1. SIGMA INSTITUTE OF ENGINEERING
Special Laplace Transforms
Prepared By:
Name
Enrollment No.
Meet Bhatt
Mitesh Darji
Jenson Joseph
Disha Parmar
Swamidas Parmar

2. Laplace Transform Laplace transform(L.T.)
; a mathematical tool that can significantly reduce the effort required to solve linear differential equation model.
Major benefit ; this transformation converts differential equations to algebraic equations, which can simplify the mathematical manipulations required to obtain a solution.

3. Laplace Transform of Representative Functions
Definition
Where the function f(t) must satisfy conditions which include being piecewise continuous for
Inverse transform ;
Property; Linear operator; it satisfies the general superposition principle.

4. Examples
1. Constant function
, ( is a constant)
2. Unit step function

5. Delta function (the first derivative of the step function)

6. 3. Derivatives
n-th order derivative
where is the i-th derivative evaluated at t=0.

7. 4. Exponential function
if b<0, the real part of s must be restricted to be larger than -b for the integrated to be finite.
5. Trigonometric functions
Euler Identity

8. 6. Rectangular pulse function
Figure 3.1. The rectangular pulse function.

9. General solution procedure
1. Take the Laplace transform of both sides of the differential equation.
2. Solve for
If the expression for dose not appear in table.
If the expression for not appear in table, go to Step 5.
3. Factor the characteristic equation polynomial.
4. Perform the Partial Fraction Expansion.
5. Use the inverse Laplace transform relations to find y(t).

10. Partial Fraction Expansion
Example) Consider
How to find and
Method 1.
Multiply both sides of (3.24) by
Equating coefficients of each power of s, we have
Solving for and simultaneously.  ,

11. Method 3. Heaviside expansion
Method 2. Since (3.24) should hold for all values of s, we can specify two values of s and solve for the two constant.
Solving,
Method 3. Heaviside expansion
The fastest and most popular method.
We multiply both sides of the equation by one of the denominator terms
and then set , which causes all terms except one to be multiplied by zero.

12. Heaviside Expansion - General form: Case 1) If all are distinct.
Case 2) Repeated Factors ; If a denominator term occur times in the denominator, terms must be included in the
expansion that incorporate the factor
in addition to the other factors.

13. Differentiation approach
Approach 2 ; Differentiation of the transform.
Multiply both sides of (3.34).
Then (3.39) is differentiated twice with respect to ,
, so that
Differentiation approach
If the denominator polynomial contains the repeated factor
, first form the quantity.
The general expression for is given as follows

14. Case 3) Complex Factors
Apply Inverse Laplace transform (3.43).
With the use of the identities,
(3.44) becomes

15. Other Laplace Transform Properties
Final value theorem
If exits,
Proof) Use the relation for the Laplace transform of a derivative.
Taking the limit as and assuming that is continuous and has a limit for all , we find
Integrating the left side and simplifying yields

16. Transform of an integral
Initial value theorem
In the analogy to the final value theorem, initial value theorem can be stated as
Proof) The proof of this theorem is similar to the case of the final value theorem.
Transform of an integral

17. Time Delay(Translation in Time)
Time delays are commonly encountered in process control problems because of the transport time required for a fluid to flow through piping.
Figure 3.2. A time function with and without time delay.
(a) Original function(no delay); (b) Function with delay.
Where is a unit step function,