Merge pull request #9 from devmotion/dde_problems

Add some DDE problems from DelayDiffEq.jl

Author: ChrisRackauckas

src/DiffEqProblemLibrary.jl

@@ -6,6 +6,7 @@ using DiffEqBase, ParameterizedFunctions, DiffEqPDEBase, JLD
 
 include("ode_premade_problems.jl")
 include("dae_premade_problems.jl")
+include("dde_premade_problems.jl")
 include("sde_premade_problems.jl")
 include("fem_premade_problems.jl")
 include("premade_meshes.jl")
@@ -26,6 +27,13 @@ export prob_ode_linear, prob_ode_bigfloatlinear, prob_ode_2Dlinear,
  #DAE Example Problems
  export prob_dae_resrob
 
+#DDE Example Problems
+export prob_dde_1delay, prob_dde_1delay_notinplace, prob_dde_1delay_scalar_notinplace,
+       prob_dde_2delays, prob_dde_2delays_notinplace, prob_dde_2delays_scalar_notinplace,
+       prob_dde_1delay_long, prob_dde_1delay_long_notinplace,
+       prob_dde_1delay_long_scalar_notinplace, prob_dde_2delays_long,
+       prob_dde_2delays_long_notinplace, prob_dde_2delays_long_scalar_notinplace
+
  #FEM Example Problems
  export  prob_femheat_moving, prob_femheat_pure, prob_femheat_diffuse,
          prob_poisson_wave, prob_poisson_noisywave, prob_femheat_birthdeath,

src/dde_premade_problems.jl

@@ -0,0 +1,305 @@
+# DDE examples with analytical solution
+
+## Single constant delay
+
+### In-place function
+
+f_1delay = function (t,u,h,du)
+    du[1] = - h(t-1)[1]
+end
+
+function (f::typeof(f_1delay))(::Type{Val{:analytic}}, t, u₀)
+    if t < 0
+        return 0
+    elseif t < 1
+        return u₀
+    elseif t < 2
+        return u₀ * (2 - t)
+    elseif t < 3
+        return u₀ * (8 - 6t + t^2) / 2
+    elseif t < 4
+        return u₀ * (51 - 45t + 12t^2 - t^3) / 6
+    elseif t < 5
+        return u₀ * (460 - 436t + 144t^2 - 20t^3 + t^4) / 24
+    elseif t < 6
+        return u₀ * (5_425 - 5_305t + 1_970t^2 - 350t^3 + 30t^4 - t^5) / 120
+    elseif t < 7
+        return u₀ * (79_206 - 78_486t + 31_260t^2 - 6_420t^3 + 720t^4 - 42t^5 + t^6) / 720
+    elseif t < 8
+        return u₀ * (1_377_985 - 1_372_945t + 571_767t^2 - 128_975t^3 + 17_045t^4 -
+                     1_323t^5 + 56t^6 - t^7) / 5040
+    elseif t < 9
+        return u₀ * (27_801_096 - 27_760_776t + 11_914_168t^2 - 2_866_808t^3 + 423_080t^4 -
+                     39_256t^5 + 2_240t^6 - 72t^7 + t^8)  / 40_320
+    elseif t ≤ 10
+        return u₀ * (637_630_353 - 637_267_473t + 279_414_396t^2 - 70_442_316t^3 +
+                     11_247_894t^4 - 1_179_990t^5 + 81_396t^6 - 3_564t^7 + 90t^8 - t^9) /
+                     362_880
+    else
+        error("This analytical solution is only valid on (-∞,10]")
+    end
+end
+
+"""
+    prob_dde_1delay(u₀)
+
+Model problem of finding a solution ``u(t)`` in the time span ``t \\in [0,10]`` to the
+delay differential equation
+
+```math
+\\frac{du}{dt} = -u(t-1)
+```
+
+with history function
+
+```math
+u(t) = \\begin{cases}
+0 & \\text{if } t < 0, \\
+u₀ & \\text{if } t = 0,
+\\end{cases}
+```
+
+for ``t \\leq 0``. Hence the problem is discontinuous at ``t = 0`` for all ``u₀ \\neq 0``.
+
+An analytical solution of this problem is provided for ``t \\in (-\\infty,10]``.
+"""
+prob_dde_1delay(u₀) = ConstantLagDDEProblem(f_1delay, t->[0.0], [u₀], [1], (0.0, 10.0);
+                                            iip=true)
+
+### Not in-place function
+
+f_1delay_notinplace = function (t,u,h)
+    - h(t-1)
+end
+
+(f::typeof(f_1delay_notinplace))(::Type{Val{:analytic}}, t, u0) = f_1delay(Val{:analytic},
+                                                                           t, u0)
+
+#### Vectorized history function
+
+"""
+    prob_dde_1delay_notinplace(u₀)
+
+Same as [`prob_dde_1delay`](@ref), but purposefully implemented with a not in-place
+function.
+"""
+prob_dde_1delay_notinplace(u₀) =
+    ConstantLagDDEProblem(f_1delay_notinplace, t->[0.0], [u₀], [1], (0.0, 10.0); iip=false)
+
+#### Scalar history function
+
+"""
+    prob_dde_1delay_scalar_notinplace(u₀)
+
+Same as [`prob_dde_1delay_notinplace`](@ref), but purposefully implemented with a scalar
+history function.
+"""
+prob_dde_1delay_scalar_notinplace(u₀) =
+    ConstantLagDDEProblem(f_1delay_notinplace, t->0.0, u₀, [1], (0.0, 10.0); iip=false)
+
+## Two constant delays
+
+### In-place function
+
+f_2delays = function (t,u,h,du)
+    du[1] = - h(t-1/3)[1] - h(t-1/5)[1]
+end
+
+function (f::typeof(f_2delays))(::Type{Val{:analytic}}, t, u₀)
+    if t < 0
+        return 0
+    elseif t < 1/5
+        return u₀
+    elseif t < 1/3
+        return u₀ * (6 - 5t) / 5
+    elseif t < 2/5
+        return u₀ * (23 - 30t) / 15
+    elseif t < 8/15
+        return u₀ * (242 - 360t + 75t^2) / 150
+    elseif t < 3/5
+        return u₀ * (854 - 1_560t + 675t^2) / 450
+    elseif t < 2/3
+        return u₀ * (4_351 - 8_205t + 4_050t^2 - 375t^3) / 2_250
+    elseif t < 11/15
+        return u₀ * (1_617 - 3_235t + 1_725t^2 - 125t^3) / 750
+    elseif t < 4/5
+        return u₀ * (7_942 - 17_280t + 11_475t^2 - 2_250t^3) / 3_375
+    elseif t < 13/15
+        return u₀ * (319_984 - 702_720t + 480_600t^2 - 108_000t^3 + 5_625t^4) / 135_000
+    elseif t < 14/15
+        return u₀ * (40_436 - 94_980t + 72_900t^2 - 19_500t^3 + 625t^4) / 15_000
+    elseif t ≤ 1
+        return u₀ * (685_796 - 1_670_388t + 1_392_660t^2 - 467_100t^3 + 50_625t^4) / 243_000
+    else
+        error("This analytical solution is only valid on (-∞,1]")
+    end
+end
+
+"""
+    prob_dde_2delays(u₀)
+
+Model problem of finding a solution ``u(t)`` in the time span ``t \\in [0,1]`` to the delay
+differential equation
+
+```math
+\\frac{du}{dt} = -u(t-1/3)-u(t-1/5)
+```
+
+with history function
+
+```math
+u(t) = \\begin{cases}
+0 & \text{if } t < 0 \\
+u₀ & \text{if } t = 0
+\end{cases}
+```
+
+for ``t \\leq 0``. Hence the problem is discontinuous at ``t = 0`` for all ``u₀ \\neq 0``.
+
+An analytical solution of this problem is provided for ``t \\in (-\\infty,1]``.
+"""
+prob_dde_2delays(u₀) = ConstantLagDDEProblem(f_2delays, t->[0.0], [u₀], [1//3, 1//5],
+                                             (0.0, 1.0); iip=true)
+
+### Not in-place function
+
+f_2delays_notinplace = function (t,u,h)
+    - h(t-1/3) - h(t-1/5)
+end
+
+(f::typeof(f_2delays_notinplace))(::Type{Val{:analytic}}, t, u0) =
+    f_2delays(Val{:analytic}, t, u0)
+
+#### Vectorized history function
+
+"""
+    prob_dde_2delays_notinplace(u₀)
+
+Same as [`prob_dde_2delays`](@ref), but purposefully implemented with a not in-place
+function.
+"""
+prob_dde_2delays_notinplace(u₀) =
+    ConstantLagDDEProblem(f_2delays_notinplace, t->[0.0], [u₀], [1//3, 1//5], (0.0, 1.0);
+                          iip=false)
+
+#### Scalar history function
+
+"""
+    prob_dde_2delays_scalar_notinplace(u₀)
+
+Same as [`prob_dde_2delays_notinplace`](@ref), but purposefully implemented with a scalar
+history function.
+"""
+prob_dde_2delays_scalar_notinplace(u₀) =
+    ConstantLagDDEProblem(f_2delays_notinplace, t->0.0, u₀, [1//3, 1//5], (0.0, 1.0);
+                          iip=false)
+
+# DDE examples without analytical solution
+
+## Single constant delay
+
+### In-place function
+
+f_1delay_long = function (t,u,h,du)
+    du[1] = - h(t-0.2)[1] + u[1]
+end
+
+"""
+Model problem of finding a solution ``u(t)`` in the time span ``t \\in [0,100]`` to the
+delay differential equation
+
+```math
+\\frac{du}{dt} = -u(t-0.2) + u(t)
+```
+
+with history function
+
+```math
+u(t) = \\begin{cases}
+0 & \\text{if } t < 0,\\
+1 & \\text{if } t = 0,
+\\end{cases}
+```
+
+for ``t \\leq 0``. Hence the problem is discontinuous at ``t = 0``.
+"""
+prob_dde_1delay_long = ConstantLagDDEProblem(f_1delay_long, t->[0.0], [1.0], [0.2],
+                                             (0.0, 100.0); iip=true)
+
+### Not in-place function
+
+f_1delay_long_notinplace = function (t,u,h)
+    - h(t-0.2) + u
+end
+
+"""
+Same as [`prob_dde_1delay_long`](@ref), but purposefully implemented with a not in-place
+function.
+"""
+prob_dde_1delay_long_notinplace =
+    ConstantLagDDEProblem(f_1delay_long_notinplace, t->[0.0], [1.0], [0.2], (0.0, 100.0);
+                          iip=false)
+
+"""
+Same as [`prob_dde_1delay_long_notinplace`](@ref), but purposefully implemented with a
+scalar history function.
+"""
+prob_dde_1delay_long_scalar_notinplace =
+    ConstantLagDDEProblem(f_1delay_long_notinplace, t->0.0, 1.0, [0.2], (0.0, 100.0);
+                          iip=false)
+
+## Two constant delays
+
+### In-place function
+
+f_2delays_long = function (t,u,h,du)
+    du[1] = - h(t-1/3)[1] - h(t-1/5)[1]
+end
+
+"""
+Model problem of finding a solution ``u(t)`` in the time span ``t \\in [0,100]`` to the
+delay differential equation
+
+```math
+\\frac{du}{dt} = -u(t-1/3) - u(1-1/5)
+```
+
+with history function
+
+```math
+u(t) = \\begin{cases}
+0 & \\text{if } t < 0,\\
+1 & \\text{if } t = 0, 
+\\end{cases}
+```
+
+for ``t < 0``. Hence the problem is discontinuous at ``t = 0``.
+"""
+prob_dde_2delays_long = ConstantLagDDEProblem(f_2delays_long, t->[0.0], [1.0], [1//3, 1//5],
+                                              (0.0, 100.0); iip=true)
+
+### Not in-place function
+
+f_2delays_long_notinplace = function (t,u,h)
+    - h(t-1/3) - h(t-1/5)
+end
+
+#### Vectorized history function
+
+"""
+Same as [`prob_dde_2delays_long`](@ref), but purposefully implemented with a not in-place
+function.
+"""
+prob_dde_2delays_long_notinplace =
+    ConstantLagDDEProblem(f_2delays_long_notinplace, t->[0.0], [1.0], [1//3, 1//5],
+                          (0.0, 100.0); iip=false)
+
+#### Scalar history function
+
+"""
+Same as [`prob_dde_2delays_long_notinplace`](@ref), but purposefully implemented with a scalar
+history function.
+"""
+prob_dde_2delays_long_scalar_notinplace =
+    ConstantLagDDEProblem(f_2delays_long_notinplace, t->0.0, 1.0, [1//3, 1//5],
+                          (0.0, 100.0); iip=false)