Solving Fuzzy Fractional Differential Equations Using Zadeh's Extension Principle

Algorithm 2

Fuzzy fractional Euler method for a nonlinear problem.

Input: Fractional Euler Equation, Fuzzy Initial Value, 𝛽, 𝑡₀, 𝑡_𝑁 and 𝑁

Descritise 𝛼←𝛼_𝑛,𝛼_𝑛−1,…,𝛼₀;

𝑡ℎ←^𝑁−𝑡₀𝑁;

for  𝑗=#(𝛼)   do

   𝑥0₁(1,𝑗)←𝑎+𝛼(𝑗)∗(𝑏−𝑎);

   𝑥0₂(1,𝑗)←𝑐−𝛼(𝑗)∗(𝑐−𝑏);

end

𝑤₁(1,∶)←fliplr(𝑥0₁(1,∶));

𝑤₂(1,∶)←fliplr(𝑥0₂(1,∶));

for  𝑖=1∶𝑁   do

  for  𝑗=1∶#(𝛼)   do

𝑤₁(𝑖+1,𝑗)←min𝑔(𝛽,ℎ,𝑡_𝑖,𝑢)|𝑢∈[𝑤₁(𝑖,1),𝑤₂(𝑖,1)];

𝑤₂(𝑖+1,𝑗)←max𝑔(𝛽,ℎ,𝑡_𝑖,𝑢)|𝑢∈[𝑤₁(𝑖,1),𝑤₂(𝑖,1)];

 end

 for  𝑖=1∶𝑁   do

for  𝑗=2∶#(𝛼)   do

   𝑝₁(𝑖,𝑗)←min𝑔(𝛽,ℎ,𝑡_𝑖,𝑢)|𝑢∈[𝑤₁(𝑖,𝑗),𝑤₂(𝑖,𝑗−1)];

   𝑞₁(𝑖,𝑗)←𝑤₁(𝑖,𝑗−1);

𝑟₁(𝑖,𝑗)←min𝑔(𝛽,ℎ,𝑡_𝑖,𝑢)|𝑢∈[𝑤₂(𝑖,𝑗−1),𝑤₂(𝑖,𝑗)];

𝑤₁(𝑖+1,𝑗)←min([𝑝₁(𝑖,𝑗),𝑞₁(𝑖,𝑗),𝑟₁(𝑖,𝑗)]);

𝑝₂(𝑖,𝑗)←max𝑔(𝛽,ℎ,𝑡_𝑖,𝑢)|𝑢∈[𝑤₁(𝑖,𝑗),𝑤₁(𝑖,𝑗−1)];

𝑞₂(𝑖,𝑗)←𝑤₂(𝑖,𝑗−1);

𝑟₂(𝑖,𝑗)←max𝑔(𝛽,ℎ,𝑡_𝑖,𝑢)|𝑢∈[𝑤₂(𝑖,𝑗−1),𝑤₂(𝑖,𝑗)];

𝑤₂(𝑖+1,𝑗)←max([𝑝₂(𝑖,𝑗),𝑞₂(𝑖,𝑗),𝑟₂(𝑖,𝑗)]);

end

 end

end

Output: 𝑤₁ and 𝑤₂