**Example 1.**

*(i) The locally integrable function f*(*t*) *is a distribution generated by the locally integrable function f*(*t*)*. Then we define f*(*t*), *ϕ*(*t*) = -Ω *f*(*t*)*ϕ*(*t*)*dt, where* Ω *is the support of ϕ*(*t*) *and ϕ*(*t*) ∈ D*.*

*(ii) The Dirac delta function is a distribution defined by δ*(*t*), *ϕ*(*t*) = *ϕ*(0) *and the support of δ*(*t*) *is* {0}*.*

A distribution *T* generated by a locally integrable function is called a regular distribution; otherwise, it is called a singular distribution.

**Definition 3.** *The kth-order derivative of a distribution T, denoted by <sup>T</sup>*(*k*)*, is defined by T*(*k*), *ϕ*(*t*)= (−<sup>1</sup>)*<sup>k</sup> T*, *ϕ*(*k*)(*t*) *for all ϕ*(*t*) ∈ D*.*
