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[英]System of differential equations with time dependent constants in arrays, using odeint
[英]Implement solution of differential equations system using exponential matrix in Julia
我嘗試在 Julia 中重現我在圖中顯示並取自Matrix Exponentiation的示例
我向您展示了我在 Julia 中復制練習的程度。 但我不知道如何引入向量 t,對於感興趣的范圍,例如 t = -3: 0.25: 3. 在矩陣中:[exp (u1 * t 0; 0 exp (u2 * t], u1 u2 特征值。
Julia>A=[0 1;1 0]
2×2 Array{Int64,2}:
0 1
1 0
F=eigen(A)
Eigen{Float64,Float64,Array{Float64,2},Array{Float64,1}}
values:
2-element Array{Float64,1}:
-1.0
1.0
vectors:
2×2 Array{Float64,2}:
-0.707107 0.707107
0.707107 0.707107
D = diagm(exp.(F.values))
2×2 Array{Float64,2}:
0.367879 0.0
0.0 2.71828
P = F.vectors
13:06:08->>2×2 Array{Float64,2}:
-0.707107 0.707107
0.707107 0.707107
在站點Fabian Dablander代碼上顯示了 R 中實現該解決方案的代碼。 這些是帶給 Julia 的腳本:
using Plots
using LinearAlgebra
#Solving differential equations using matrix exponentials
A=[-0.20 -1;1 0] #[-0.40 -1;1 0.45] A=[0 1;1 0]
x0=[1 1]# [1 1] x0=[0.25 0.25] x0=[1 0]
tmax=20
n=1000
ts=LinRange(0,tmax,n)
x = Array{Float64}(undef, 0, 0)
x=x0
for i in 1:n
x=vcat(x,x0*exp(A*ts[i]))
end
plot(x)
plot(x[:,1],x[:,2])
#Solving differential equations finding eigenvalues and eigenvectors
A=[-0.20 -1;1 0] #A=[-0.40 -1;1 0.45] A=[0 1;1 0]
x0=[1 1]# [1 1] x0=[0.25 0.25] x0=[1 0]
tmax=20
n=500
# compute eigenvectors and eigenvalues
eig = eigen(A)
E =eig.vectors
λ=eig.values
# solve for the initial conditon
C =E\x0'
# create time steps
ts=LinRange(0,tmax,n)
x = Array{Float64}(undef,n, size(A,2))
for i in 1:n
x[i,:]=real.(C'*diagm(exp.(λ*ts[i]))*E)
end
plot(x)
plot(x[:,1],x[:,2])
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