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

Table 1

The fundamental operations of two-dimensional differential transform method.

Original function	Transformed function
𝑤(𝑥,𝑡)=𝑢(𝑥,𝑡)±𝑣(𝑥,𝑡)	𝑊(𝑚,𝑛)=𝑈(𝑚,𝑛)±𝑉(𝑚,𝑛)
𝑤(𝑥,𝑡)=𝑐𝑢(𝑥,𝑡)	𝑊(𝑚,𝑛)=𝑐𝑈(𝑚,𝑛)
𝜕𝑤(𝑥,𝑡)=𝜕𝑥𝑢(𝑥,𝑡)	𝑊(𝑚,𝑛)=(𝑚+1)𝑈(𝑚+1,𝑛)
𝜕𝑤(𝑥,𝑡)=𝜕𝑡𝑢(𝑥,𝑡)	𝑊(𝑚,𝑛)=(𝑚+1)𝑈(𝑚,𝑛+1)
𝜕𝑤(𝑥,𝑡)=^𝑟+𝑠𝜕𝑥^𝑟𝜕𝑡^𝑠𝑢(𝑥,𝑡)	𝑊(𝑚,𝑛)=(𝑚+𝑟)!(𝑛+𝑠)!𝑚!𝑛!𝑈(𝑚+𝑟,𝑛+𝑠)
𝑤(𝑥,𝑡)=𝑢(𝑥,𝑡)𝑣(𝑥,𝑡)	𝑊(𝑚,𝑛)=∑^𝑚𝑛𝑟=0∑𝑠=0𝑈(𝑟,𝑛−𝑠)𝑉(𝑚−𝑟,𝑠)
𝑤(𝑥,𝑡)=𝑥^𝛼𝑡^𝛽	𝑊(𝑚,𝑛)=𝛿(𝑚−𝛼,𝑛−𝛽)=1𝑚=𝛼,𝑛=𝛽0otherwise