Temperature Variations Analysis for Condensed Matter Micro- and Nanoparticles Combustion Burning in Gaseous Oxidizing Media by DTM and BPES

Table 1

Some fundamental operations of the differential transformation method.

Origin function	Transformed function
𝑥(𝑡)=𝛼𝑓(𝑥)±𝛽𝑔(𝑡)	𝑋(𝑘)=𝛼𝐹(𝑘)±𝛽𝐺(𝑘)
𝑑𝑥(𝑡)=^𝑚𝑓(𝑡)𝑑𝑡^𝑚	𝑋(𝑘)=(𝑘+𝑚)!𝐹(𝑘+𝑚)𝑘!
𝑥(𝑡)=𝑓(𝑡)𝑔(𝑡)	𝑋(𝑘)=∑^𝑘𝑙=0𝐹(𝑙)𝐺(𝑘−𝑙)
𝑥(𝑡)=𝑡^𝑚	𝑋(𝑘)=𝛿(𝑘−𝑚)=1,if𝑘=𝑚,0,if𝑘≠𝑚
𝑥(𝑡)=exp(𝑡)	1𝑋(𝑘)=𝑘!
𝑥(𝑡)=sin(𝜔𝑡+𝛼)	𝜔𝑋(𝑘)=^𝑘𝑘!sin𝑘𝜋2+𝛼
𝑥(𝑡)=cos(𝜔𝑡+𝛼)	𝜔𝑋(𝑘)=^𝑘𝑘!cos𝑘𝜋2+𝛼