Riley hobson bence · Problem 13.27

Integral transforms

Mathematics — Methods · laplace-transforms-convolution

Problem

The function

f_a(x)

is defined as unity for

0 < x < a

and zero otherwise. Find its Laplace transform

\bar{f}_a(s)

and deduce that the transform of

xf_a(x)

\frac{1}{s^2}\left[1-(1+as)e^{-sa}\right].

Write

f_a(x)

in terms of Heaviside functions and hence obtain an explicit expression for

g_a(x) = \int_0^x f_a(y)f_a(x-y)\,dy.

Use the expression to write

\bar{g}_a(s)

in terms of the functions

\bar{f}_a(s)

and

\bar{f}_{2a}(s)

, and their derivatives, and hence show that

\bar{g}_a(s)

is equal to the square of

\bar{f}_a(s)

, in accordance with the convolution theorem.