplace transform of a 1 F 1 t plus a 2 F 2 t this is equals Laplace transform

of a 1 into Laplace transform of F 1 t plus a 2 into Laplace transform of F 2 t . So,

I can remove this and again we can go back to earlier one .

So, just the proof whatever I told you just I am showing here. So, that you can understand

it in better way same thing we have written 0 to infinity e power minus s t into a 1 F

1 t plus a 2 F 2 t this is equals this. And a 1 into 0 to infinity e power minus s t F

1 t d t is Laplace transform of F 1 t plus a 2 into Laplace transform of F 2 t .

Now, come to a piecewise continuous function, a function F t is said to be piecewise continuous

or sectionally continuous on a closed interval a to b if it is defined on that is interval

and in such that the interval can be subdivided into a finite number of intervals. In each

of which F t is continuous at has finite right and left hand limit. The meaning is that a

function is defined in a interval a to b where the function F t is defined and t lies in

between a to b. The function may be continuous throughout

in a to b or the function may be piecewise continuous ; that means, if the entire domain

I ship divided into n subdomains then in each of the subdomains the function F t will be

continuous. Then we call it the function F t as a piecewise continuous function, say

for this one you have . So, this is you are defining x sorry this

is your t this is your 0 on this side you are defining say any function y equals f x.

Your entire domain maybe from here to here a is here b is here in between something like

that is there. Say a to b 1 the function behaves like this this is say a 1 from this point

b again it has something like this, up to this. Then again from here it is coming something

like this way and it is reaching here. So, if I take the entire point a to b in a

to b the function f x may not be fully continuous , but if I subdivide this one into some smaller

subsections say a to a 1, a 1 to b 1, b 1 to b 2, b 2 to b then in each of these subdomains

the function is continuous then we can tell that the function is piecewise continuous.

I hope it is clear that a function may not be continuous in a particular domain a to

b, but if I subdivide the function a to b into n subdomains and if in each subdomain

the function is continuous we call the function as piecewise continuous function. Again let

us go back . So, now come to the other one that is function

of exponential order a function F t is said to be of exponential order as t approaches

infinity if there exists a positive constant real number capital M and a number a and a

finite number t 0 such that absolute value of f t modulus of F t less than m into e power

a t. So, please note this one, a function F t is said to be exponential order if whenever

t approaches infinity if there exists a positive constant real number M one a some number a

which is greater than 0 and a finite number t 0 such that absolute value of F t is less

than M into a power t. Or I can write down absolute value of e power minus a t into F

t less than M for all t greater than equals t 0 .

So, please note this one that e power minus a t f t is absolute value of this must be

less than M for some t greater than t 0 then we say that the function is of exponential

order. Sometimes we write this like this way also. So, what we do sometimes limit t approaches

infinity e power minus a t into f t. If this value is a finite constant if you get for

some a greater than where a is greater than 0. Then we can say that F t is of exponential

order a then we say that f t is of exponential order a .

So, in the other way also as t approaches infinity e power minus a t F t if this limiting

value is a finite value then also we say that F t is of exponential order or we can say

that absolute value of e power minus a t f t is less than equal m whenever t approaches

infinity. So, we can define it in any one of these two ways .

So, these already we are writing . Now, if a function f t is of exponential order a and

it is also of order b please note this one a function F t is of exponential order a.

And also it is of order b where b is greater than a then we write sometimes F t equals

order of e e power a t as t approaches infinity. That means, whenever it is of exponential

order a and if I know that is a b where b is greater than a then also we can tell the

same thing that f t is an exponential order alpha where we can write down order of e power

a t and please note that as t approaches infinity then only this condition should be satisfied

. Example say we have to show that t power n

is of exponential order as t approaches infinity n being any positive integer . So, what we

have to do we have to find out limit t approaches infinity e power minus a t into t power n

because we want to find out the its whether t power n is of exponential order alpha or

not where a is greater than 0. Now what is happening this you can write down limit t

approaches infinity e power t power n by e power a t . So, that you are bringing it in

the form of infinity by infinity . So, once it is coming in the form of infinity

by infinity to evaluate this limit you can use L hospital rule n times. If I use the

L hospital rule n times in that case I can obtain this is equals limit t t approaches

infinity factorial n by a power n into e power a t.

So, once we are doing this after successively we are derivating finding the derivative using

L hospital rules ultimately you will get this one. Factorial n by a power n into e power

a t and whose value this lower portion always goes to infinity so, that the value becomes

0. The value limiting value of this as t approaches infinity always will be 0 therefore, since

limit t approaches infinity e power minus a t t power n as a as a finite value 0 we

can say that t power n is of exponential order on the .

So, this only I am just showing here you can just see factorial n by infinity and 0 . So,

this example shows that t power n is of exponential order alpha. So, I hope now you can understand

whether a function is of exponential order alpha or not for that one we have to just

find out limit t approaches infinity e power minus a t into F t d t. And if the value is

a finite value we say that this function is of exponential order alpha .

Next example you see that so that F t equals e power t square is not of exponential order

as t approaches infinity n being the of positive integer. Means the opposite one now we are

showing that e power t square is not of exponential order alpha. So, for this one again we have

to find out the limiting value that is limit t approaches infinity, it will be e power

t into t minus a. If you calculate the value e power minus a t into f t f t is e power

t square . So, e power t square minus a t and if I try to find the limiting value of

this always this will be infinity if of course, a is greater than 0.

So, once I am getting this limit t approaches infinity e power t into t minus a this limiting

value is infinity . So, I can say that e power t a e power t squared function is not of exponential

order alpha. . So, simply we can show this that limit is this e power minus a t into

t square which we can write down as a power t into t minus a for a greater than 0 and

the value is infinity . So, by this way we will we can find out whether

a function is of exponential order or not. In the next lecture what we will study what

is the use of this exponential order or how it relates in the transform calculus if a

function is of exponential order alpha then what will be the effect that we will see in

the next class.