Extension of Zhou's Method to Neutral Functional-Differential Equation with Proportional Delays

Table 1

Fundamental operations of differential transformation.

Original function	Transformed function
𝑢(𝑡)=𝑓(𝑡)±𝑔(𝑡)	𝑈(𝑘)=𝐹(𝑘)±𝐺(𝑘)
𝑢(𝑡)=𝑐𝑓(𝑡)	𝑈(𝑘)=𝑐𝐹(𝑘),𝑐∈ℛ
𝑢(𝑡)=𝑓(𝑡)𝑔(𝑡)	𝑈(𝑘)=∑^𝑘ℓ=0𝐹(ℓ)𝐺(𝑘−ℓ)
𝑢(𝑡)=𝑓1(𝑡)𝑓2(𝑡)⋯𝑓𝑚−1(𝑡)𝑓𝑚(𝑡)	𝑈(𝑘)=×∑𝐹^𝑘ℓ𝑚−1=02∑^ℓℓ𝑚−1𝑚−2=0⋯∑^ℓ3ℓ2=0×∑^ℓ2ℓ1=0𝐹1(ℓ1)(ℓ2−ℓ1)⋯𝐹𝑚−1(ℓ𝑚−1−ℓ𝑚−1)𝐹(𝑘−ℓ𝑚−1)
𝑢(𝑡)=(𝑑^𝑛/𝑑𝑡^𝑛)𝑓(𝑡)	𝑈(𝑘)=((𝑘+𝑛)!/𝑘!)𝐹𝑚(𝑘)
𝑢(𝑡)=𝑡^𝑚	𝑈(𝑘)=𝛿(𝑘−𝑚)=^1,𝑘=𝑚;0,𝑘≠𝑚