Nonlinear Klein-Gordon and Schrödinger Equations by the Projected Differential Transform Method

Table 1

Fundamental operations for the three-dimensional DTM.

Original function 𝑤(𝑥,𝑦,𝑡)	Transformed function 𝑊(𝑘,ℎ,𝑚)
𝑢(𝑥,𝑦,𝑡)±𝑣(𝑥,𝑦,𝑡)	𝑈(𝑘,ℎ,𝑚)±𝑉(𝑘,ℎ,𝑚)
𝑐𝑢(𝑥,𝑦,𝑡)	𝑐𝑈(𝑘,ℎ,𝑚)
𝜕^𝑟+𝑠+𝑝𝑢(𝑥,𝑦,𝑡)𝜕𝑥^𝑟𝜕𝑦^𝑠𝜕𝑡^𝑝	(𝑘+𝑟)!𝑘!(ℎ+𝑠)!𝑠!(𝑚+𝑝)!𝑝!𝑈(𝑘+𝑟,ℎ+𝑠,𝑚+𝑝)
𝜕𝑢(𝑥,𝑦,𝑡)𝜕𝑥𝜕𝑣(𝑥,𝑦,𝑡)𝜕𝑦	∑^𝑘𝑟=0∑^ℎ𝑠=0∑^𝑚𝑝=0(𝑘−𝑟+1)(ℎ−𝑠+1)𝑈(𝑘−𝑟+1,𝑠,𝑝)𝑈(𝑟,ℎ−𝑠+1,𝑚−𝑝)
𝑢(𝑥,𝑦,𝑡)𝑣(𝑥,𝑦,𝑡)	∑^𝑗𝑟=0∑^ℎ𝑠=0∑^𝑚𝑝=0𝑈(𝑟,ℎ−𝑠,𝑚−𝑝)𝑉(𝑘−𝑟,𝑠,𝑝)
𝑢(𝑥,𝑦,𝑡)𝑣(𝑥,𝑦,𝑡)𝑔(𝑥,𝑦,𝑡)	∑^𝑘𝑟=0∑^𝑘−𝑟𝜏=0∑^ℎ𝑠=0∑^ℎ−𝑠𝑝=0∑^𝑚𝑙=0∑^𝑚−𝑙𝑞=0𝑈(𝑟,ℎ−𝑠−𝑝,𝑙)𝑉(𝜏,𝑠,𝑚−𝑙−𝑞)𝐺(𝑘−𝑟−𝜏,𝑝,𝑞)