Hi,
I am trying to solve this PDE system and getting the error below. Any ideas what I am missing here?!
Thank you.
Modeling code:
using DifferentialEquations, ModelingToolkit, MethodOfLines, DomainSets
@parameters t r
Dt = Differential(t)
Dr = Differential(r)
Drr = Differential(r)^2
Rcap = 8.0
Rkrogh = 72.0
Rec=1e5
Av=6.023e23
D=1.3e-7
P=2.8e-7
Γ=1.0
kon=6e-3
KD=0.1
ϵ=0.24
Rs=1.0
kinT=0.01
kinB=0.01
keT=0.03
keB=0.03
feT=0.5
feB=0.5
Rcell=10.0
A=60.0
B=40.0
ka=0.6*60
kb=0.17*24*60
Cplasma0=1*0.02/2
# derived quantities
Vcell = (4.0/3.0)*pi*(Rcell^3)
koff = KD*kon
Cplasma = Cplasma0 * ((A*exp(-ka*t)) + B*exp(-kb*t))
# variables
@variables u(..) T(t)=Rec/(Av * Vcell) DT(t)=0.0 Te(t)=0.0 DTe(t)=0.0
Rxn = kon*(u(t, r)/ϵ)*((T/ϵ)*Γ) - koff*DT
Cplasma = Cplasma0 * ((A*exp(-ka*t)) + B*exp(-kb*t))
eqs = [
Dt(u(t, r)) ~ D*(1/r*Dr(u(t, r)) + Drr(u(t, r))) - Rxn,
Dt(T) ~ Rs - Rxn - kinT*T + keT*feT*Te,
Dt(DT) ~ Rxn - kinB*DT + keB*feB*DTe,
Dt(Te) ~ kinT*T - keT*Te,
Dt(DTe) ~ kinB*DT - keB*DTe
]
# initial and boundary conditions
# Space and time domains
domains = [t ∈ Interval(0.0, 50.0*24.0*60.0),
r ∈ Interval(Rcap, Rkrogh)]
bcs = [u(0.0, r) ~ 0.0,
D*Dr(u(t, Rcap)) ~ P*(Cplasma - (u(t, Rcap)/ϵ)),
Dr(u(t, Rkrogh)) ~ 0.0]
# define PDE system
@named pdesys = PDESystem(eqs, bcs, domains, [t, r], [u(t, r), T, DT, Te, DTe])
# Method of lines discretization
dr = 8.0
discretization = MOLFiniteDifference([r => dr], t)
# Convert the PDE problem into an ODE problem and solve
prob = discretize(pdesys, discretization)
sol = solve(prob, Tsit5())
Error message:
The system of equations is:
Equation[-1.3e-7(0.015625(u(t))[1] + 0.0625(-0.125(u(t))[1] + 0.125(u(t))[2]) - 0.03125(u(t))[2] + 0.0156
25(u(t))[3]) - 0.0006000000000000001DT(t) + Differential(t)((u(t))[2]) + 0.10416666666666669(u(t))[2]*T(t) ~ 0, -1.3e-7(0.041666666666666664(-0.125(u(t))[2] + 0.125(u(t))[3]) + 0.015625(u(t))[2] - 0.03125(u(t))[3] + 0.015625(u(t))[4]) - 0.0006000000000000001DT(t) + Differential(t)((u(t))[3]) + 0.10416666666666669(u(t))[3]*T(t) ~ 0, -1.3e-7(0.015625(u(t))[3] + 0.03125(-0.125(u(t))[3] + 0.125(u(t))[4]) - 0.03125(u(t))[4] + 0.015625(u(t))[5]) - 0.0006000000000000001DT(t) + Differential(t)((u(t))[4]) + 0.10416666666666669(u(t))[4]*T(t) ~ 0, -1.3e-7(0.025(-0.125(u(t))[4] + 0.125(u(t))[5]) + 0.015625(u(t))[4] - 0.03125(u(t))[5] + 0.015625(u(t))[6]) + Differential(t)((u(t))[5]) - 0.0006000000000000001DT(t) + 0.10416666666666669(u(t))[5]*T(t) ~ 0, -1.3e-7(0.015625(u(t))[5] + 0.020833333333333332(-0.125(u(t))[5] + 0.125(u(t))[6]) - 0.03125(u(t))[6] + 0.015625(u(t))[7]) - 0.0006000000000000001DT(t) + Differential(t)((u(t))[6]) + 0.10416666666666669(u(t))[6]*T(t) ~ 0, -1.3e-7(0.017857142857142856(-0.125(u(t))[6] + 0.125(u(t))[7]) + 0.015625(u(t))[6] - 0.03125(u(t))[7] + 0.015625(u(t))[8]) - 0.0006000000000000001DT(t) + Differential(t)((u(t))[7]) + 0.10416666666666669(u(t))[7]*T(t) ~ 0, -1.3e-7(0.015625(-0.125(u(t))[7] + 0.125(u(t))[8]) + 0.015625(u(t))[7] - 0.03125(u(t))[8] + 0.015625(u(t))[9]) - 0.0006000000000000001DT(t) + Differential(t)((u(t))[8]) + 0.10416666666666669(u(t))[8]*T(t) ~ 0, -1.0 - 0.0006000000000000001DT(t) + 0.01T(t) - 0.015Te(t) + Differential(t)(T(t)) + 0.10416666666666669T(t)*u(t, r) ~ 0, 0.0106DT(t) + Differential(t)(DT(t)) - 0.015DTe(t) - 0.10416666666666669T(t)*u(t, r) ~ 0, -0.01T(t) + 0.03Te(t) + Differential(t)(Te(t)) ~ 0, Differential(t)(DTe(t)) - 0.01DT(t) + 0.03DTe(t) ~ 0, 1.3e-7(-0.1875(u(t))[1] + 0.25(u(t))[2] - 0.0625(u(t))[3]) ~ 2.8e-7(-4.166666666666667(u(t))[1] + 0.01(40.0exp(-244.8t) + 60.0exp(-36.0t))), 0.0625(u(t))[7] - 0.25(u(t))[8] + 0.1875(u(t))[9] ~ 0.0]
There are 13 variables and 13 equations.
There are 11 time derivatives.
The variables without time derivatives are:
SymbolicUtils.BasicSymbolic{Real}[(u(t))[1], (u(t))[9]]
The equations without time derivatives are:
Equation[1.3e-7(-0.1875(u(t))[1] + 0.25(u(t))[2] - 0.0625(u(t))[3]) ~ 2.8e-7(-4.166666666666667(u(t))[1]
+ 0.01(40.0exp(-244.8t) + 60.0exp(-36.0t))), 0.0625(u(t))[7] - 0.25(u(t))[8] + 0.1875(u(t))[9] ~ 0.0] ERROR: ExtraVariablesSystemException: The system is unbalanced. There are 14 highest order derivative var
iables and 13 equations. More variables than equations, here are the potential extra variable(s):
(u(t))[2]
(u(t))[3]
(u(t))[4]
(u(t))[5]
(u(t))[6]
(u(t))[7]
(u(t))[8]
T(t)
DT(t)
Te(t)
DTe(t)
u(t, r)
Stacktrace:
[1] error_reporting(state::TearingState{…}, bad_idxs::Vector{…}, n_highest_vars::Int64, iseqs::Bool, or
ig_inputs::Set{…}) @ ModelingToolkit.StructuralTransformations ~/.julia/packages/ModelingToolkit/dCa81/src/structural_tr
ansformation/utils.jl:50 [2] check_consistency(state::TearingState{ODESystem}, orig_inputs::Set{Any})
@ ModelingToolkit.StructuralTransformations ~/.julia/packages/ModelingToolkit/dCa81/src/structural_tr
ansformation/utils.jl:85 [3] _structural_simplify!(state::TearingState{…}, io::Nothing; simplify::Bool, check_consistency::Bool,
fully_determined::Bool, kwargs::@Kwargs{}) @ ModelingToolkit ~/.julia/packages/ModelingToolkit/dCa81/src/systems/systemstructure.jl:609
[4] _structural_simplify!
@ ~/.julia/packages/ModelingToolkit/dCa81/src/systems/systemstructure.jl:597 [inlined]
[5] structural_simplify!(state::TearingState{…}, io::Nothing; simplify::Bool, check_consistency::Bool,
fully_determined::Bool, kwargs::@Kwargs{}) @ ModelingToolkit ~/.julia/packages/ModelingToolkit/dCa81/src/systems/systemstructure.jl:563
[6] __structural_simplify(sys::ODESystem, io::Nothing; simplify::Bool, kwargs::@Kwargs{})
@ ModelingToolkit ~/.julia/packages/ModelingToolkit/dCa81/src/systems/systems.jl:56
[7] __structural_simplify
@ ~/.julia/packages/ModelingToolkit/dCa81/src/systems/systems.jl:36 [inlined]
[8] structural_simplify(sys::ODESystem, io::Nothing; simplify::Bool, kwargs::@Kwargs{})
@ ModelingToolkit ~/.julia/packages/ModelingToolkit/dCa81/src/systems/systems.jl:21
[9] structural_simplify
@ ~/.julia/packages/ModelingToolkit/dCa81/src/systems/systems.jl:19 [inlined]
[10] structural_simplify(sys::ODESystem)
@ ModelingToolkit ~/.julia/packages/ModelingToolkit/dCa81/src/systems/systems.jl:19
[11] discretize(pdesys::PDESystem, discretization::MOLFiniteDifference{…}; analytic::Nothing, kwargs::@K
wargs{}) @ PDEBase ~/.julia/packages/PDEBase/Aqj4G/src/discretization_state.jl:60
[12] discretize(pdesys::PDESystem, discretization::MOLFiniteDifference{…})
@ PDEBase ~/.julia/packages/PDEBase/Aqj4G/src/discretization_state.jl:55
[13] top-level scope
@ REPL[93]:1
Some type information was truncated. Use `show(err)` to see complete types.