Laplace Transform: Sufficient Conditions For Existence

LAPLACE TRANSFORM ‒ SUFFICIENT CONDITIONS FOR EXISTENCE

Definition: The Laplace Transformation

Let a function f(t) be continuous and defined for positive values of 't'. The Laplace transformation of f(t) associates a function s defined by the equation

 L [f(t)] = ϕ(s) = F(s) = ₀ʃ ^∞ e^‒st f(t) dt

Here, F(s) is said to be the Laplace transform of f(t) and it is written as L[f(t)] or L[f]. Thus, F(s) L[f(t)]

 L[f(t)] = ₀ʃ ^∞ e^−st f(t) at, t >0

PROBLEMS BASED ON LAPLACE TRANSFORM – SUFFICIENT CONDITIONS FOR EXISTENCE

Example 1: State the conditions under which the Laplace Transform of f(t) exists.

Solution:

If a function f(t) is continuous or (piecewise) sectionally continuous in a closed interval a ≤ t ≤ b and is of exponential order, then its Laplace Transform exists.

(OR)

 (i) f(t) should be continuous or piecewise continuous in the given closed interval [a, b], where a > 0.

(ii) f(t) should be of exponential order.

Examples

(1) L[] does not exist, since  is not of any exponential order.

(2) L[ cot t ] does not exist, since cott is not piecewise continuous.

 i.e., cot t has an infinite number of infinite discontinuities at  0, ±π, ± 2π, ……

(3) L(tant) does not exist, since tant is not piecewise continuous.

i.e., tant has infinite number of infinite discontinuities at π/2, 3π/2, 5π/2, …

Definition: Exponential order

A function f(t) is said to be of exponential order if

Example 2: Show that xⁿ is of exponential order as x→ ∞, n > 0.

Solution:

Hence, xⁿ is of exponential order.

Example 3: Show that t² is of exponential order.

Solution:

Hence, t² is of exponential order.

Example 4: Show that the function f(t) = e^t2 is not of exponential order.

Solution:

So, f(t)=e^t2 is not of exponential order.

Example 5: Define function of class A.

Solution:

A function which is sectionally continuous over any finite interval and is of exponential order is known as a function of class A.

EXERCISE

1. Show that tⁿ is of exponential order.

2. Show that √t is of exponential order.

3. State the conditions under which the Laplace Transform of f(1) exists.

4. Write any two uses of Laplace Transform.

Important Result