A Domain Decomposition Method for Time Fractional Reaction-Diffusion Equation

Algorithm 1

Domain decomposition algorithm for time fractional reaction-diffusion equation.

(1) [ℎ!]   input: 𝑀,𝑃,𝑥_𝑅,𝑇,𝜇,𝐾,𝜖 et al.

 Output: 𝑈

(2) 𝐼←𝑀/𝑃

(3) Allocate memory space 𝑈^𝑁_𝐼,𝑃,𝐴_𝐼,3,𝑃,𝐹^𝑁_𝐼,𝑃,𝑉³_𝐼,𝑃 et al.

(4) Init matrices 𝑈,𝐴,𝐹,𝑏,𝑟 et al.

(5) Declare local variables 𝛿_𝑃,𝛿

(6) 𝑉^1→3_{1→𝐼,1→𝑃}←0

(7) Get 𝑈⁰_{1→𝐼,1→𝑃} with initial boundary.

(8) 𝑡𝑜𝑡𝑎𝑙𝐼𝑡𝑒𝑟𝑎𝑡𝑖𝑜𝑛←0

(9) for  𝑛=0  to  𝑁−1 step by 1 do

(10)  for 𝑝=1  to  𝑃 step by 1 do

(11)     𝑈^𝑛+1_{1→𝐼,𝑝}←𝑏_𝑛𝑈⁰_{1→𝐼,𝑝}+𝐾𝐹^𝑛+1_{1→𝐼,𝑝}

(12)     for  𝑘=1  to  𝑛 step by 1 do

(13)     𝑈^𝑛+1_{1→𝐼,𝑝}←𝑈^𝑛_{1→𝐼,𝑝}+𝑈^𝑘_{1→𝐼,𝑝}𝑟_𝑛−𝑘

(14)     𝛿←1.0

(15)     𝑙𝑜𝑐𝑎𝑙𝐼𝑡𝑒𝑟𝑎𝑡𝑖𝑜𝑛←0

(16)     while 𝛿>𝜖 do

(17)     for 𝑝=1  to  𝑃 step by 1 do

(18)        𝑉²_{1→𝐼,𝑝}←𝑈^𝑛+1_{1→𝐼,𝑝}

(19)        if 𝑝>1 then

(20)        𝑉²_1,𝑝←𝑉²_1,𝑝+𝑉¹_{𝐼,𝑝−1}/ℎ²

(21)        if 𝑝<𝑃 then

(22)        𝑉²_𝐼,𝑝←𝑉²_𝐼,𝑝+𝑉¹_𝐼,𝑝+1/ℎ²

(23)     for  𝑝=1  to  𝑃 step by 1 do

(24)        solve 𝐴_{1→𝐼,3,𝑝}𝑉³_{1→𝐼,𝑝}=𝑉²_{1→𝐼,𝑝}

(25)        𝛿_𝑝=max𝑉¹_𝑖,𝑝−𝑉³_𝑖,𝑝^𝐼_𝑖=1

(26)        𝛿=max𝛿,𝛿_𝑝

(27)     𝑙𝑜𝑐𝑎𝑙𝐼𝑡𝑒𝑟𝑎𝑡𝑖𝑜𝑛←𝑙𝑜𝑐𝑎𝑙𝐼𝑡𝑒𝑟𝑎𝑡𝑖𝑜𝑛+1

(28)     𝑡𝑜𝑡𝑎𝑙𝐼𝑡𝑒𝑟𝑎𝑡𝑖𝑜𝑛←𝑡𝑜𝑡𝑎𝑙𝐼𝑡𝑒𝑟𝑎𝑡𝑖𝑜𝑛+𝑐𝑜𝑢𝑛𝑡

(29)     for 𝑝=1  to 𝑃 step by 1 do

(30)     𝑈^𝑛+1_{1→𝐼,𝑝}←𝑉³_{1→𝐼,𝑝}

(31) Output the information

(32) Free memory space