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Laplace Transform is used to convert differential equations into algebraic equations. The inverse Laplace transform is the equation that transforms a Laplace transform into a function of time. There are often situations and cases where the differential equations need to be converted into algebraic equations to make it easy to study and analyse. They are often used in engineering systems, electrical or control systems.
Key Terms: Laplace Transform, Inverse Laplace Transform, Differential Equations, Algebraic Equations, Mellin’s and Post’s Inverse Formula
Laplace Transforms
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Laplace Transform is a method which transforms a differential equation into an algebraic equation. It is widely used in control engineering and in electrical circuit analysis. It is not only used to write differential equations to describe conditions but also to write directly in terms of the Laplace transform. Here, the input is referred to as being the time domain while the output is said to be in the s-domain. Thus, this equation takes information about a system in the time domain and uses a machine to transform it into information in the s-domain. Differential equations that describe the behaviour of a system in the time domain are converted into algebraic equations in the s-domain, thus considerably simplifying the solution.
The Laplace transform equation is given by;
\(\bigtriangledown²f = 0, or \bigtriangledown f = 0\)
Here,
\(\bigtriangledown = \bigtriangledown •\bigtriangledown = \bigtriangledown²\)
\(\bigtriangledown²\) is the Laplace operator
\(\bigtriangledown•\) is the divergence operator
\(\bigtriangledown\) is the gradient operator
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Inverse Laplace Transform
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The inverse Laplace transform is the equation that transforms a Laplace transform into a function of time. If L{f(t)} = F(s) is the Laplace transform of F(s), the inverse being written as,
f(t) = L-1 {F(s)}
The inverse laplace transform for a few functions can be seen from the table below:
| Function y(a) | Transform Y(b) | b |
|---|---|---|
| 1 | 1/b | b>0 |
| a | 1/b² | b>0 |
| Ai , i = integer | i!/s(i+1) | b>0 |
| exp (ta), where t = constant | 1/(b−t) | b>t |
| cos (sa), s= constant | b/b²+s² | b>0 |
| Sin (sa), s = constant | t/b²+s² | b>0 |
| exp(ta)cos(sa) | b−t/(b−t)²+s² | b>t |
| exp(ta)sin(sa) | s/(b−t)²+s² | b>t |
There are two important integral parts of inverse laplace transforms, they are:
- Mellin’s Inverse Formula: It is one of the two crucial parts of inverse laplace transforms. It can be represented as:
- Post’s Inverse Formula: It is an easy, simple but impractical formula. It can be represented as:
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Determinants Detailed Video Explanation:
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Properties of Inverse Laplace Transform
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The following are basic properties of inverse laplace transform:
- Additive Properties: If we have a Laplace transform as the sum of two separate terms then we can take the inverse of each separately and the sum of the two inverse Laplace transforms is the inverse of their sum.
- First Shift Theorem: The first shift theorem can be written in inverse form as:
L-1 {F(s-a)} = eatf(t)
where f(t) is the inverse transform of F{s}.
- Second Shift Theorem: The second shift theorem can be written in inverse form as:
L-1 {e-stF(s-a)} = f(t-T)u(t-T)
How to Solve Inverse Laplace Transform Based Questions
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To solve inverse laplace transform based questions, one may follow the following steps:
- Note down all the denominators and numerators.
- Factor out all the numbers.
- Note down all the exponentials.
- Then solve the questions.
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Things to Remember
- Laplace Transform is used to transform a differential equation into an algebraic equation.
- The inverse Laplace transform is the equation that transforms a Laplace transform into a function of time.
- There are three basic properties of inverse laplace transform, they are: additive property, first shift theorem, and second shift theorem.
- The laplace and inverse laplace transform can be used in conversion of differential equations into matrix form.
- Double laplace transform is applied in crossflow heat exchangers.
Sample Questions
Ques. Find the inverse transform of the following term:
Y(b) = 6/b - 1/(b-8) - 5/(b-3) (2 Marks)
Ans. Y(b)= 6(1/b) - 1/(b-8) - 5 [1/(b-3)]
Y(a) = 6(1) – ea8 +5 (e3t)
Y(a) = 6 – ea8 +5e3t
Ques. Find the inverse transform of the following function:
Y(b) = 7/b - 1/(b-5) - 5/(b-3) (2 Marks)
Ans. Y(b)= 7(1/b) - 1/(b-5) - 5 [1/(b-3)]
Y(a) = 7(1) – ea5 +5 (e3t)
Y(a) = 7 – ea5 +5e3t
Ques. Find the inverse transform of the following:
Y(b) = 6/b - 1/(b-8) - 8/(b-3) (2 Marks)
Ans. Y(b)= 6(1/b) - 1/(b-8) - 8 [1/(b-3)]
Y(a) = 6(1) – ea8 +8(e3t)
Y(a) = 6 – ea8 +8e3t
Ques. Find the inverse transform of the following term:
Y(b) = 4/b - 1/(b-8) - 4/(b-3) (2 Marks)
Ans. Y(b)= 4(1/b) - 1/(b-8) - 4 [1/(b-3)]
Y(a) = 6(1) – ea8 +4 (e3t)
Y(a) = 6 – ea8 +4e3t
Ques. Find the inverse transform of the following function:
Y(b) = 6/b - 1/(b-9) - 1/(b-3) (2 Marks)
Ans. Y(b)= 6(1/b) - 1/(b-9) - 1/(b-3)
Y(a) = 6(1) – ea9 + 1(e3t)
Y(a) = 6 – ea9 +e3t
Ques. Find the inverse transform of the following:
Y(b) = 1/(b-8) - 5/(b-3) (2 Marks)
Ans. Y(b)= 1/(b-8) - 5 [1/(b-3)]
Y(a) = ea8 +5 (e3t)
Y(a) = ea8 +5e3t
Ques. Find the inverse transform of the following:
Y(b) = 6/b - 1/(b-8) (2 Marks)
Ans. Y(b)= 6(1/b) - 1/(b-8)
Y(a) = 6(1) – ea8
Y(a) = 6 – ea8
Ques. Find the inverse transform of the following:
Y(b) = 8/b - 1/(b-8) - 5/(b-3) (2 Marks)
Ans. Y(b)= 6(1/b) - 1/(b-8)
Y(a) = 8(1) – ea8
Y(a) = 8 – ea8
Ques. Find the inverse transform of the following:
Y(b) = 10/b - 1/(b-1) - 5/(b-3) (2 Marks)
Ans. Y(b)= 10(1/b) - 1/(b-1) - 5 [1/(b-3)]
Y(a) = 10(1) – ea +5 (e3t)
Y(a) = 10 – ea +5e3t
Ques. Find the inverse transform of the following equation:
Y(b) = 7/b (2 Marks)
Ans. Y(b)= 7(1/b)
Y(a) = 7(1)
Y(a) = 7
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