julia - 微分方程求解()不起作用( Julia )

Question

我对此非常陌生，我在网上找不到任何帮助，如果之前已经回答过，我很抱歉。我试图在常微分方程上遵循这个简单的例子

using DifferentialEquations
f(t,u) = 1.01*u
u0=1/2
tspan = (0.0,1.0)
prob = ODEProblem(f,u0,tspan)
sol = solve(prob,Tsit5(),reltol=1e-8,abstol=1e-8)
using Plots
plot(sol,linewidth=5,title="Solution to the linear ODE with a thick line",
xaxis="Time (t)",yaxis="u(t) (in μm)",label="My Thick Line!") # legend=false
plot!(sol.t, t->0.5*exp(1.01t),lw=3,ls=:dash,label="True Solution!")

但是当我调用solve函数时出现这个错误

MethodError: no method matching f(::Float64, ::DiffEqBase.NullParameters, ::Float64)
Closest candidates are:
f(::Any, ::Any) at /Users/diogomiguez/.julia/pluto_notebooks/Cute science.jl#==#dbde1a90-407c-11eb-15ad-394234f71852:1
(::DiffEqBase.ODEFunction{false,typeof(Main.workspace465.f),LinearAlgebra.UniformScaling{Bool},Nothing,Nothing,Nothing,Nothing,Nothing,Nothing,Nothing,Nothing,Nothing,Nothing,Nothing,Nothing})(::Float64, ::Vararg{Any,N} where N)@diffeqfunction.jl:248
initialize!(::OrdinaryDiffEq.ODEIntegrator{OrdinaryDiffEq.CompositeAlgorithm{Tuple{OrdinaryDiffEq.Tsit5,OrdinaryDiffEq.Rosenbrock23{0,false,DiffEqBase.DefaultLinSolve,DataType}},OrdinaryDiffEq.AutoSwitchCache{OrdinaryDiffEq.Tsit5,OrdinaryDiffEq.Rosenbrock23{0,false,DiffEqBase.DefaultLinSolve,DataType},Rational{Int64},Int64}},false,Float64,Nothing,Float64,DiffEqBase.NullParameters,Float64,Float64,Float64,Array{Float64,1},OrdinaryDiffEq.ODECompositeSolution{Float64,1,Array{Float64,1},Nothing,Nothing,Array{Float64,1},Array{Array{Float64,1},1},DiffEqBase.ODEProblem{Float64,Tuple{Float64,Float64},false,DiffEqBase.NullParameters,DiffEqBase.ODEFunction{false,typeof(Main.workspace465.f),LinearAlgebra.UniformScaling{Bool},Nothing,Nothing,Nothing,Nothing,Nothing,Nothing,Nothing,Nothing,Nothing,Nothing,Nothing,Nothing},Base.Iterators.Pairs{Union{},Union{},Tuple{},NamedTuple{(),Tuple{}}},DiffEqBase.StandardODEProblem},OrdinaryDiffEq.CompositeAlgorithm{Tuple{OrdinaryDiffEq.Tsit5,OrdinaryDiffEq.Rosenbrock23{0,false,DiffEqBase.DefaultLinSolve,DataType}},OrdinaryDiffEq.AutoSwitchCache{OrdinaryDiffEq.Tsit5,OrdinaryDiffEq.Rosenbrock23{0,false,DiffEqBase.DefaultLinSolve,DataType},Rational{Int64},Int64}},OrdinaryDiffEq.CompositeInterpolationData{DiffEqBase.ODEFunction{false,typeof(Main.workspace465.f),LinearAlgebra.UniformScaling{Bool},Nothing,Nothing,Nothing,Nothing,Nothing,Nothing,Nothing,Nothing,Nothing,Nothing,Nothing,Nothing},Array{Float64,1},Array{Float64,1},Array{Array{Float64,1},1},OrdinaryDiffEq.CompositeCache{Tuple{OrdinaryDiffEq.Tsit5ConstantCache{Float64,Float64},OrdinaryDiffEq.Rosenbrock23ConstantCache{Float64,DiffEqBase.TimeDerivativeWrapper{DiffEqBase.ODEFunction{false,typeof(Main.workspace465.f),LinearAlgebra.UniformScaling{Bool},Nothing,Nothing,Nothing,Nothing,Nothing,Nothing,Nothing,Nothing,Nothing,Nothing,Nothing,Nothing},Float64,DiffEqBase.NullParameters},DiffEqBase.UDerivativeWrapper{DiffEqBase.ODEFunction{false,typeof(Main.workspace465.f),LinearAlgebra.UniformScaling{Bool},Nothing,Nothing,Nothing,Nothing,Nothing,Nothing,Nothing,Nothing,Nothing,Nothing,Nothing,Nothing},Float64,DiffEqBase.NullParameters},Float64,Float64,DiffEqBase.DefaultLinSolve}},OrdinaryDiffEq.AutoSwitchCache{OrdinaryDiffEq.Tsit5,OrdinaryDiffEq.Rosenbrock23{0,false,DiffEqBase.DefaultLinSolve,DataType},Rational{Int64},Int64}}},DiffEqBase.DEStats},DiffEqBase.ODEFunction{false,typeof(Main.workspace465.f),LinearAlgebra.UniformScaling{Bool},Nothing,Nothing,Nothing,Nothing,Nothing,Nothing,Nothing,Nothing,Nothing,Nothing,Nothing,Nothing},OrdinaryDiffEq.CompositeCache{Tuple{OrdinaryDiffEq.Tsit5ConstantCache{Float64,Float64},OrdinaryDiffEq.Rosenbrock23ConstantCache{Float64,DiffEqBase.TimeDerivativeWrapper{DiffEqBase.ODEFunction{false,typeof(Main.workspace465.f),LinearAlgebra.UniformScaling{Bool},Nothing,Nothing,Nothing,Nothing,Nothing,Nothing,Nothing,Nothing,Nothing,Nothing,Nothing,Nothing},Float64,DiffEqBase.NullParameters},DiffEqBase.UDerivativeWrapper{DiffEqBase.ODEFunction{false,typeof(Main.workspace465.f),LinearAlgebra.UniformScaling{Bool},Nothing,Nothing,Nothing,Nothing,Nothing,Nothing,Nothing,Nothing,Nothing,Nothing,Nothing,Nothing},Float64,DiffEqBase.NullParameters},Float64,Float64,DiffEqBase.DefaultLinSolve}},OrdinaryDiffEq.AutoSwitchCache{OrdinaryDiffEq.Tsit5,OrdinaryDiffEq.Rosenbrock23{0,false,DiffEqBase.DefaultLinSolve,DataType},Rational{Int64},Int64}},OrdinaryDiffEq.DEOptions{Float64,Float64,Float64,Float64,typeof(DiffEqBase.ODE_DEFAULT_NORM),typeof(LinearAlgebra.opnorm),DiffEqBase.CallbackSet{Tuple{},Tuple{}},typeof(DiffEqBase.ODE_DEFAULT_ISOUTOFDOMAIN),typeof(DiffEqBase.ODE_DEFAULT_PROG_MESSAGE),typeof(DiffEqBase.ODE_DEFAULT_UNSTABLE_CHECK),DataStructures.BinaryHeap{Float64,Base.Order.ForwardOrdering},DataStructures.BinaryHeap{Float64,Base.Order.ForwardOrdering},Nothing,Nothing,Int64,Tuple{},Tuple{},Tuple{}},Float64,Float64,Nothing,OrdinaryDiffEq.DefaultInit}, ::OrdinaryDiffEq.Tsit5ConstantCache{Float64,Float64})@low_order_rk_perform_step.jl:565
initialize!(::OrdinaryDiffEq.ODEIntegrator{OrdinaryDiffEq.CompositeAlgorithm{Tuple{OrdinaryDiffEq.Tsit5,OrdinaryDiffEq.Rosenbrock23{0,false,DiffEqBase.DefaultLinSolve,DataType}},OrdinaryDiffEq.AutoSwitchCache{OrdinaryDiffEq.Tsit5,OrdinaryDiffEq.Rosenbrock23{0,false,DiffEqBase.DefaultLinSolve,DataType},Rational{Int64},Int64}},false,Float64,Nothing,Float64,DiffEqBase.NullParameters,Float64,Float64,Float64,Array{Float64,1},OrdinaryDiffEq.ODECompositeSolution{Float64,1,Array{Float64,1},Nothing,Nothing,Array{Float64,1},Array{Array{Float64,1},1},DiffEqBase.ODEProblem{Float64,Tuple{Float64,Float64},false,DiffEqBase.NullParameters,DiffEqBase.ODEFunction{false,typeof(Main.workspace465.f),LinearAlgebra.UniformScaling{Bool},Nothing,Nothing,Nothing,Nothing,Nothing,Nothing,Nothing,Nothing,Nothing,Nothing,Nothing,Nothing},Base.Iterators.Pairs{Union{},Union{},Tuple{},NamedTuple{(),Tuple{}}},DiffEqBase.StandardODEProblem},OrdinaryDiffEq.CompositeAlgorithm{Tuple{OrdinaryDiffEq.Tsit5,OrdinaryDiffEq.Rosenbrock23{0,false,DiffEqBase.DefaultLinSolve,DataType}},OrdinaryDiffEq.AutoSwitchCache{OrdinaryDiffEq.Tsit5,OrdinaryDiffEq.Rosenbrock23{0,false,DiffEqBase.DefaultLinSolve,DataType},Rational{Int64},Int64}},OrdinaryDiffEq.CompositeInterpolationData{DiffEqBase.ODEFunction{false,typeof(Main.workspace465.f),LinearAlgebra.UniformScaling{Bool},Nothing,Nothing,Nothing,Nothing,Nothing,Nothing,Nothing,Nothing,Nothing,Nothing,Nothing,Nothing},Array{Float64,1},Array{Float64,1},Array{Array{Float64,1},1},OrdinaryDiffEq.CompositeCache{Tuple{OrdinaryDiffEq.Tsit5ConstantCache{Float64,Float64},OrdinaryDiffEq.Rosenbrock23ConstantCache{Float64,DiffEqBase.TimeDerivativeWrapper{DiffEqBase.ODEFunction{false,typeof(Main.workspace465.f),LinearAlgebra.UniformScaling{Bool},Nothing,Nothing,Nothing,Nothing,Nothing,Nothing,Nothing,Nothing,Nothing,Nothing,Nothing,Nothing},Float64,DiffEqBase.NullParameters},DiffEqBase.UDerivativeWrapper{DiffEqBase.ODEFunction{false,typeof(Main.workspace465.f),LinearAlgebra.UniformScaling{Bool},Nothing,Nothing,Nothing,Nothing,Nothing,Nothing,Nothing,Nothing,Nothing,Nothing,Nothing,Nothing},Float64,DiffEqBase.NullParameters},Float64,Float64,DiffEqBase.DefaultLinSolve}},OrdinaryDiffEq.AutoSwitchCache{OrdinaryDiffEq.Tsit5,OrdinaryDiffEq.Rosenbrock23{0,false,DiffEqBase.DefaultLinSolve,DataType},Rational{Int64},Int64}}},DiffEqBase.DEStats},DiffEqBase.ODEFunction{false,typeof(Main.workspace465.f),LinearAlgebra.UniformScaling{Bool},Nothing,Nothing,Nothing,Nothing,Nothing,Nothing,Nothing,Nothing,Nothing,Nothing,Nothing,Nothing},OrdinaryDiffEq.CompositeCache{Tuple{OrdinaryDiffEq.Tsit5ConstantCache{Float64,Float64},OrdinaryDiffEq.Rosenbrock23ConstantCache{Float64,DiffEqBase.TimeDerivativeWrapper{DiffEqBase.ODEFunction{false,typeof(Main.workspace465.f),LinearAlgebra.UniformScaling{Bool},Nothing,Nothing,Nothing,Nothing,Nothing,Nothing,Nothing,Nothing,Nothing,Nothing,Nothing,Nothing},Float64,DiffEqBase.NullParameters},DiffEqBase.UDerivativeWrapper{DiffEqBase.ODEFunction{false,typeof(Main.workspace465.f),LinearAlgebra.UniformScaling{Bool},Nothing,Nothing,Nothing,Nothing,Nothing,Nothing,Nothing,Nothing,Nothing,Nothing,Nothing,Nothing},Float64,DiffEqBase.NullParameters},Float64,Float64,DiffEqBase.DefaultLinSolve}},OrdinaryDiffEq.AutoSwitchCache{OrdinaryDiffEq.Tsit5,OrdinaryDiffEq.Rosenbrock23{0,false,DiffEqBase.DefaultLinSolve,DataType},Rational{Int64},Int64}},OrdinaryDiffEq.DEOptions{Float64,Float64,Float64,Float64,typeof(DiffEqBase.ODE_DEFAULT_NORM),typeof(LinearAlgebra.opnorm),DiffEqBase.CallbackSet{Tuple{},Tuple{}},typeof(DiffEqBase.ODE_DEFAULT_ISOUTOFDOMAIN),typeof(DiffEqBase.ODE_DEFAULT_PROG_MESSAGE),typeof(DiffEqBase.ODE_DEFAULT_UNSTABLE_CHECK),DataStructures.BinaryHeap{Float64,Base.Order.ForwardOrdering},DataStructures.BinaryHeap{Float64,Base.Order.ForwardOrdering},Nothing,Nothing,Int64,Tuple{},Tuple{},Tuple{}},Float64,Float64,Nothing,OrdinaryDiffEq.DefaultInit}, ::OrdinaryDiffEq.CompositeCache{Tuple{OrdinaryDiffEq.Tsit5ConstantCache{Float64,Float64},OrdinaryDiffEq.Rosenbrock23ConstantCache{Float64,DiffEqBase.TimeDerivativeWrapper{DiffEqBase.ODEFunction{false,typeof(Main.workspace465.f),LinearAlgebra.UniformScaling{Bool},Nothing,Nothing,Nothing,Nothing,Nothing,Nothing,Nothing,Nothing,Nothing,Nothing,Nothing,Nothing},Float64,DiffEqBase.NullParameters},DiffEqBase.UDerivativeWrapper{DiffEqBase.ODEFunction{false,typeof(Main.workspace465.f),LinearAlgebra.UniformScaling{Bool},Nothing,Nothing,Nothing,Nothing,Nothing,Nothing,Nothing,Nothing,Nothing,Nothing,Nothing,Nothing},Float64,DiffEqBase.NullParameters},Float64,Float64,DiffEqBase.DefaultLinSolve}},OrdinaryDiffEq.AutoSwitchCache{OrdinaryDiffEq.Tsit5,OrdinaryDiffEq.Rosenbrock23{0,false,DiffEqBase.DefaultLinSolve,DataType},Rational{Int64},Int64}})@composite_perform_step.jl:39
#__init#399(::Tuple{}, ::Tuple{}, ::Tuple{}, ::Nothing, ::Bool, ::Bool, ::Bool, ::Bool, ::Nothing, ::Bool, ::Bool, ::Float64, ::Nothing, ::Float64, ::Bool, ::Bool, ::Rational{Int64}, ::Nothing, ::Nothing, ::Rational{Int64}, ::Int64, ::Int64, ::Int64, ::Rational{Int64}, ::Bool, ::Int64, ::Nothing, ::Nothing, ::Int64, ::typeof(DiffEqBase.ODE_DEFAULT_NORM), ::typeof(LinearAlgebra.opnorm), ::typeof(DiffEqBase.ODE_DEFAULT_ISOUTOFDOMAIN), ::typeof(DiffEqBase.ODE_DEFAULT_UNSTABLE_CHECK), ::Bool, ::Bool, ::Bool, ::Bool, ::Bool, ::Bool, ::Bool, ::Int64, ::String, ::typeof(DiffEqBase.ODE_DEFAULT_PROG_MESSAGE), ::Nothing, ::Bool, ::Bool, ::Bool, ::Bool, ::OrdinaryDiffEq.DefaultInit, ::Base.Iterators.Pairs{Symbol,Bool,Tuple{Symbol,Symbol},NamedTuple{(:default_set, :second_time),Tuple{Bool,Bool}}}, ::typeof(DiffEqBase.__init), ::DiffEqBase.ODEProblem{Float64,Tuple{Float64,Float64},false,DiffEqBase.NullParameters,DiffEqBase.ODEFunction{false,typeof(Main.workspace465.f),LinearAlgebra.UniformScaling{Bool},Nothing,Nothing,Nothing,Nothing,Nothing,Nothing,Nothing,Nothing,Nothing,Nothing,Nothing,Nothing},Base.Iterators.Pairs{Union{},Union{},Tuple{},NamedTuple{(),Tuple{}}},DiffEqBase.StandardODEProblem}, ::OrdinaryDiffEq.CompositeAlgorithm{Tuple{OrdinaryDiffEq.Tsit5,OrdinaryDiffEq.Rosenbrock23{0,false,DiffEqBase.DefaultLinSolve,DataType}},OrdinaryDiffEq.AutoSwitch{OrdinaryDiffEq.Tsit5,OrdinaryDiffEq.Rosenbrock23{0,false,DiffEqBase.DefaultLinSolve,DataType},Rational{Int64},Int64}}, ::Tuple{}, ::Tuple{}, ::Tuple{}, ::Type{Val{true}})@solve.jl:429
#__solve#398@solve.jl:4[inlined]
#__solve#1(::Bool, ::Base.Iterators.Pairs{Symbol,Bool,Tuple{Symbol},NamedTuple{(:second_time,),Tuple{Bool}}}, ::typeof(DiffEqBase.__solve), ::DiffEqBase.ODEProblem{Float64,Tuple{Float64,Float64},false,DiffEqBase.NullParameters,DiffEqBase.ODEFunction{false,typeof(Main.workspace465.f),LinearAlgebra.UniformScaling{Bool},Nothing,Nothing,Nothing,Nothing,Nothing,Nothing,Nothing,Nothing,Nothing,Nothing,Nothing,Nothing},Base.Iterators.Pairs{Union{},Union{},Tuple{},NamedTuple{(),Tuple{}}},DiffEqBase.StandardODEProblem}, ::Nothing)@default_solve.jl:7
#__solve#463@solve.jl:230[inlined]
__solve@solve.jl:217[inlined]
#solve_call#451@solve.jl:65[inlined]
solve_call@solve.jl:52[inlined]
#solve_up#453@solve.jl:89[inlined]
solve_up@solve.jl:79[inlined]
#solve#452@solve.jl:74[inlined]
solve@solve.jl:72[inlined]
top-level scope@Local: 1[inlined]

也可能有帮助:

->Pkg.status("DifferentialEquations")
Status `~/.julia/environments/v1.5/Project.toml`
  [0c46a032] DifferentialEquations v6.15.0

Answer 1

看来您一直在关注 this example来自 DifferentialEquations.jl文件；奇怪的是，Google 似乎优先考虑 v2.0 的文档(请参阅链接中的 URL)。
v6.15 的文档是 here ; API 现在已更改为预期 f 的三个参数.将您的代码更改为

f(u,p,t) = 1.01*u

你应该没有问题。正如您在 MethodError 中看到的那样, 需要一个带有三个参数的函数，这就是您遇到问题的原因。