# DDE Problems

`SciMLBase.DDEProblem`

— TypeDefines a delay differential equation (DDE) problem. Documentation Page: https://diffeq.sciml.ai/stable/types/dde_types/

**Mathematical Specification of a DDE Problem**

To define a DDE Problem, you simply need to give the function $f$, the initial condition $u_0$ at time point $t_0$, and the history function $h$ which together define a DDE:

\[\frac{du}{dt} = f(u,h,p,t) \qquad (t \geq t_0)\]

\[u(t_0) = u_0,\]

\[u(t) = h(t) \qquad (t < t_0).\]

$f$ should be specified as `f(u, h, p, t)`

(or in-place as `f(du, u, h, p, t)`

), $u_0$ should be an AbstractArray (or number) whose geometry matches the desired geometry of `u`

, and $h$ should be specified as described below. The history function `h`

is accessed for all delayed values. Note that we are not limited to numbers or vectors for $u_0$; one is allowed to provide $u_0$ as arbitrary matrices / higher dimension tensors as well.

**Functional Forms of the History Function**

The history function `h`

can be called in the following ways:

`h(p, t)`

: out-of-place calculation`h(out, p, t)`

: in-place calculation`h(p, t, deriv::Type{Val{i}})`

: out-of-place calculation of the`i`

th derivative`h(out, p, t, deriv::Type{Val{i}})`

: in-place calculation of the`i`

th derivative`h(args...; idxs)`

: calculation of`h(args...)`

for indices`idxs`

Note that a dispatch for the supplied history function of matching form is required for whichever function forms are used in the user derivative function `f`

.

**Declaring Lags**

Lags are declared separately from their use. One can use any lag by simply using the interpolant of `h`

at that point. However, one should use caution in order to achieve the best accuracy. When lags are declared, the solvers can more efficiently be more accurate and thus this is recommended.

**Neutral and Retarded Delay Differential Equations**

Note that the history function specification can be used to specify general retarded arguments, i.e. `h(p,α(u,t))`

. Neutral delay differential equations can be specified by using the `deriv`

value in the history interpolation. For example, `h(p,t-τ, Val{1})`

returns the first derivative of the history values at time `t-τ`

.

Note that algebraic equations can be specified by using a singular mass matrix.

**Problem Type**

**Constructors**

```
DDEProblem(f[, u0], h, tspan[, p]; <keyword arguments>)
DDEProblem{isinplace}(f[, u0], h, tspan[, p]; <keyword arguments>)
```

Parameter `isinplace`

optionally sets whether the function is inplace or not. This is determined automatically, but not inferred.

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 DiffEqFunctions page.

**Arguments**

`f`

: The function in the DDE.`u0`

: The initial condition. Defaults to the value`h(p, first(tspan))`

of the history function evaluated at the initial time point.`h`

: The history function for the DDE before`t0`

.`tspan`

: The timespan for the problem.`p`

: The parameters with which function`f`

is called. Defaults to`NullParameters`

.`constant_lags`

: A collection of constant lags used by the history function`h`

. Defaults to`()`

.`dependent_lags`

A tuple of functions`(u, p, t) -> lag`

for the state-dependent lags used by the history function`h`

. Defaults to`()`

.`neutral`

: If the DDE is neutral, i.e., if delays appear in derivative terms.`order_discontinuity_t0`

: The order of the discontinuity at the initial time point. Defaults to`0`

if an initial condition`u0`

is provided. Otherwise it is forced to be greater or equal than`1`

.`kwargs`

: The keyword arguments passed onto the solves.

**Dynamical Delay Differential Equations**

Much like Dynamical ODEs, a Dynamical DDE is a Partitioned DDE of the form:

\[\frac{dv}{dt} = f_1(u,t,h) \\ \frac{du}{dt} = f_2(v,h) \\\]

**Constructors**

```
DynamicalDDEProblem(f1, f2[, v0, u0], h, tspan[, p]; <keyword arguments>)
DynamicalDDEProblem{isinplace}(f1, f2[, v0, u0], h, tspan[, p]; <keyword arguments>)
```

Parameter `isinplace`

optionally sets whether the function is inplace or not. This is determined automatically, but not inferred.

**Arguments**

`f`

: The function in the DDE.`v0`

and`u0`

: The initial condition. Defaults to the values`h(p, first(tspan))...`

of the history function evaluated at the initial time point.`h`

: The history function for the DDE before`t0`

. Must return an object with the indices 1 and 2, with the values of`v`

and`u`

respectively.`tspan`

: The timespan for the problem.`p`

: The parameters with which function`f`

is called. Defaults to`NullParameters`

.`constant_lags`

: A collection of constant lags used by the history function`h`

. Defaults to`()`

.`dependent_lags`

A tuple of functions`(v, u, p, t) -> lag`

for the state-dependent lags used by the history function`h`

. Defaults to`()`

.`neutral`

: If the DDE is neutral, i.e., if delays appear in derivative terms.`order_discontinuity_t0`

: The order of the discontinuity at the initial time point. Defaults to`0`

if an initial condition`u0`

is provided. Otherwise it is forced to be greater or equal than`1`

.`kwargs`

: The keyword arguments passed onto the solves.

The for dynamical and second order DDEs, the history function will return an object with the indicies 1 and 2 defined, where `h(p, t_prev)[1]`

is the value of $f_2(v, u, h, p, t_{\mathrm{prev}})$ and `h(p, t_prev)[2]`

is the value of $f_1(v, u, h, p, t_{\mathrm{prev}})$ (this is for consistency with the ordering of the intitial conditions in the constructor). The supplied history function must also return such a 2-index object, which can be accomplished with a tuple `(v,u)`

or vector `[v,u]`

.

**2nd Order Delay Differential Equations**

To define a 2nd Order DDE Problem, you simply need to give the function $f$ and the initial condition $u_0$ which define an DDE:

\[u'' = f(u',u,h,p,t)\]

`f`

should be specified as `f(du,u,p,t)`

(or in-place as `f(ddu,du,u,p,t)`

), and `u₀`

should be an AbstractArray (or number) whose geometry matches the desired geometry of `u`

. Note that we are not limited to numbers or vectors for `u₀`

; one is allowed to provide `u₀`

as arbitrary matrices / higher dimension tensors as well.

From this form, a dynamical ODE:

\[v' = f(v,u,h,p,t) \\ u' = v \\\]

**Constructors**

```
SecondOrderDDEProblem(f, [, du0, u0], h, tspan[, p]; <keyword arguments>)
SecondOrderDDEProblem{isinplace}(f, [, du0, u0], h, tspan[, p]; <keyword arguments>)
```

Parameter `isinplace`

optionally sets whether the function is inplace or not. This is determined automatically, but not inferred.

**Arguments**

`f`

: The function in the DDE.`du0`

and`u0`

: The initial condition. Defaults to the values`h(p, first(tspan))...`

of the history function evaluated at the initial time point.`h`

: The history function for the DDE before`t0`

. Must return an object with the indices 1 and 2, with the values of`v`

and`u`

respectively.`tspan`

: The timespan for the problem.`p`

: The parameters with which function`f`

is called. Defaults to`NullParameters`

.`constant_lags`

: A collection of constant lags used by the history function`h`

. Defaults to`()`

.`dependent_lags`

A tuple of functions`(v, u, p, t) -> lag`

for the state-dependent lags used by the history function`h`

. Defaults to`()`

.`neutral`

: If the DDE is neutral, i.e., if delays appear in derivative terms.`order_discontinuity_t0`

: The order of the discontinuity at the initial time point. Defaults to`0`

if an initial condition`u0`

is provided. Otherwise it is forced to be greater or equal than`1`

.`kwargs`

: The keyword arguments passed onto the solves.

As above, the history function will return an object with indices 1 and 2, with the values of `du`

and `u`

respectively. The supplied history function must also match this return type, e.g. by returning a 2-element tuple or vector.

**Example Problems**

Example problems can be found in DiffEqProblemLibrary.jl.

To use a sample problem, such as `prob_ode_linear`

, you can do something like:

```
#] add DiffEqProblemLibrary
using DiffEqProblemLibrary.ODEProblemLibrary
# load problems
ODEProblemLibrary.importodeproblems()
prob = ODEProblemLibrary.prob_ode_linear
sol = solve(prob)
```

`SciMLBase.DDEFunction`

— Type`DDEFunction{iip,F,TMM,Ta,Tt,TJ,JVP,VJP,JP,SP,TW,TWt,TPJ,S,O,TCV} <: AbstractDDEFunction{iip}`

A representation of a DDE function `f`

, defined by:

\[M \frac{du}{dt} = f(u,h,p,t)\]

and all of its related functions, such as the Jacobian of `f`

, its gradient with respect to time, and more. For all cases, `u0`

is the initial condition, `p`

are the parameters, and `t`

is the independent variable.

**Constructor**

```
DDEFunction{iip,recompile}(f;
mass_matrix = __has_mass_matrix(f) ? f.mass_matrix : I,
analytic = __has_analytic(f) ? f.analytic : nothing,
tgrad= __has_tgrad(f) ? f.tgrad : nothing,
jac = __has_jac(f) ? f.jac : nothing,
jvp = __has_jvp(f) ? f.jvp : nothing,
vjp = __has_vjp(f) ? f.vjp : nothing,
jac_prototype = __has_jac_prototype(f) ? f.jac_prototype : nothing,
sparsity = __has_sparsity(f) ? f.sparsity : jac_prototype,
paramjac = __has_paramjac(f) ? f.paramjac : nothing,
syms = __has_syms(f) ? f.syms : nothing,
indepsym= __has_indepsym(f) ? f.indepsym : nothing,
colorvec = __has_colorvec(f) ? f.colorvec : nothing,
sys = __has_sys(f) ? f.sys : nothing)
```

Note that only the function `f`

itself is required. This function should be given as `f!(du,u,h,p,t)`

or `du = f(u,h,p,t)`

. See the section on `iip`

for more details on in-place vs out-of-place handling. The histroy function `h`

acts as an interpolator over time, i.e. `h(t)`

with options matching the solution interface, i.e. `h(t; save_idxs = 2)`

.

All of the remaining functions are optional for improving or accelerating the usage of `f`

. These include:

`mass_matrix`

: the mass matrix`M`

represented in the ODE function. Can be used to determine that the equation is actually a differential-algebraic equation (DAE) if`M`

is singular. Note that in this case special solvers are required, see the DAE solver page for more details: https://diffeq.sciml.ai/stable/solvers/dae_solve/. Must be an AbstractArray or an AbstractSciMLOperator.`analytic(u0,p,t)`

: used to pass an analytical solution function for the analytical solution of the ODE. Generally only used for testing and development of the solvers.`tgrad(dT,u,h,p,t)`

or dT=tgrad(u,p,t): returns $\frac{\partial f(u,p,t)}{\partial t}$`jac(J,u,h,p,t)`

or`J=jac(u,p,t)`

: returns $\frac{df}{du}$`jvp(Jv,v,h,u,p,t)`

or`Jv=jvp(v,u,p,t)`

: returns the directional derivative$\frac{df}{du} v$`vjp(Jv,v,h,u,p,t)`

or`Jv=vjp(v,u,p,t)`

: returns the adjoint derivative$\frac{df}{du}^\ast v$`jac_prototype`

: a prototype matrix matching the type that matches the Jacobian. For example, if the Jacobian is tridiagonal, then an appropriately sized`Tridiagonal`

matrix can be used as the prototype and integrators will specialize on this structure where possible. Non-structured sparsity patterns should use a`SparseMatrixCSC`

with a correct sparsity pattern for the Jacobian. The default is`nothing`

, which means a dense Jacobian.`paramjac(pJ,h,u,p,t)`

: returns the parameter Jacobian $\frac{df}{dp}$.`syms`

: the symbol names for the elements of the equation. This should match`u0`

in size. For example, if`u0 = [0.0,1.0]`

and`syms = [:x, :y]`

, this will apply a canonical naming to the values, allowing`sol[:x]`

in the solution and automatically naming values in plots.`indepsym`

: the canonical naming for the independent variable. Defaults to nothing, which internally uses`t`

as the representation in any plots.`colorvec`

: a color vector according to the SparseDiffTools.jl definition for the sparsity pattern of the`jac_prototype`

. This specializes the Jacobian construction when using finite differences and automatic differentiation to be computed in an accelerated manner based on the sparsity pattern. Defaults to`nothing`

, which means a color vector will be internally computed on demand when required. The cost of this operation is highly dependent on the sparsity pattern.

**iip: In-Place vs Out-Of-Place**

For more details on this argument, see the ODEFunction documentation.

**recompile: Controlling Compilation and Specialization**

For more details on this argument, see the ODEFunction documentation.

**Fields**

The fields of the DDEFunction type directly match the names of the inputs.

## Solution Type

`DDEProblem`

solutions return an `ODESolution`

. For more information, see the ODE problem definition page for the `ODESolution`

docstring.

## Example Problems

Example problems can be found in DiffEqProblemLibrary.jl.

To use a sample problem, such as `prob_ode_linear`

, you can do something like:

```
#] add DiffEqProblemLibrary
using DiffEqProblemLibrary.ODEProblemLibrary
# load problems
ODEProblemLibrary.importodeproblems()
prob = ODEProblemLibrary.prob_ode_linear
sol = solve(prob)
```

### DDEs with 1 constant delay

`DDEProblemLibrary.prob_dde_constant_1delay_ip`

— Constant`prob_dde_constant_1delay_ip`

Delay differential equation

\[u'(t) = -u(t - 1)\]

for $t \in [0, 1]$ with history function $\phi(t) = 0$ if $t < 0$ and $\phi(0) = 1$.

**Solution**

The analytical solution for $t \in [0, 10]$ can be obtained by the method of steps and is provided in this implementation.

`DDEProblemLibrary.prob_dde_constant_1delay_oop`

— Constant`prob_dde_constant_1delay_oop`

Same delay differential equation as `prob_dde_constant_1delay_ip`

, but purposefully implemented with an out-of-place function.

`DDEProblemLibrary.prob_dde_constant_1delay_scalar`

— Constant`prob_dde_constant_1delay_scalar`

Same delay differential equation as `prob_dde_constant_1delay_ip`

, but purposefully implemented with a scalar function.

`DDEProblemLibrary.prob_dde_constant_1delay_long_ip`

— Constant`prob_dde_constant_1delay_long_ip`

Delay differential equation

\[u'(t) = u(t) - u(t - 1/5)\]

for $t \in [0, 100]$ with history function $\phi(t) = 0$ if $t < 0$ and $\phi(0) = 1$.

`DDEProblemLibrary.prob_dde_constant_1delay_long_oop`

— Constant`prob_dde_constant_1delay_long_oop`

Same delay differential equation as `prob_dde_constant_1delay_long_ip`

, but purposefully implemented with an out-of-place function.

`DDEProblemLibrary.prob_dde_constant_1delay_long_scalar`

— Constant`prob_dde_constant_1delay_long_scalar`

Same delay differential equation as `prob_dde_constant_1delay_long_ip`

, but purposefully implemented with a scalar function.

### DDEs with 2 constant delays

`DDEProblemLibrary.prob_dde_constant_2delays_ip`

— Constant`prob_dde_constant_2delays_ip`

Delay differential equation

\[u'(t) = -u(t - 1/3) - u(t - 1/5)\]

for $t \in [0, 1]$ with history function $\phi(t) = 0$ if $t < 0$ and $\phi(0) = 1$.

**Solution**

The analytical solution for $t \in [0, 10]$ can be obtained by the method of steps and is provided in this implementation.

`DDEProblemLibrary.prob_dde_constant_2delays_oop`

— Constant`prob_dde_constant_2delays_oop`

Same delay differential equation as `prob_dde_constant_2delays_ip`

, but purposefully implemented with an out-of-place function.

`DDEProblemLibrary.prob_dde_constant_2delays_scalar`

— Constant`prob_dde_constant_2delays_scalar`

Same delay differential equation as `prob_dde_constant_2delays_ip`

, but purposefully implemented with a scalar function.

`DDEProblemLibrary.prob_dde_constant_2delays_long_ip`

— Constant`prob_dde_constant_2delays_long_ip`

Delay differential equation

\[u'(t) = - u(t - 1/3) - u(t - 1/5)\]

for $t \in [0, 100]$ with history function $\phi(t) = 0$ if $t < 0$ and $\phi(0) = 1$.

`DDEProblemLibrary.prob_dde_constant_2delays_long_oop`

— Constant`prob_dde_constant_2delays_long_oop`

Same delay differential equation as `prob_dde_constant_2delays_long_ip`

, but purposefully implemented with an out-of-place function.

`DDEProblemLibrary.prob_dde_constant_2delays_long_scalar`

— Constant`prob_dde_constant_2delays_long_scalar`

Same delay differential equation as `prob_dde_constant_2delays_long_ip`

, but purposefully implemented with a scalar function.

### DDETest Problems

Some details:

```
# DDEs with time dependent delays
prob_dde_DDETST_A1, prob_dde_DDETST_A2,
# DDEs with vanishing time dependent delays
prob_dde_DDETST_B1, prob_dde_DDETST_B2,
# DDEs with state dependent delays
prob_dde_DDETST_C1, prob_dde_DDETST_C2, prob_dde_DDETST_C3, prob_dde_DDETST_C4,
# DDEs with vanishing state dependent delays
prob_dde_DDETST_D1, prob_dde_DDETST_D2,
# neutral DDEs with time dependent delays
prob_dde_DDETST_E1, prob_dde_DDETST_E2,
# neutral DDEs with vanishing time dependent delays
prob_dde_DDETST_F1, prob_dde_DDETST_F2, prob_dde_DDETST_F3, prob_dde_DDETST_F4, prob_dde_DDETST_F5,
# neutral DDEs with state dependent delays
prob_dde_DDETST_G1, prob_dde_DDETST_G2,
# neutral DDEs with vanishing state dependent delays
prob_dde_DDETST_H1, prob_dde_DDETST_H2, prob_dde_DDETST_H3, prob_dde_DDETST_H4
```

`DDEProblemLibrary.prob_dde_DDETST_A1`

— Constant`prob_dde_DDETST_A1`

Delay differential equation model of blood production, given by

\[u'(t) = \frac{0.2 u(t - 14)}{1 + u(t - 14)^{10}} - 0.1 u(t)\]

for $t \in [0, 500]$ and history function $\phi(t) = 0.5$ for $t \leq 0$.

**References**

Mackey, M. C. and Glass, L. (1977). Oscillation and chaos in physiological control systems, Science (197), pp. 287-289.

`DDEProblemLibrary.prob_dde_DDETST_A2`

— Constant`prob_dde_DDETST_A2`

Delay differential equation model of chronic granulocytic leukemia, given by

\[u_1'(t) = \frac{1.1}{1 + \sqrt{10} u_1(t - 20)^{5/4}} - \frac{10 u_1(t)}{1 + 40 u_2(t)},\]

\[u_2'(t) = \frac{100 u_1(t)}{1 + 40 u_2(t)} - 2.43 u_2(t),\]

for $t \in [0, 100]$ and history function

\[\phi_1(t) = 1.05767027/3,\]

\[\phi_2(t) = 1.030713491/3,\]

for $t \leq 0$.

**References**

Wheldon, T., Kirk, J. and Finlay, H. (1974). Cyclical granulopoiesis in chronic granulocytic leukemia: A simulation study., Blood (43), pp. 379-387.

`DDEProblemLibrary.prob_dde_DDETST_B1`

— Constant`prob_dde_DDETST_B1`

Delay differential equation

\[u'(t) = 1 - u(\exp(1 - 1/t))\]

for $t \in [0.1, 10]$ with history function $\phi(t) = \log t$ for $t \in (0, 0.1]$.

**Solution**

The analytical solution for $t \in [0.1, 10]$ is

\[u(t) = \log t.\]

**References**

Neves, K. W. (1975). Automatic integration of functional differential equations: An approach, ACM Trans. Math. Soft. (1), pp. 357-368.

`DDEProblemLibrary.prob_dde_DDETST_B2`

— Constant`prob_dde_DDETST_B2`

Delay differential equation

\[u'(t) = - 1 - u(t) + 2 [u(t / 2) < 0]\]

for $t \in [0, 2 \log 66]$ with history function $\phi(0) = 1$.

**Solution**

The analytical solution for $t \in [0, 2 \log 66]$ is

\[u(t) = \begin{cases} 2 \exp(-t) - 1 & \text{if } t \in [0, 2 \log 2], \\ 1 - 6 \exp(-t) & \text{if } t \in (2 \log 2, 2 \log 6], \\ 66 \exp(-t) - 1 & \text{if } t \in (2 \log 6, 2 \log 66]. \end{cases}\]

**References**

Neves, K. W. and Thompson, S. (1992). Solution of systems of functional differential equations with state dependent delays, Technical Report TR-92-009, Computer Science, Radford University.

`DDEProblemLibrary.prob_dde_DDETST_C1`

— Constant`prob_dde_DDETST_C1`

Delay differential equation

\[u'(t) = - 2 u(t - 1 - |u(t)|) (1 - u(t)^2)\]

for $t \in [0, 30]$ with history function $\phi(t) = 0.5$ for $t \leq 0$.

**References**

Paul, C. A. H. (1994). A test set of functional differential equations, Technical Report 249, The Department of Mathematics, The University of Manchester, Manchester, England.

`DDEProblemLibrary.prob_dde_DDETST_C2`

— Constant`prob_dde_DDETST_C2`

Delay differential equation

\[u_1'(t) = - 2 u_1(t - u_2(t)),\]

\[u_₂'(t) = \frac{|u_1(t - u_2(t))| - |u_1(t)|}{1 + |u_1(t - u_2(t))|},\]

for $t \in [0, 40]$ with history function

\[\phi_1(t) = 1,\]

\[\phi_2(t) = 0.5,\]

for $t \leq 0$.

**References**

Paul, C. A. H. (1994). A test set of functional differential equations, Technical Report 249, The Department of Mathematics, The University of Manchester, Manchester, England.

`DDEProblemLibrary.prob_dde_DDETST_C3`

— Constant`prob_dde_DDETST_C3`

Delay differential equation model of hematopoiesis, given by

\[u_1'(t) = \hat{s}_0 u_2(t - T_1) - \gamma u_1(t) - Q,\]

\[u_2'(t) = f(u_1(t)) - k u_2(t),\]

\[u_3'(t) = 1 - \frac{Q \exp(\gamma u_3(t))}{\hat{s}_0 u_2(t - T_1 - u_3(t))},\]

for $t \in [0, 300]$ with history function $\phi_1(0) = 3.325$, $\phi_3(0) = 120$, and

\[\phi_2(t) = \begin{cases} 10 & \text{if } t \in [- T_1, 0],\\ 9.5 & \text{if } t < - T_1, \end{cases}\]

where $f(y) = a / (1 + K y^r)$, $\hat{s}_0 = 0.0031$, $T_1 = 6$, $\gamma = 0.001$, $Q = 0.0275$, $k = 2.8$, $a = 6570$, $K = 0.0382$, and $r = 6.96$.

**References**

Mahaffy, J. M., Belair, J. and Mackey, M. C. (1996). Hematopoietic model with moving boundary condition and state dependent delay, Private communication.

`DDEProblemLibrary.prob_dde_DDETST_C4`

— Constant`prob_dde_DDETST_C4`

Delay differential equation model of hematopoiesis, given by the same delay differential equation as `prob_dde_DDETST_C3`

\[u_1'(t) = \hat{s}_0 u_2(t - T_1) - \gamma u_1(t) - Q,\]

\[u_2'(t) = f(u_1(t)) - k u_2(t),\]

\[u_3'(t) = 1 - \frac{Q \exp(\gamma u_3(t))}{\hat{s}_0 u_2(t - T_1 - u_3(t))},\]

for $t \in [0, 100]$ with history function $\phi_1(0) = 3.5$, $\phi_3(0) = 50$, and $\phi_2(t) = 10$ for $t \leq 0$, where $f(y) = a / (1 + K y^r)$, $\hat{s}_0 = 0.00372$, $T_1 = 3$, $\gamma = 0.1$, $Q = 0.00178$, $k = 6.65$, $a = 15600$, $K = 0.0382$, and $r = 6.96$.

**References**

Mahaffy, J. M., Belair, J. and Mackey, M. C. (1996). Hematopoietic model with moving boundary condition and state dependent delay, Private communication.

`DDEProblemLibrary.prob_dde_DDETST_D1`

— Constant`prob_dde_DDETST_D1`

Delay differential equation

\[u_1'(t) = u_2(t), \]

\[u_2'(t) = - u_2(\exp(1 - u_2(t))) u_2(t)^2 \exp(1 - u_2(t)),\]

for $t \in [0.1, 5]$ with history function

\[\phi_1(t) = \log t, \]

\[\phi_2(t) = 1 / t,\]

for $t \in (0, 0.1]$.

**Solution**

The analytical solution for $t \in [0.1, 5]$ is

\[u_1(t) = \log t, \]

\[u_2(t) = 1 / t.\]

**References**

Neves, K. W. (1975). Automatic integration of functional differential equations: An approach, ACM Trans. Math. Soft. (1), pp. 357-368.

`DDEProblemLibrary.prob_dde_DDETST_D2`

— Constant`prob_dde_DDETST_D2`

Delay differential equation model of antigen antibody dynamics with fading memory, given by

\[u_1'(t) = - r_1 u_1(t) u_2(t) + r_2 u_3(t), \]

\[u_2'(t) = - r_1 u_1(t) u_2(t) + \alpha r_1 u_1(t - u_4(t)) u_2(t - u_4(t)),\]

\[u_3'(t) = r_1 u_1(t) u_2(t) - r_2 u_3(t), \]

\[u_4'(t) = 1 + \frac{3 \delta - u_1(t) u_2(t) - u_3(t)}{u_1(t - u_4(t)) u_2(t - u_4(t)) + u_3(t - u_4(t))} \exp(\delta u_4(t)),\]

for $t \in [0, 40]$ with history function

\[\phi_1(t) = 5, \]

\[\phi_2(t) = 0.1, \]

\[\phi_3(t) = 0, \]

\[\phi_4(t) = 0,\]

for $t \leq 0$, where $r_1 = 0.02$, $r_2 = 0.005$, $\alpha = 3$, and $\delta = 0.01$.

**References**

Gatica, J. and Waltman, P. (1982). A threshold model of antigen antibody dynamics with fading memory, in Lakshmikantham (ed.), Nonlinear phenomena in mathematical science, Academic Press, New York, pp. 425-439.

`DDEProblemLibrary.prob_dde_DDETST_E1`

— Constant`prob_dde_DDETST_E1`

Delay differential equation model of a food-limited population, given by

\[u(t) = r u(t) (1 - u(t - 1) - c u'(t - 1))\]

for $t \in [0, 40]$ with history function $\phi(t) = 2 + t$ for $t \leq 0$, where $r = \pi / \sqrt{3} + 1/20$ and $c = \sqrt{3} / (2 \pi) - 1 / 25$.

**References**

Kuang, Y. and Feldstein, A. (1991). Boundedness of solutions of a nonlinear nonautonomous neutral delay equation, J. Math. Anal. Appl. (156), pp. 293-304.

`DDEProblemLibrary.prob_dde_DDETST_E2`

— Constant`prob_dde_DDETST_E2`

Delay differential equation model of a logistic Gauss-type predator-prey system, given by

\[u_1'(t) = u_1(t) (1 - u_1(t - \tau) - \rho u_1'(t - \tau)) - \frac{u_2(t) u_1(t)^2}{u_1(t)^2 + 1}, \]

\[u_2'(t) = u_2(t) \left(\frac{u_1(t)^2}{u_1(t)^2 + 1} - \alpha\right),\]

for $t \in [0, 2]$ with history function

\[\phi_1(t) = 0.33 - t / 10, \]

\[\phi_2(t) = 2.22 + t / 10,\]

for $t \leq 0$, where $\alpha = 0.1$, $\rho = 2.9$, and $\tau = 0.42$.

**References**

Kuang, Y. (1991). On neutral delay logistics Gauss-type predator-prey systems, Dyn. Stab. Systems (6), pp. 173-189.

`DDEProblemLibrary.prob_dde_DDETST_F1`

— Constant`prob_dde_DDETST_F1`

Delay differential equation

\[u'(t) = 2 \cos(2t) u(t / 2)^{2 \cos t} + \log(u'(t / 2)) - \log(2 \cos t) - \sin t\]

for $t \in [0, 1]$ with history function $\phi(0) = 1$ and $\phi'(0) = 2$.

**Solution**

The analytical solution for $t \in [0, 1]$ is

\[u(t) = \exp(\sin(2t)).\]

**References**

Jackiewicz, Z. (1981). One step methods for the numerical solution of Volterra functional differential equations of neutral type, Applicable Anal. (12), pp. 1-11.

`DDEProblemLibrary.prob_dde_DDETST_F2`

— Constant`prob_dde_DDETST_F2`

Delay differential equation

\[u'(t) = u'(2t - 0.5)\]

for $t \in [0.25, 0.499]$ with history function $\phi(t) = \exp(-t^2)$ and $\phi'(t) = -2t \exp(-t^2)$ for $t \leq 0.25$.

**Solution**

The analytical solution for $t \in [0.25, 0.499]$ is

\[u(t) = u_i(t) = \exp(-4^i t^2 + B_i t + C_i) / 2^i + K_i\]

if $t \in [x_i, x_{i + 1}]$, where

\[x_i = (1 - 2^{-i}) / 2, \]

\[B_i = 2 (4^{i-1} + B_{i-1}), \]

\[C_i = - 4^{i-2} - B_{i-1} / 2 + C_{i-1}, \]

\[K_i = - \exp(-4^i x_i^2 + B_i x_i + C_i) / 2^i + u_{i-1}(x_i),\]

and $B_0 = C_0 = K_0 = 0$.

**References**

Neves, K. W. and Thompson, S. (1992). Solution of systems of functional differential equations with state dependent delays, Technical Report TR-92-009, Computer Science, Radford University.

`DDEProblemLibrary.prob_dde_DDETST_F3`

— Constant`prob_dde_DDETST_F3`

Delay differential equation

\[u'(t) = \exp(-u(t)) + L_3 \left[\sin(u'(\alpha(t))) - \sin\left(\frac{1}{3 + \alpha(t)}\right)\right]\]

for $t \in [0, 10]$ with history function $\phi(0) = \log 3$ and $\phi'(0) = 1 / 3$, where $\alpha(t) = 0.5 t (1 - \cos(2 \pi t))$ and $L_3 = 0.2$.

**Solution**

The analytical solution for $t \in [0, 10]$ is

\[u(t) = \log(t + 3).\]

`DDEProblemLibrary.prob_dde_DDETST_F4`

— Constant`prob_dde_DDETST_F4`

Same delay differential equation as `prob_dde_DDETST_F3`

with $L_3 = 0.4$.

`DDEProblemLibrary.prob_dde_DDETST_F5`

— Constant`prob_dde_DDETST_F5`

Same delay differential equation as `prob_dde_DDETST_F3`

with $L_3 = 0.6$.

`DDEProblemLibrary.prob_dde_DDETST_G1`

— Constant`prob_dde_DDETST_G1`

Delay differential equation

\[u'(t) = - u'(t - u(t)^2 / 4)\]

for $t \in [0, 1]$ with history function $\phi(t) = 1 - t$ for $t \leq 0$ and $\phi'(t) = -1$ for $t < 0$.

**Solution**

The analytical solution for $t \in [0, 1]$ is

\[u(t) = t + 1.\]

**References**

El'sgol'ts, L. E. and Norkin, S. B. (1973). Introduction to the Theory and Application of Differential Equations with Deviating Arguments, Academic Press, New York, p. 44.

`DDEProblemLibrary.prob_dde_DDETST_G2`

— Constant`prob_dde_DDETST_G2`

Delay differential equation

\[u'(t) = - u'(u(t) - 2)\]

for $t \in [0, 1]$ with history function $\phi(t) = 1 - t$ for $t \leq 0$ and $\phi'(t) = -1$ for $t < 0$.

**Solution**

The analytical solution for $t \in [0, 1]$ is

\[u(t) = t + 1.\]

El'sgol'ts, L. E. and Norkin, S. B. (1973). Introduction to the Theory and Application of Differential Equations with Deviating Arguments, Academic Press, New York, pp. 44-45.

`DDEProblemLibrary.prob_dde_DDETST_H1`

— Constant`prob_dde_DDETST_H1`

Delay differential equation

\[u'(t) = - \frac{4 t u(t)^2}{4 + \log(\cos(2t))^2} + \tan(2t) + 0.5 \arctan\left(u'\left(\frac{t u(t)^2}{1 + u(t)^2}\right)\right)\]

for $t \in [0, 0.225 \pi]$ with history function $\phi(0) = 0$ and $\phi'(0) = 0$.

**Solution**

The analytical solution for $t \in [0, 0.225 \pi]$ is

\[u(t) = - \log(\cos(2t)) / 2.\]

**References**

Castleton, R. N. and Grimm, L. J. (1973). A first order method for differential equations of neutral type, Math. Comput. (27), pp. 571-577.

`DDEProblemLibrary.prob_dde_DDETST_H2`

— Constant`prob_dde_DDETST_H2`

Delay differential equation

\[u'(t) = \cos(t) (1 + u(t u(t)^2)) + L_3 u(t) u'(t u(t)^2) + (1 - L_3) \sin(t) \cos(t \sin(t)^2) - \sin(t + t \sin(t)^2)\]

for $t \in [0, \pi]$ with history function $\phi(0) = 0$ and $\phi'(0) = 1$, where $L_3 = 0.1$.

**Solution**

The analytical solution for $t \in [0, \pi]$ is

\[u(t) = \sin(t).\]

**References**

Hayashi, H. (1996). Numerical solution of retarded and neutral delay differential equations using continuous Runge-Kutta methods, PhD thesis, Department of Computer Science, University of Toronto, Toronto, Canada.

`DDEProblemLibrary.prob_dde_DDETST_H3`

— Constant`prob_dde_DDETST_H3`

Same delay differential equation as `prob_dde_DDETST_H2`

with $L_3 = 0.3$.

**References**

Hayashi, H. (1996). Numerical solution of retarded and neutral delay differential equations using continuous Runge-Kutta methods, PhD thesis, Department of Computer Science, University of Toronto, Toronto, Canada.

`DDEProblemLibrary.prob_dde_DDETST_H4`

— Constant`prob_dde_DDETST_H4`

Same delay differential equation as `prob_dde_DDETST_H2`

with $L_3 = 0.5$.

**References**

Hayashi, H. (1996). Numerical solution of retarded and neutral delay differential equations using continuous Runge-Kutta methods, PhD thesis, Department of Computer Science, University of Toronto, Toronto, Canada.

### Radar5 Test Problems

`DDEProblemLibrary.prob_dde_RADAR5_oregonator`

— Constant`prob_dde_RADAR5_oregonator`

Delay differential equation model from chemical kinetics, given by

\[ u_1'(t) = - k_1 A u_2(t) - k_2 u_1(t) u_2(t - \tau) + k_3 B u_1(t) - 2 k_4 u_1(t)^2, \]

\[ u_2'(t) = - k_1 A u_2(t) - k_2 u_1(t) u_2(t - \tau) + f k_3 B u_1(t),\]

for $t \in [0, 100.5]$ with history function

\[ \phi_1(t) = 1e-10, \]

\[ \phi_2(t) = 1e-5,\]

for $t \leq 0$, where $k_1 = 1.34$, $k_2 = 1.6e9$, $k_3 = 8000$, $k_4 = 4e7$, $k_5 = 1$, $f = 1$, $A = 0.06$, $B = 0.06$, and $\tau = 0.15$.

**References**

Epstein, I. and Luo, Y. (1991). Differential delay equations in chemical kinetics. Nonlinear models, Journal of Chemical Physics (95), pp. 244-254.

`DDEProblemLibrary.prob_dde_RADAR5_robertson`

— Constant`prob_dde_RADAR5_robertson`

Delay differential equation model of a chemical reaction with steady state solution, given by

\[ u_1'(t) = - a u_1(t) + b u_2(t - \tau) u_3(t), \]

\[ u_2'(t) = a u_1(t) - b u_2(t - \tau) u_3(t) - c u_2(t)^2, \]

\[ u_3'(t) = c u_2(t)^2,\]

for $t \in [0, 10e10]$ with history function $\phi_1(0) = 1$, $\phi_2(t) = 0$ for $t \in [-\tau, 0]$, and $\phi_3(0) = 0$, where $a = 0.04$, $b = 10_000$, $c = 3e7$, and $\tau = 0.01$.

**References**

Guglielmi, N. and Hairer, E. (2001). Implementing Radau IIA methods for stiff delay differential equations, Computing (67), pp. 1-12.

`DDEProblemLibrary.prob_dde_RADAR5_waltman`

— Constant`prob_dde_RADAR5_waltman`

Delay differential equation model of antibody production, given by

\[ u_1'(t) = - r u_1(t) u_2(t) - s u_1(t) u_4(t), \]

\[ u_2'(t) = - r u_1(t) u_2(t) + \alpha r u_1(u_5(t)) u_2(u_5(t)) [t \geq t_0], \]

\[ u_3'(t) = r u_1(t) u_2(t), \]

\[ u_4'(t) = - s u_1(t) u_4(t) - \gamma u_4(t) + \beta r u_1(u_6(t)) u_2(u_6(t)) [t > t_1], \]

\[ u_5'(t) = [t \geq t_0] \frac{u_1(t) u_2(t) + u_3(t)}{u_1(u_5(t)) u_2(u_5(t)) + u_3(u_5(t))}, \]

\[ u_6'(t) = [t \geq t_1] \frac{1e-12 + u_2(t) + u_3(t)}{1e-12 + u_2(u_6(t)) + u_3(u_6(t))},\]

for $t \in [0, 300]$ with history function

\[ \phi_1(t) = \phi_0, \]

\[ \phi_2(t) = 1e-15, \]

\[ \phi_3(t) = 0, \]

\[ \phi_4(t) = 0, \]

\[ \phi_5(t) = 0, \]

\[ \phi_6(t) = 0,\]

for $t \leq 0$, where $\alpha = 1.8$, $\beta = 20$, $\gamma = 0.002$, $r = 5e4$, $s = 1e5$, $t_0 = 32$, $t_1 = 119$, and $\phi_0 = 0.75e-4$.

**References**

Waltman, P. (1978). A threshold model of antigen-stimulated antibody production, Theoretical Immunology (8), pp. 437-453.

### QS Example

`DDEProblemLibrary.prob_dde_qs`

— Constant`prob_dde_qs`

Delay differential equation model of Quorum Sensing (QS) of Pseudomonas putida IsoF in continuous cultures.

**References**

Buddrus-Schiemann et al. (2014). Analysis of N-Acylhomoserine Lactone Dynamics in Continuous Cultures of Pseudomonas Putida IsoF By Use of ELISA and UHPLC/qTOF-MS-derived Measurements and Mathematical Models, Analytical and Bioanalytical Chemistry.