The Laplace transform is an integral transform that converts a function of time into a function of complex frequency. It is defined as the integral of the function multiplied by e-st from 0 to infinity. The Laplace transform is used to solve differential equations by converting them to algebraic equations. Some key properties of the Laplace transform include linearity, shifting theorems, differentiation and integration formulas, and methods for periodic and anti-periodic functions.
its ppt for the laplace transform which part of Advance maths engineering. its contains the main points and one example solved in it and have the application related the chemical engineering
It is the ppt on Laplace Transform and it's applications.This topic is taken out from Advance Engineering Mathematics comes in 3rd semester of engineering.
Laplace transforms
Definition of Laplace Transform
First Shifting Theorem
Inverse Laplace Transform
Convolution Theorem
Application to Differential Equations
Laplace Transform of Periodic Functions
Unit Step Function
Second Shifting Theorem
Dirac Delta Function
its ppt for the laplace transform which part of Advance maths engineering. its contains the main points and one example solved in it and have the application related the chemical engineering
It is the ppt on Laplace Transform and it's applications.This topic is taken out from Advance Engineering Mathematics comes in 3rd semester of engineering.
Laplace transforms
Definition of Laplace Transform
First Shifting Theorem
Inverse Laplace Transform
Convolution Theorem
Application to Differential Equations
Laplace Transform of Periodic Functions
Unit Step Function
Second Shifting Theorem
Dirac Delta Function
laplace transform and inverse laplace, properties, Inverse Laplace Calculatio...Waqas Afzal
Laplace Transform
-Proof of common function
-properties
-Initial Value and Final Value Problems
Inverse Laplace Calculations
-by identification
-Partial fraction
Solution of Ordinary differential using Laplace and inverse Laplace
Z Transform And Inverse Z Transform - Signal And SystemsMr. RahüL YøGi
The z-transform is the most general concept for the transformation of discrete-time series.
The Laplace transform is the more general concept for the transformation of continuous time processes.
For example, the Laplace transform allows you to transform a differential equation, and its corresponding initial and boundary value problems, into a space in which the equation can be solved by ordinary algebra.
The switching of spaces to transform calculus problems into algebraic operations on transforms is called operational calculus. The Laplace and z transforms are the most important methods for this purpose.
laplace transform and inverse laplace, properties, Inverse Laplace Calculatio...Waqas Afzal
Laplace Transform
-Proof of common function
-properties
-Initial Value and Final Value Problems
Inverse Laplace Calculations
-by identification
-Partial fraction
Solution of Ordinary differential using Laplace and inverse Laplace
Z Transform And Inverse Z Transform - Signal And SystemsMr. RahüL YøGi
The z-transform is the most general concept for the transformation of discrete-time series.
The Laplace transform is the more general concept for the transformation of continuous time processes.
For example, the Laplace transform allows you to transform a differential equation, and its corresponding initial and boundary value problems, into a space in which the equation can be solved by ordinary algebra.
The switching of spaces to transform calculus problems into algebraic operations on transforms is called operational calculus. The Laplace and z transforms are the most important methods for this purpose.
This chapter provides complete solution of different circuits using Laplace transform method and also provides information about applications of Laplace transforms.
Some types of matrices, Eigen value , Eigen vector, Cayley- Hamilton Theorem & applications, Properties of Eigen values, Orthogonal matrix , Pairwise orthogonal, orthogonal transformation of symmetric matrix, denationalization of a matrix by orthogonal transformation (or) orthogonal deduction, Quadratic form and Canonical form , conversion from Quadratic to Canonical form, Order, Index Signature, Nature of canonical form.
Basic concepts of integration, definite and indefinite integrals,properties of definite integral, problem based on properties,method of integration, substitution, partial fraction, rational , irrational function integration, integration by parts, reduction formula, improper integral, convergent and divergent of integration
Partial differentiation, total differentiation, Jacobian, Taylor's expansion, stationary points,maxima & minima (Extreme values),constraint maxima & minima ( Lagrangian multiplier), differentiation of implicit functions.
critical points/ stationary points , turning points,Increasing, decreasing functions, absolute maxima & Minima, Local Maxima & Minima , convex upward & convex downward - first & second derivative tests.
Periodic Function, Dirichlet's Condition, Fourier series, Even & Odd functions, Euler's Formula for Fourier Coefficients, Change of Interval, Fourier series in the intervals (0,2l), (-l,l) , (-pi, pi), (0, 2pi), Half Range Cosine & Sine series Root mean square, Complex Form of Fourier series, Parseval's Identity
To find the complete solution to the second order PDE
(i.e) To find the Complementary Function & Particular Integral for a second order (Higher Order) PDE
Cauchy's integral theorem, Cauchy's integral formula, Cauchy's integral formula for derivatives, Taylor's Series, Maclaurin’s Series,Laurent's Series,Singularities and zeros, Cauchy's Residue theorem,Evaluation various types of complex integrals.
Complementary function, particular integral,homogeneous linear functions with constant variables, Euler Cauchy's equation, Legendre's equation, Method of variation of parameters,Simultaneous first order linear differential equation with constant coefficients,
Methods of integration, integration of rational algebraic functions, integration of irrational algebraic functions, definite integrals, properties of definite integral, integration by parts, Bernoulli's theorem, reduction formula
Analytic Function, C-R equation, Harmonic function, laplace equation, Construction of analytic function, Critical point, Invariant point , Bilinear Transformation
Operation “Blue Star” is the only event in the history of Independent India where the state went into war with its own people. Even after about 40 years it is not clear if it was culmination of states anger over people of the region, a political game of power or start of dictatorial chapter in the democratic setup.
The people of Punjab felt alienated from main stream due to denial of their just demands during a long democratic struggle since independence. As it happen all over the word, it led to militant struggle with great loss of lives of military, police and civilian personnel. Killing of Indira Gandhi and massacre of innocent Sikhs in Delhi and other India cities was also associated with this movement.
Read| The latest issue of The Challenger is here! We are thrilled to announce that our school paper has qualified for the NATIONAL SCHOOLS PRESS CONFERENCE (NSPC) 2024. Thank you for your unwavering support and trust. Dive into the stories that made us stand out!
Model Attribute Check Company Auto PropertyCeline George
In Odoo, the multi-company feature allows you to manage multiple companies within a single Odoo database instance. Each company can have its own configurations while still sharing common resources such as products, customers, and suppliers.
We all have good and bad thoughts from time to time and situation to situation. We are bombarded daily with spiraling thoughts(both negative and positive) creating all-consuming feel , making us difficult to manage with associated suffering. Good thoughts are like our Mob Signal (Positive thought) amidst noise(negative thought) in the atmosphere. Negative thoughts like noise outweigh positive thoughts. These thoughts often create unwanted confusion, trouble, stress and frustration in our mind as well as chaos in our physical world. Negative thoughts are also known as “distorted thinking”.
The Art Pastor's Guide to Sabbath | Steve ThomasonSteve Thomason
What is the purpose of the Sabbath Law in the Torah. It is interesting to compare how the context of the law shifts from Exodus to Deuteronomy. Who gets to rest, and why?
How to Create Map Views in the Odoo 17 ERPCeline George
The map views are useful for providing a geographical representation of data. They allow users to visualize and analyze the data in a more intuitive manner.
The French Revolution, which began in 1789, was a period of radical social and political upheaval in France. It marked the decline of absolute monarchies, the rise of secular and democratic republics, and the eventual rise of Napoleon Bonaparte. This revolutionary period is crucial in understanding the transition from feudalism to modernity in Europe.
For more information, visit-www.vavaclasses.com
Palestine last event orientationfvgnh .pptxRaedMohamed3
An EFL lesson about the current events in Palestine. It is intended to be for intermediate students who wish to increase their listening skills through a short lesson in power point.
1. Laplace transform is an
integral transform named after the great
French mathematician, Pierre Simon De
Laplace. It regularly used to transforms a
function of a real variable t to a function of a
complex variable s (complex frequency). The
Laplace transform changes one signal into
another according to some fixed set of rules
or equations. It is used to convert differential
equations into algebraic equations.
2. There are certain steps which need to be followed in order to do a
Laplace transform of a time function. In order to transform a given
function of time f(t) into its corresponding Laplace transform, we have
to follow the following steps:
First multiply f(t) by 𝑒−𝑠𝑡
, s being a complex number (s = a + i b).
Integrate this product w.r.t time with limits as zero and infinity. This
integration results in Laplace transformation of f(t), which is denoted
by F(s).
The Laplace transform L, of a function f(t) , t > 0 is defined by
0
∞
𝑓 𝑡 𝑒−𝑠𝑡
𝑑𝑡 , the resulting expression is a function s and denoted by
F(s). The Laplace transform of f(t) is denoted by L[f(t)]
𝑖. 𝑒 𝐿 𝑓 𝑡 = 0
∞
𝑓 𝑡 𝑒−𝑠𝑡
𝑑𝑡 = F(s)
3. The time function f(t) is obtained back from the Laplace
transform by a process called inverse Laplace
transformation and denoted by 𝐿−1
The inverse transform of F(s) = 𝐿−1
𝐹 𝑠 =
𝐿−1
𝐿 𝑓 𝑡 = 𝑓(𝑡)
4. The main properties of Laplace Transform as follows:
1. Linearity:
Let C1, C2 be constants. f(t), g(t) be the functions of time, t,
then
𝐿 𝐶1 𝑓 𝑡 + 𝐶2 𝑔 𝑡 = 𝐶1 𝐿 𝑓 𝑡 + 𝐶2 𝐿[𝑔 𝑡 ]
5. 2. First shifting Theorem: (s- shifting)
If 𝐿 𝑓 𝑡 = 𝐹 𝑠 then 𝐿 𝑒 𝑎𝑡 𝑓 𝑡 = 𝐹(𝑠 − 𝑎) and 𝐿 𝑒−𝑎𝑡 𝑓 𝑡 = 𝐹(𝑠 + 𝑎)
Also 𝐿−1
𝐹 𝑠 − 𝑎 = 𝑒 𝑎𝑡
𝑓 𝑡 and 𝐿−1
𝐹 𝑠 + 𝑎 = 𝑒−𝑎𝑡
𝑓 𝑡
3. Second shifting Theorem: (t- shifting, time shifting)
If 𝐿 𝑓 𝑡 = 𝐹 𝑠 then the Laplace Transform of f(t) after the delay of
time, T is equal to the product of Laplace Transform of f(t) and e-st
𝑖. 𝑒 𝐿 𝑓 𝑡 − 𝑇 𝑢 𝑡 − 𝑇 = 𝑒−𝑠𝑡
𝐹(𝑠) , where 𝑢(𝑡 − 𝑇) is the unit
step function.
Note: 𝑢 𝑛(𝑡) is the Heaviside step function which is given by
𝑢 𝑡 − 𝑛 =
0 𝑖𝑓 𝑡 < 𝑛
1𝑖𝑓 𝑡 ≥ 𝑛
In particular, 𝑢 𝑡 =
0, 𝑖𝑓 𝑡 < 0
1, 𝑖𝑓 𝑡 ≥ 0
9. Procedure:
1. Take the Laplace transform on both side of the given
differential equation.
2. Use the initial conditions , which gives an algebraic equation.
3. Solve the algebraic equation and get the value of L(y) in terms
of s, which is F(s).(𝑖. 𝑒 𝐿 𝑦 = 𝐹(𝑠))
4. Find y by taking inverse Laplace transformation,𝑦 = 𝐿−1[𝐹 𝑠 ],
which is the required solution.
10. Periodic Function:
A function f(t) is said to be periodic with period T (> 0) 𝑖𝑓 𝑓(𝑡 + 𝑇) = 𝑓(𝑡)
Anti-Periodic function:
A function f(t) is said to be anti-periodic with period T, if 𝑓(𝑡 + 𝑇) = −𝑓(𝑡)
for all t
𝑖. 𝑒 𝑓 𝑡 + 2𝑇 = 𝑓 𝑡 + 𝑇 + 𝑇 = −𝑓 𝑡 + 𝑇 = − −𝑓 𝑡 = 𝑓(𝑡)
Example :
1.𝑆𝑖𝑛 2𝜋 + 𝑡 = 𝑠𝑖𝑛𝑡, hence 𝑠𝑖𝑛𝑡 is periodic function with period 2𝜋.
2.𝑆𝑖𝑛 𝜋 + 𝑡 = −𝑠𝑖𝑛𝑡, hence 𝑠𝑖𝑛𝑡 is anti periodic function with period
𝜋.
11. Laplace Transform of Periodic Functions
If f(t) is periodic with period T > 0, then 𝐿 𝑓 𝑡 =
1
1−𝑒−𝑠𝑇 0
𝑇
𝑒−𝑠𝑡 𝑓 𝑡 𝑑𝑡
Laplace Transform of Anti-Periodic Functions
If f(t) is anti-periodic with period T > 0, then 𝐿 𝑓 𝑡 =
1
1+𝑒−𝑠𝑇 0
𝑇
𝑒−𝑠𝑡 𝑓 𝑡 𝑑𝑡