Update FDDEProblem

Signed-off-by: ErikQQY <2283984853@qq.com>

src/types/problems.jl

@@ -221,11 +221,87 @@ end
 struct StandardFDDEProblem end
 
 """
-    FDDEProblem(f, ϕ, α, τ, tspan)
+Defines a fractional delay differential equation (FDDE) problem.
 
-  - `f`: The function describing fractional delay differential equations.
+## Mathematical Specification of an FDDE problem
+
+To define an FDDE problem, you simply need to given the function ``f``, the history function ``ϕ``, the fractional order ``α``, the delay ``τ`` and the initial condition ``u₀``  which define an FDDE:
+
+```math
+D^\\alpha_t y(t)=f(t,y(t),y(t-τ)),t ≥ 0
+```
+
+```math
+y(t)=ϕ(t), t ≤ 0
+```
+
+There are two different ways of specifying `f`:
+
+  - `f(du,u,h,p,t)`: The function describing fractional delay differential equations.
+  - `f(u,h,p,t)`: returning `du`. Less memory-efficient way, particularly suitable when mutation is not allowed (e.g. with certain automatic differentiation packages such as Zygote).
   - `ϕ`: History function
-    Construct a fractional delayed differential equation problem.
+  - `α`: The fractional order of the differential equations, commensurate and non-commensurate is both supported.
+  - `τ`: The time delay of the differential equations.
+
+## Problem Type
+
+### Constructors
+
+`FODEProblem` can be constructed by first building an `ODEFunction` or
+by simply passing the FODE right-hand side to the constructor. The constructors
+are:
+
+  - `FDDEProblem(f::ODEFunction,order,u0,ϕ,tspan,p=NullParameters();kwargs...)`
+  - `FDDEProblem{isinplace,specialize}(f,order,u0,ϕ,tspan,p=NullParameters();kwargs...)` :
+    Defines the FODE with the specified functions. `isinplace` optionally sets whether
+    the function is inplace or not. This is determined automatically, but not inferred.
+    `specialize` optionally controls the specialization level. See the
+    [specialization levels section of the SciMLBase documentation](https://docs.sciml.ai/SciMLBase/stable/interfaces/Problems/#Specialization-Levels)
+    for more details. The default is `AutoSpecialize`.
+
+For more details on the in-place and specialization controls, see the ODEFunction
+documentation.
+
+Parameters are optional, and if not given, then a `NullParameters()` singleton
+will be used which will throw nice errors if you try to index non-existent
+parameters. Any extra keyword arguments are passed on to the solvers. For example,
+if you set a `callback` in the problem, then that `callback` will be added in
+every solve call.
+
+For specifying Jacobians and mass matrices, see the `ODEFunction` documentation.
+
+### Fields
+
+  - `f`: The function in the FDDE.
+  - `ϕ`: The history function in the FDDE.
+  - `order`: The order of the FDDE.
+  - `τ`: The time delay of the FDDE.
+  - `tspan`: The timespan for the problem.
+  - `p`: The parameters.
+  - `kwargs`: The keyword arguments passed onto the solves.
+
+## Example Problem
+
+```julia
+function ϕ(p, t)
+    if t == 0
+        return 19.00001
+    else
+        return 19.0
+    end
+end
+function f(y, ϕ, p, t)
+    return 3.5 * y * (1 - ϕ / 19)
+end
+τ = [0.8]
+order = 0.97
+u0 = 19.00001
+tspan = (0.0, 2.0)
+dt = 0.5
+prob = FDDEProblem(f, order, u0, ϕ, constant_lags = τ, tspan)
+sol = solve(prob, DelayPIEX(), dt = dt)
+```
+
 """
 struct FDDEProblem{uType, tType, oType, lType, isinplace, P, F, H, K, PT} <:
        SciMLBase.AbstractDDEProblem{uType, tType, lType, isinplace}