The Telegraph Equation and Its Solution by Reduced Differential Transform Method

Table 1

Fundamental operations of the reduced differential transform method.

Original function	Reduced differential transformed function
𝑅𝐷[𝑢(𝑥,𝑡)𝑣(𝑥,𝑡)]	𝑈𝑘(𝑥)⊗𝑉𝑘(𝑥)=∑^𝑘𝑟=0𝑈𝑟(𝑥)𝑉𝑘−𝑟(𝑥)
𝑅𝐷𝛼𝑢(𝑥,𝑡)±𝛽𝑣(𝑥,𝑡)	𝛼𝑈𝑘(𝑥)±𝛽𝑉𝑘(𝑥)
𝑅𝐷𝜕𝑢(𝜕𝑥𝑥,𝑡)	𝜕𝑈𝜕𝑥^𝑘(𝑥)
𝑅𝐷𝜕𝑢(𝜕𝑡𝑥,𝑡)	(𝑘+1)𝑈𝑘+1(𝑥)
𝑅𝐷𝜕^𝑟+𝑠𝜕𝑥^𝑟𝜕𝑡^𝑠𝑢(𝑥,𝑡)	(𝑘+𝑠)!𝜕𝑘!𝜕𝑥^𝑟𝑟𝑈𝑘+𝑠(𝑥)
𝑅𝐷𝑥^𝑚𝑡^𝑛	𝑥^𝑚,𝑘=𝑛0,otherwise
𝑅𝐷𝑒^𝜆𝑡	𝜆^𝑘𝑘!
𝑅𝐷[sin(𝑤𝑡+𝛼)]	𝑤^𝑘𝑘!sin𝜋𝑘2!+𝛼
𝑅𝐷[cos(𝑤𝑡+𝛼)]	𝑤^𝑘𝑘!cos𝜋𝑘2!+𝛼