Numerical Simulation of Fractional Fornberg-Whitham Equation by Differential Transformation Method

Table 1

Operations of the two-dimensional differential transform.

Original function	Transformed function
𝑢(𝑥,𝑦)=𝑓(𝑥,𝑦)∓𝑔(𝑥,𝑦)	𝑈𝛼,𝛽(𝑘,ℎ)=𝐹𝛼,𝛽(𝑘,ℎ)∓𝐺(𝑘,ℎ)
𝑢(𝑥,𝑦)=𝜉𝑓(𝑥,𝑦)	𝑈𝛼,𝛽(𝑘,ℎ)=𝜉𝐹𝛼,𝛽(𝑘,ℎ)
𝑢(𝑥,𝑦)=𝜕𝑓(𝑥,𝑦)/𝜕𝑥	𝑈𝛼,𝛽(𝑘,ℎ)=(𝑘+1)𝐹(𝑘+1,ℎ)
𝑢(𝑥,𝑦)=𝐷𝛼∗𝑥0𝑓(𝑥,𝑦),0<𝛼≤1	𝑈𝛼,𝛽(𝑘,ℎ)=(Γ(𝛼(𝑘+1)+1)/Γ(𝛼𝑘+1))𝐹𝛼,𝛽(𝑘+1,ℎ)
𝑢(𝑥,𝑦)=𝐷𝛼∗𝑦0𝑓(𝑥,𝑦),0<𝛼≤1	𝑈𝛼,𝛽(𝑘,ℎ)=(Γ(𝛼(ℎ+1)+1)/Γ(𝛼ℎ+1))𝐹𝛼,𝛽(𝑘,ℎ+1)
𝑢(𝑥,𝑦)=(𝑥−𝑥0)𝑚𝛼(𝑦−𝑦0)𝑛𝛽	𝑈𝛼,𝛽(𝑘,ℎ)=𝛿(𝑘−𝑚,ℎ−𝑛)={{{{{1,𝑘=𝑟,ℎ=𝑠0,otherwise
𝑢(𝑥,𝑦)=𝑓(𝑥,𝑦)𝑔(𝑥,𝑦)	𝑈𝛼,𝛽(𝑘,ℎ)=∑𝑘𝑚=0∑ℎ𝑛=0𝐹𝛼,𝛽(𝑚,ℎ−𝑛)𝐺𝛼,𝛽(𝑘−𝑚,𝑛)
𝑢(𝑥,𝑦)=𝑓(𝑥,𝑦)𝑔(𝑥,𝑦)ℎ(𝑥,𝑦)	𝑈𝛼,𝛽(𝑘,ℎ)=∑𝑘𝑘4=0∑𝑘−𝑘4𝑘3=0∑ℎ𝑘2=0∑ℎ−𝑘2𝑘1=0𝐹𝛼,𝛽(𝑘4,ℎ−𝑘2−𝑘1)
	𝐺𝛼,𝛽(𝑘3,𝑘2)𝐻𝛼,𝛽(𝑘−𝑘4−𝑘3,𝑘1)