International Scholarly Research Notices

Research Article

Comparing Numerical Methods for Solving Time-Fractional Reaction-Diffusion Equations

Table 1

The operations of the GDTM.
Original functions Transformed functions
𝑢 ( 𝑥 , 𝑦 ) = 𝜈 ( 𝑥 , 𝑦 ) ± 𝑤 ( 𝑥 , 𝑦 ) 𝑈 𝛼 , 𝛽 ( 𝑘 , ℎ ) = 𝑉 𝛼 , 𝛽 ( 𝑘 , ℎ ) ± 𝑊 𝛼 , 𝛽 ( 𝑘 , ℎ )
  
𝑢 ( 𝑥 , 𝑦 ) = 𝜆 𝜈 ( 𝑥 , 𝑦 ) 𝑈 𝛼 , 𝛽 ( 𝑘 , ℎ ) = 𝜆 𝑉 𝛼 , 𝛽 ( 𝑘 , ℎ )
  
𝑢 ( 𝑥 , 𝑦 ) = 𝐷 𝛼 𝑥 0 𝜈 ( 𝑥 , 𝑦 ) 𝑈 𝛼 , 𝛽 ( 𝑘 , ℎ ) = Γ ( 𝛼 ( 𝑘 + 1 ) + 1 ) 𝑉 Γ ( 𝛼 𝑘 + 1 ) 𝛼 , 𝛽 ( 𝑘 + 1 , ℎ ) ,   0 < 𝛼 ≤ 1
  
𝑢 ( 𝑥 , 𝑦 ) = 𝐷 𝛽 𝑦 0 𝜈 ( 𝑥 , 𝑦 ) 𝑈 𝛼 , 𝛽 ( 𝑘 , ℎ ) = Γ ( 𝛽 ( ℎ + 1 ) + 1 ) 𝑉 Γ ( 𝛽 ℎ + 1 ) 𝛼 , 𝛽 ( 𝑘 , ℎ + 1 ) ,   0 < 𝛽 ≤ 1
  
𝑢 ( 𝑥 , 𝑦 ) = 𝐷 𝛼 𝑥 0 𝐷 𝛽 𝑦 0 𝜈 ( 𝑥 , 𝑦 ) 𝑈 𝛼 , 𝛽 ( 𝑘 , ℎ ) = Γ ( 𝛼 ( 𝑘 + 1 ) + 1 ) Γ ( 𝛽 ( ℎ + 1 ) + 1 ) 𝑉 Γ ( 𝛼 𝑘 + 1 ) Γ ( 𝛽 ℎ + 1 ) 𝛼 , 𝛽 ( 𝑘 + 1 , ℎ + 1 ) ,   0 < 𝛼 , 𝛽 ≤ 1
  
𝑢 ( 𝑥 , 𝑦 ) = 𝐷 𝛾 𝑥 0 𝜈 ( 𝑥 , 𝑦 ) 𝑈 𝛼 , 𝛽 ( 𝑘 , ℎ ) = Γ ( 𝛼 𝑘 + 𝛾 + 1 ) 𝑉 Γ ( 𝛼 𝑘 + 1 ) 𝛼 , 𝛽 𝛾 𝑘 + 𝛼 , ℎ ,   𝑚 − 1 < 𝛾 ≤ 1
  
𝑢 ( 𝑥 , 𝑦 ) = 𝐷 𝛾 𝑥 0 𝐷 𝛿 𝑦 0 𝜈 ( 𝑥 , 𝑦 ) 𝑈 𝛼 , 𝛽 ( 𝑘 , ℎ ) = Γ ( 𝛼 𝑘 + 𝛾 + 1 ) Γ ( 𝛽 ℎ + 𝛿 + 1 ) 𝑉 Γ ( 𝛼 𝑘 + 1 ) Γ ( 𝛽 ℎ + 1 ) 𝛼 , 𝛽 𝛾 𝑘 + 𝛼 𝛿 , ℎ + 𝛽
  
𝑢 ( 𝑥 , 𝑦 ) = ( 𝑥 − 𝑥 0 ) 𝑘 𝛼 ( 𝑥 − 𝑥 0 ) ℎ 𝛽 𝑈 𝛼 , 𝛽 ( 𝑘 , ℎ ) = 𝛿 ( 𝑘 − 𝑛 ) 𝛿 ( ℎ − 𝑚 )
  
𝑢 ( 𝑥 , 𝑦 ) = 𝜈 ( 𝑥 , 𝑦 ) 𝑤 ( 𝑥 , 𝑦 ) 𝑈 𝛼 , 𝛽 ( 𝑘 , ℎ ) = 𝑘 ∑ 𝑟 = 0   ℎ ∑ 𝑠 = 0 𝑉 𝛼 , 𝛽 ( 𝑟 , ℎ − 𝑠 ) 𝑊 𝛼 , 𝛽 ( 𝑘 − 𝑟 , 𝑠 )
  
𝑢 ( 𝑥 , 𝑦 ) = 𝜈 ( 𝑥 , 𝑦 ) 𝑤 ( 𝑥 , 𝑦 ) 𝑞 ( 𝑥 , 𝑦 ) 𝑈 𝛼 , 𝛽 ( 𝑘 , ℎ ) = 𝑘 ∑ 𝑟 = 0   𝑘 − 𝑟 ∑ 𝑡 = 0   ℎ ∑ 𝑠 = 0   ℎ − 𝑠 ∑ 𝑝 = 0 𝑉 𝛼 , 𝛽 ( 𝑟 , ℎ − 𝑠 − 𝑝 ) 𝑊 𝛼 , 𝛽 ( 𝑡 , 𝑠 ) 𝑄 𝛼 , 𝛽 ( 𝑘 − 𝑟 − 𝑡 , 𝑝 )