Numerical Study of Two-Dimensional Volterra Integral Equations by RDTM and Comparison with DTM

Table 2

The fundamental operations of two-dimensional RDTM.

Original function	Reduced transformed function
𝑤(𝑥,𝑡)=𝑢(𝑥,𝑡)±𝑣(𝑥,𝑡)	𝑊𝑘(𝑥)=𝑈𝑘(𝑥)±𝑉𝑘(𝑥)
𝜕𝑤(𝑥,𝑡)=𝜕𝑥𝑢(𝑥,𝑡)	𝑊𝑘𝜕(𝑥)=𝑈𝜕𝑥^𝑘(𝑥)
𝜕𝑤(𝑥,𝑡)=𝜕𝑡𝑢(𝑥,𝑡)	𝑊𝑘(𝑥)=(𝑘+1)𝑈𝑘+1(𝑥)
𝜕𝑤(𝑥,𝑡)=^𝑟+𝑠𝜕𝑥^𝑟𝜕𝑡^𝑠𝑢(𝑥,𝑡)	𝑊𝑘(𝑥)=(𝑘+𝑠)!𝜕𝑘!𝜕𝑥^𝑟𝑟𝑈𝑘+𝑠(𝑥)
𝑤(𝑥,𝑡)=𝑢(𝑥,𝑡)𝑣(𝑥,𝑡)	𝑊𝑘(𝑥)=∑^𝑘𝑟=0𝑈𝑟(𝑥)𝑉𝑘−𝑟(𝑥)
𝑤(𝑥,𝑡)=𝑥^𝑚𝑡^𝑛	𝑊𝑘(𝑥)=𝑥^𝑚𝑥𝛿(𝑘−𝑛)=^𝑚𝑘=𝑛0otherwise