DDE Problems

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.

prob_dde_constant_1delay_oop

Same delay differential equation as prob_dde_constant_1delay_ip, but purposefully implemented with an out-of-place function.

prob_dde_constant_1delay_scalar

Same delay differential equation as prob_dde_constant_1delay_ip, but purposefully implemented with a scalar function.

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 $\phi(0) = 1$.

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.

prob_dde_constant_1delay_long_scalar

Same delay differential equation as prob_dde_constant_1delay_long_ip, but purposefully implemented with a scalar function.

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.

prob_dde_constant_2delays_oop

Same delay differential equation as prob_dde_constant_2delays_ip, but purposefully implemented with an out-of-place function.

prob_dde_constant_2delays_scalar

Same delay differential equation as prob_dde_constant_2delays_ip, but purposefully implemented with a scalar function.

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 $\phi(0) = 1$.

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.

prob_dde_constant_2delays_long_scalar

Same delay differential equation as prob_dde_constant_2delays_long_ip, but purposefully implemented with a scalar function.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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

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

prob_dde_DDETST_F4

Same delay differential equation as prob_dde_DDETST_F3 with $L_3 = 0.4$.

prob_dde_DDETST_F5

Same delay differential equation as prob_dde_DDETST_F3 with $L_3 = 0.6$.

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.

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.

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.

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.

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.

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.

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.

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.

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

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

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

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

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

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

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.

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.