The document summarizes key concepts in differential equations and their applications. It discusses: 1. The history of differential equations, which were independently invented by Isaac Newton and Gottfried Leibniz. 2. The definition of a differential equation as an equation containing the derivative of one or more dependent variables with respect to one or more independent variables. 3. Some common applications of differential equations in computer science, such as modeling temperature changes, numerical solutions, computer algebra systems, and numerical software packages.

1. Final Year Defense
Presented by Supervised by
ENGINEERING MATHEMATICS
Md.Abdullah Al Noman
Md.Abu Taleb
Md. Saidur Rahman
Md.Mahmudul Hasan
Sadia Tasmim Tumpa
Mahbubul Islam
Md.Habibur Rahman
Minhaz Halim Zim
Hasna Hena
Lecturer
Daffodil International
University
Class Presentation
PRESENTATION OF

2. Final Year Defense
Differential Equation
Invention Of Differential Equation :-
In mathematics the history of Differential Equations traces the development of differential Equation
from calculus, which itself was independently invented by English Physicist Isaac
Newton and German mathematician Gottfried Leibniz.
Isaac Newton Gottfried Leibniz.

3. Final Year Defense
Differential Equation :-
Differential Equation is an equation containing the derivative of one or more dependent variables
with respect to one or more independent variables.
For example:-

4. Final Year Defense
Uses of Differential Equation in CSE
a. General Applications like as temperature of population calculating.
b. Numerical solution.
c. List of Computer Algebra system.
d. Numerical Software package.
e. Constraint logic programming. Etc.
Computer science is such a thing we say that mathematics is the backbone of computer science and
DE is a a great part of this . So we can say application of ordinary differential in CSE is very important.
And the application field is so large.

5. Final Year Defense
MD. ABU TALEB
ID: 172-15-9887

6. Final Year Defense
Fourier Series:
• History Fourier Series?
Baron Jean Baptiste Joseph
Fourier (1768−1830) introduc
ed
the idea that any periodic
function can be represented
by
a series of sines and cosines
which are harmonically
related.

7. Final Year Defense
Definition of Fourier Series :
• Let F(x) be a function which is
definied on –L<x<L and F(x)
and F’(x) are sectionally
continuous in –L<x<L and
F(x+2L)=F(x).
That is F(x) is periodic function
with period 2L.

8. Final Year Defense
Why do we use Fourier Series ?
• Fourier series is just a means to
represent a periodic signal as an
infinite sum of sine wave
components. A periodic signal is
just a signal that repeats its
pattern at some period. The
primary reason that we use
Fourier series is that we can
better analyze a signal in
another domain rather in the
original domain. Sometimes a
signal reveals itself more in
another domain

9. Final Year Defense
MD. MAHBUBUL ISLAM
ID: 172-15-9912

10. Final Year Defense
Laplace Transform:
• What is Laplace Transform?
The Laplace transform is a useful tool for dealing with linear
systems described by ODEs. As mentioned in another answer,
the Laplace transform is defined for a larger class of functions
than the related Fourier transform.

11. Final Year Defense
Why do we use Laplace transform?
• The Laplace transform is a widely used integral
transform with many applications in physics and
engineering. It will help you to solve Differential
Equation of higher order which is the most
widely used application of Laplace transform.
.Also evaluating integral, boundary value
problems, circuit solving etc, Like the
Fourier transform, the Laplace transform is used
for solving differential and integral equations. In
physics and engineering, it is used for analysis of
linear time-invariant systems such as electrical
circuits, harmonic oscillators, optical devices,
and mechanical systems also used in signal
processing to access the frequency spectrum of
the signal in consideration There is flow chart
given below how Laplace equation solves.

12. Final Year Defense
Md. Mahmudul Hasan
ID: 172-15-10023

13. Final Year Defense
Laplace Transfrom:
Let F(t) be a function of t>0, the laplace trans from of F(t)
denoted by ℒ{F(t) and is defined by
=f(s).
Property of laplace:
1.Linearity Property: if ℒ {F(t)}=f(s).
2.First translation or shifting property: Then ℒ{𝑒 𝑎𝑡
F t = f s − a ,
3.Second Shifting Property:
4.Laplace transform of derivative:

14. Final Year Defense
5.Laplace transformation of integrals:
6.Multiplication by 𝑡 𝑛
:
if ℒ{F(t)}=f(s), then
where n= 1,2,3

15. Final Year Defense
Name: Minhaz Halim
ID No:- 172-15-9939

16. Final Year Defense
Definition: Let be a function of specified for . Then the Laplace transform of , denoted by
, is defined by
∫
where we assume at present that the parameter is real. Later it will be found useful to
consider complex.
The Laplace transform of is said to exist if the integral (1) converges for some value of ;
otherwise it
does not exist. For sufficient condition under which the Laplace Transform does exist.
Notation: If a function of t is indicated in terms of a capital letter, such as , etc., the Laplace
transform of the function is denoted by the corresponding lower case letter, i.e. etc.

17. Final Year Defense
Definition: If the Laplace Transform of a function is , i.e. if { } , then is called an
inverse
Laplace transform of and we write symbolically
{ } where
is called the inverse Laplace
transformation operator.

18. Final Year Defense
Saidur Rahman
172-15-9962

19. Final Year Defense
The convolution theorem: Definition
The Convolution of piecewise continuous functions f, g :R→ R
Is the function f∗g : R→R given by
Remarks:
 f∗g is also called the generalized product of f and g.
 The definition of convolution of two functions also holds in
the case that one of the functions is a generalized function,
like Dirac’s delta.

20. Final Year Defense
Name : Sadiatasmeem Tumpa
ID: 172-15-9798

21. Final Year Defense
Laplace Transform
where f(t) is defined for t >=0 (Abramowitz and Stegun 1972). The unilateral Laplace
transform is almost always what is meant by "the" Laplace transform, although a bilateral
Laplace transform is sometimes also defined as

22. Final Year Defense
In general Laplace Transform formula is given by
Standard Laplace Transform Formulas are given below which may be helpful for you to solve instantly instead of
making your problem lengthy.

23. Final Year Defense