2857:
2568:
778:
controllers: one could think that the simplest approach would consist in replacing them by some finite-dimensional approximations. Unfortunately, ignoring effects which are adequately represented by DDEs is not a general alternative: in the best situation (constant and known delays), it leads to the
3864:
3694:
757:
Aftereffect is an applied problem: it is well known that, together with the increasing expectations of dynamic performances, engineers need their models to behave more like the real process. Many processes include aftereffect phenomena in their inner dynamics. In addition,
770:
that are now involved in feedback control loops introduce such delays. Finally, besides actual delays, time lags are frequently used to simplify very high order models. Then, the interest for DDEs keeps on growing in all scientific areas and, especially, in control
4183:
more involved. The spectrum does however have some properties which can be exploited in the analysis. For instance, even though there are an infinite number of eigenvalues, there are only a finite number of eigenvalues in any vertical strip of the complex plane.
3498:
3137:
4167:
2424:
1258:
3699:
1085:
4031:
1601:
4707:
3503:
2852:{\displaystyle {\begin{aligned}x(t)=x(\tau )+\int _{s=\tau }^{t}{\frac {d}{dt}}x(s)\,ds&=(a\tau +1)+a\int _{s=\tau }^{t}\left(a(s-\tau )+1\right)ds\\&=(a\tau +1)+a\int _{s=0}^{t-\tau }\left(as+1\right)ds,\end{aligned}}}
3345:
1396:
2979:
3225:
2089:
3355:
1668:
3002:
1757:
1448:
4565:
4191:
and there are many methods to compute the spectrum numerically. In some special situations it is possible to solve the characteristic equation explicitly. Consider, for example, the following DDE:
1875:
929:
2573:
4457:
4040:
2135:. This can be continued for the successive intervals by using the solution to the previous interval as inhomogeneous term. In practice, the initial value problem is often solved numerically.
1933:
2224:
989:
2266:
4260:
1130:
4305:
839:
741:, systems with aftereffect or dead-time, hereditary systems, equations with deviating argument, or differential-difference equations. They belong to the class of systems with the
4357:
3885:
1455:
2133:
1117:
2520:
4594:
4589:
3230:
2259:
779:
same degree of complexity in the control design. In worst cases (time-varying delays, for instance), it is potentially disastrous in terms of stability and oscillations.
2470:
1265:
1994:
708:
2561:
1014:
449:
4736:
1009:
1965:
3859:{\displaystyle y=\int _{-\infty }^{0}x(t+\tau )\cos(\alpha \tau +\beta )\,d\tau ,\quad z=\int _{-\infty }^{0}x(t+\tau )\sin(\alpha \tau +\beta )\,d\tau .}
737:
in which the derivative of the unknown function at a certain time is given in terms of the values of the function at previous times. DDEs are also called
2002:
192:
2864:
323:
3142:
5047:
Appeltans, Pieter; Michiels, Wim (2023-04-29). "Analysis and controller-design of time-delay systems using TDS-CONTROL. A tutorial and manual".
3689:{\displaystyle {\frac {d}{dt}}x(t)=f(t,x,y),\quad {\frac {d}{dt}}y(t)=\cos(\beta )x+\alpha z,\quad {\frac {d}{dt}}z(t)=\sin(\beta )x-\alpha y,}
364:
1676:
5031:
4996:
4961:
4852:
1788:
DDEs are mostly solved in a stepwise fashion with a principle called the method of steps. For instance, consider the DDE with a single delay
254:
4469:
1791:
4179:. Because of the exponential in the characteristic equation, the DDE has, unlike the ODE case, an infinite number of eigenvalues, making a
701:
156:
844:
5354:
575:
229:
5467:
5438:
5409:
5388:
5364:
5276:
5249:
4379:
3879:
1606:
250:
202:
5266:
4194:
5144:"Mathematical Model for the Epidemiology of Tuberculosis, with Estimates of the Reproductive Number and Infection-Delay Function"
4777:
742:
318:
237:
212:
4265:
5513:
4175:
of the characteristic equation are called characteristic roots or eigenvalues and the solution set is often referred to as the
750:
694:
629:
354:
3875:
1401:
790:
746:
272:
182:
369:
5454:. Surveys and Tutorials in the Applied Mathematical Sciences. Vol. 3. New York, NY: Springer Science+Business Media.
197:
187:
4314:
3493:{\displaystyle {\frac {d}{dt}}x(t)=f\left(t,x(t),\int _{-\infty }^{0}x(t+\tau )\cos(\alpha \tau +\beta )\,d\tau \right)}
753:(ODEs) having a finite dimensional state vector. Four points may give a possible explanation of the popularity of DDEs:
644:
495:
398:
285:
207:
1882:
245:
535:
2146:
934:
767:
403:
5111:. Selected Papers, The Third International Conference on the Numerical Solutions of Volterra and Delay Equations.
789:
In spite of their complexity, DDEs often appear as simple infinite-dimensional models in the very complex area of
506:
3132:{\displaystyle {\frac {d}{dt}}x(t)=f\left(t,x(t),\int _{-\infty }^{0}x(t+\tau )e^{\lambda \tau }\,d\tau \right).}
649:
484:
500:
408:
800:
562:
5508:
5105:"Mathematical models and software tools for the glucose-insulin regulatory system and diabetes: an overview"
580:
570:
518:
359:
4802:
Richard, Jean-Pierre (2003). "Time Delay
Systems: An overview of some recent advances and open problems".
4188:
4034:
393:
4162:{\displaystyle \det(-\lambda I+A_{0}+A_{1}e^{-\tau _{1}\lambda }+\dotsb +A_{m}e^{-\tau _{m}\lambda })=0.}
2989:
1997:
734:
639:
624:
513:
456:
438:
277:
33:
2094:
1090:
2419:{\displaystyle x(t)=x(0)+\int _{s=0}^{t}{\frac {d}{dt}}x(s)\,ds=1+a\int _{s=0}^{t}\phi (s-\tau )\,ds,}
5192:
5104:
4908:
4176:
2475:
1253:{\displaystyle {\frac {d}{dt}}x(t)=f\left(t,x(t),\int _{-\infty }^{0}x(t+\tau )\,d\mu (\tau )\right)}
525:
461:
434:
5193:"Construction of Lyapunov functionals for delay differential equations in virology and epidemiology"
4896:
4618:
5485:
5375:
4762:
557:
542:
443:
307:
146:
113:
104:
5018:. Advances in Design and Control. Society for Industrial and Applied Mathematics. pp. 33–56.
4572:
5331:
5305:
5069:
5048:
4948:. Advances in Design and Control. Society for Industrial and Applied Mathematics. pp. 3–32.
4924:
4858:
4834:
4782:
4373:
552:
547:
430:
17:
2229:
5401:
Linear
Parameter-Varying and Time-Delay Systems: Analysis, Observation, Filtering & Control
2431:
5463:
5434:
5405:
5384:
5360:
5323:
5272:
5245:
5212:
5173:
5165:
5124:
5027:
4992:
4957:
4848:
1970:
1776:
1080:{\displaystyle \mathbb {R} \times \mathbb {R} ^{n}\times C^{1}(\mathbb {R} ,\mathbb {R} ^{n})}
664:
609:
388:
123:
4983:. Advances in Design and Control. Society for Industrial and Applied Mathematics. p. 9.
5455:
5426:
5315:
5237:
5204:
5155:
5116:
5019:
4984:
4949:
4916:
4840:
4811:
4026:{\displaystyle {\frac {d}{dt}}x(t)=A_{0}x(t)+A_{1}x(t-\tau _{1})+\dots +A_{m}x(t-\tau _{m})}
1596:{\displaystyle {\frac {d}{dt}}x(t)=A_{0}x(t)+A_{1}x(t-\tau _{1})+\dots +A_{m}x(t-\tau _{m})}
674:
659:
5449:
5420:
5231:
2988:
In some cases, differential equations can be represented in a format that looks like delay
2525:
5013:
4978:
4943:
4739:
4712:
4180:
614:
530:
57:
5359:. Numerical Mathematics and Scientific Computation. Oxford, UK: Oxford University Press.
669:
4912:
4901:
Proceedings of the Royal
Society of London. Series A, Mathematical and Physical Sciences
5160:
5143:
4702:{\displaystyle u(t)={\begin{cases}F(t+1),\quad |t|<1\\0,\quad |t|\geq 1\end{cases}}}
994:
783:
634:
619:
425:
413:
132:
4815:
1938:
5502:
5335:
4928:
4895:
Ockendon, John
Richard; Tayler, A. B.; Temple, George Frederick James (1971-05-04).
5490:
4862:
4757:
3340:{\displaystyle {\frac {d}{dt}}x(t)=f(t,x,y),\quad {\frac {d}{dt}}y(t)=x-\lambda y.}
654:
604:
490:
118:
4877:
3882:. The characteristic equation associated with the linear DDE with discrete delays
1391:{\displaystyle {\frac {d}{dt}}x(t)=f(t,x(t),x(t-\tau _{1}),\dots ,x(t-\tau _{m}))}
5399:
5233:
Stability and
Oscillations in Delay Differential Equations of Population Dynamics
5208:
2974:{\textstyle x(t)=(a\tau +1)+a(t-\tau )\left({\frac {1}{2}}{a(t-\tau )}+1\right).}
5120:
722:
62:
5319:
5236:. Mathematics and Its Applications. Dordrecht, NL: Kluwer Academic Publishers.
4830:
5459:
5430:
5425:. Applied Mathematical Sciences. Vol. 20. New York, NY: Springer-Verlag.
5241:
679:
5327:
5293:
5216:
5169:
5128:
4844:
3878:, many properties of linear DDEs can be characterized and analyzed using the
5023:
4988:
4953:
3220:{\displaystyle y(t)=\int _{-\infty }^{0}x(t+\tau )e^{\lambda \tau }\,d\tau }
420:
141:
84:
74:
4920:
5177:
5271:. Mathematics in Science and Engineering. San Diego, CA: Academic Press.
4752:
991:
represents the trajectory of the solution in the past. In this equation,
759:
5383:. Mathematics in Science and Engineering. New York, NY: Academic Press.
4897:"The dynamics of a current collection system for an electric locomotive"
4307:
There are an infinite number of solutions to this equation for complex
95:
90:
79:
1775:< 1. This equation and some more general forms are named after the
5268:
Delay
Differential Equations with Applications in Population Dynamics
763:
5404:. Advances in Delays and Dynamics. Heidelberg, DE: Springer-Verlag.
5310:
5191:
Kajiwara, Tsuyoshi; Sasaki, Toru; Takeuchi, Yasuhiro (2012-08-01).
5074:
5053:
2084:{\displaystyle {\frac {d}{dt}}\psi (t)=f(\psi (t),\phi (t-\tau )),}
4831:"Simple delay-based implementation of continuous-time controllers"
5068:
Juan Arias de Reyna (2017). "Arithmetic of the Fabius function".
2261:. Then the initial value problem can be solved with integration,
1663:{\displaystyle A_{0},\dotsc ,A_{m}\in \mathbb {R} ^{n\times n}}
4829:
Lavaei, Javad; Sojoudi, Somayeh; Murray, Richard M. (2010).
5089:
4695:
797:
A general form of the time-delay differential equation for
5103:
Makroglou, Athena; Li, Jiaxu; Kuang, Yang (2006-03-01).
1752:{\displaystyle {\frac {d}{dt}}x(t)=ax(t)+bx(\lambda t),}
5142:
Salpeter, Edwin E.; Salpeter, Shelley R. (1998-02-15).
1443:{\displaystyle \tau _{1}>\dots >\tau _{m}\geq 0.}
4560:{\displaystyle {\frac {d}{dt}}u(t)=2u(2t+1)-2u(2t-1).}
2867:
1870:{\displaystyle {\frac {d}{dt}}x(t)=f(x(t),x(t-\tau ))}
5294:"On an electrodynamic origin of quantum fluctuations"
4715:
4597:
4575:
4472:
4382:
4317:
4268:
4197:
4043:
3888:
3702:
3506:
3358:
3233:
3145:
3005:
2571:
2528:
2478:
2434:
2269:
2232:
2149:
2097:
2005:
1973:
1941:
1885:
1794:
1679:
1609:
1458:
1404:
1268:
1133:
1093:
1017:
997:
937:
847:
803:
749:(PDEs) which are infinite dimensional, as opposed to
924:{\displaystyle {\frac {d}{dt}}x(t)=f(t,x(t),x_{t}),}
4836:
Proceedings of the 2010 American
Control Conference
5356:Numerical Methods for Delay Differential Equations
4730:
4701:
4583:
4559:
4451:
4351:
4299:
4254:
4161:
4025:
3858:
3688:
3492:
3339:
3219:
3131:
2973:
2851:
2555:
2514:
2464:
2418:
2253:
2218:
2127:
2083:
1988:
1959:
1927:
1869:
1751:
1662:
1595:
1442:
1390:
1252:
1111:
1079:
1003:
983:
923:
833:
5015:Stability and Stabilization of Time-Delay Systems
4980:Stability and Stabilization of Time-Delay Systems
4945:Stability and Stabilization of Time-Delay Systems
782:Voluntary introduction of delays can benefit the
5012:Michiels, Wim; Niculescu, Silviu-Iulian (2007).
4977:Michiels, Wim; Niculescu, Silviu-Iulian (2007).
4942:Michiels, Wim; Niculescu, Silviu-Iulian (2007).
4452:{\displaystyle x(t)=x(0)\,e^{W_{k}(-1)\cdot t}.}
4044:
1928:{\displaystyle \phi \colon \to \mathbb {R} ^{n}}
2219:{\displaystyle f(x(t),x(t-\tau ))=ax(t-\tau )}
984:{\displaystyle x_{t}=\{x(\tau ):\tau \leq t\}}
4878:"The pantograph equation in quantum calculus"
702:
8:
5374:Bellman, Richard; Cooke, Kenneth L. (1963).
4255:{\displaystyle {\frac {d}{dt}}x(t)=-x(t-1).}
2563:we integrate and fit the initial condition,
978:
951:
5197:Nonlinear Analysis: Real World Applications
1996:which is the solution to the inhomogeneous
4300:{\displaystyle -\lambda -e^{-\lambda }=0.}
2472:, where the initial condition is given by
774:Delay systems are still resistant to many
709:
695:
175:
50:
29:
5422:Ordinary and Delay Differential Equations
5353:Bellen, Alfredo; Zennaro, Marino (2003).
5309:
5159:
5073:
5052:
4714:
4681:
4673:
4651:
4643:
4613:
4596:
4577:
4576:
4574:
4473:
4471:
4420:
4415:
4410:
4381:
4328:
4316:
4282:
4267:
4198:
4196:
4139:
4131:
4121:
4097:
4089:
4079:
4066:
4042:
4014:
3992:
3970:
3948:
3923:
3889:
3887:
3846:
3798:
3790:
3769:
3721:
3713:
3701:
3626:
3565:
3507:
3505:
3478:
3430:
3422:
3359:
3357:
3292:
3234:
3232:
3210:
3201:
3173:
3165:
3144:
3114:
3105:
3077:
3069:
3006:
3004:
2937:
2927:
2866:
2802:
2791:
2707:
2696:
2654:
2627:
2621:
2610:
2572:
2570:
2527:
2477:
2433:
2406:
2382:
2371:
2348:
2321:
2315:
2304:
2268:
2231:
2148:
2096:
2006:
2004:
1972:
1940:
1919:
1915:
1914:
1884:
1795:
1793:
1680:
1678:
1648:
1644:
1643:
1633:
1614:
1608:
1584:
1562:
1540:
1518:
1493:
1459:
1457:
1428:
1409:
1403:
1376:
1342:
1269:
1267:
1229:
1205:
1197:
1134:
1132:
1100:
1096:
1095:
1092:
1068:
1064:
1063:
1055:
1054:
1045:
1032:
1028:
1027:
1019:
1018:
1016:
996:
942:
936:
909:
848:
846:
825:
821:
820:
802:
834:{\displaystyle x(t)\in \mathbb {R} ^{n}}
4794:
590:
341:
263:
221:
164:
131:
103:
53:
39:
32:
7:
5451:Applied Delay Differential Equations
1935:. Then the solution on the interval
157:List of named differential equations
4352:{\displaystyle \lambda =W_{k}(-1),}
230:Dependent and independent variables
5161:10.1093/oxfordjournals.aje.a009463
4187:This characteristic equation is a
3794:
3717:
3426:
3169:
3073:
1201:
25:
5377:Differential-Difference Equations
2128:{\displaystyle \psi (0)=\phi (0)}
1112:{\displaystyle \mathbb {R} ^{n}.}
18:Differential-difference equations
5148:American Journal of Epidemiology
4778:Functional differential equation
365:Carathéodory's existence theorem
5292:López, Álvaro G. (2020-09-01).
4672:
4642:
4262:The characteristic equation is
3779:
3625:
3564:
3291:
2515:{\displaystyle x(0)=\phi (0)=1}
751:ordinary differential equations
5486:"Delay-Differential Equations"
4876:Griebel, Thomas (2017-01-01).
4725:
4719:
4682:
4674:
4652:
4644:
4636:
4624:
4607:
4601:
4551:
4536:
4524:
4509:
4497:
4491:
4435:
4426:
4407:
4401:
4392:
4386:
4343:
4334:
4246:
4234:
4222:
4216:
4150:
4047:
4020:
4001:
3976:
3957:
3938:
3932:
3913:
3907:
3843:
3828:
3819:
3807:
3766:
3751:
3742:
3730:
3668:
3662:
3650:
3644:
3607:
3601:
3589:
3583:
3558:
3540:
3531:
3525:
3475:
3460:
3451:
3439:
3412:
3406:
3383:
3377:
3316:
3310:
3285:
3267:
3258:
3252:
3194:
3182:
3155:
3149:
3098:
3086:
3059:
3053:
3030:
3024:
2953:
2941:
2919:
2907:
2898:
2883:
2877:
2871:
2778:
2763:
2733:
2721:
2683:
2668:
2651:
2645:
2600:
2594:
2585:
2579:
2550:
2535:
2522:. Similarly, for the interval
2503:
2497:
2488:
2482:
2444:
2438:
2403:
2391:
2345:
2339:
2294:
2288:
2279:
2273:
2242:
2236:
2213:
2201:
2189:
2186:
2174:
2165:
2159:
2153:
2122:
2116:
2107:
2101:
2075:
2072:
2060:
2051:
2045:
2039:
2030:
2024:
1983:
1977:
1954:
1942:
1910:
1907:
1892:
1864:
1861:
1849:
1840:
1834:
1828:
1819:
1813:
1743:
1734:
1722:
1716:
1704:
1698:
1590:
1571:
1546:
1527:
1508:
1502:
1483:
1477:
1385:
1382:
1363:
1348:
1329:
1320:
1314:
1302:
1293:
1287:
1242:
1236:
1226:
1214:
1187:
1181:
1158:
1152:
1074:
1051:
1011:is a functional operator from
963:
957:
915:
899:
893:
881:
872:
866:
813:
807:
791:partial differential equations
747:partial differential equations
452: / Integral solutions
1:
5109:Applied Numerical Mathematics
4816:10.1016/S0005-1098(03)00167-5
1879:with given initial condition
27:Type of differential equation
5209:10.1016/j.nonrwa.2011.12.011
4584:{\displaystyle \mathbb {R} }
1452:Linear with discrete delays
727:delay differential equations
496:Exponential response formula
242:Coupled / Decoupled
5121:10.1016/j.apnum.2005.04.023
3870:The characteristic equation
5530:
5419:Driver, Rodney D. (1977).
5320:10.1007/s11071-020-05928-5
2254:{\displaystyle \phi (t)=1}
5460:10.1007/978-0-387-74372-1
5431:10.1007/978-1-4684-9467-9
5242:10.1007/978-94-015-7920-9
4767:Classical electrodynamics
2465:{\displaystyle x(t)=at+1}
1771:are constants and 0 <
630:Józef Maria Hoene-Wroński
576:Undetermined coefficients
485:Method of characteristics
370:Cauchy–Kowalevski theorem
5398:Briat, Corentin (2015).
4845:10.1109/ACC.2010.5530439
3227:to get a system of ODEs
1989:{\displaystyle \psi (t)}
355:Picard–Lindelöf theorem
349:Existence and uniqueness
5448:Erneux, Thomas (2009).
5024:10.1137/1.9780898718645
4989:10.1137/1.9780898718645
4954:10.1137/1.9780898718645
3880:characteristic equation
581:Variation of parameters
571:Separation of variables
360:Peano existence theorem
5514:Differential equations
5230:Gopalsamy, K. (1992).
4921:10.1098/rspa.1971.0078
4839:. pp. 5781–5788.
4732:
4703:
4585:
4561:
4453:
4353:
4301:
4256:
4189:nonlinear eigenproblem
4163:
4035:exponential polynomial
4027:
3860:
3690:
3494:
3341:
3221:
3133:
2990:differential equations
2975:
2853:
2557:
2516:
2466:
2420:
2255:
2220:
2129:
2085:
1990:
1961:
1929:
1871:
1753:
1664:
1597:
1444:
1392:
1254:
1113:
1081:
1005:
985:
925:
835:
768:communication networks
650:Carl David Tolmé Runge
193:Differential-algebraic
34:Differential equations
5484:Skip Thompson (ed.).
4733:
4704:
4586:
4562:
4454:
4354:
4302:
4257:
4164:
4028:
3861:
3691:
3495:
3342:
3222:
3134:
2999:Consider an equation
2976:
2854:
2558:
2556:{\displaystyle t\in }
2517:
2467:
2421:
2256:
2221:
2130:
2086:
1998:initial value problem
1991:
1962:
1930:
1872:
1754:
1665:
1598:
1445:
1393:
1255:
1114:
1082:
1006:
986:
926:
836:
735:differential equation
640:Augustin-Louis Cauchy
625:Joseph-Louis Lagrange
457:Numerical integration
439:Exponential stability
302:Relation to processes
4731:{\displaystyle F(t)}
4713:
4595:
4573:
4569:Have as solution in
4470:
4380:
4315:
4311:. They are given by
4266:
4195:
4041:
3886:
3700:
3504:
3356:
3231:
3143:
3003:
2865:
2569:
2526:
2476:
2432:
2267:
2230:
2147:
2095:
2003:
1971:
1939:
1883:
1792:
1677:
1673:Pantograph equation
1607:
1456:
1402:
1266:
1131:
1091:
1015:
995:
935:
845:
801:
462:Dirac delta function
198:Integro-differential
4913:1971RSPSA.322..447O
4763:Population dynamics
4466:The following DDE:
3803:
3726:
3435:
3178:
3082:
2813:
2712:
2626:
2387:
2320:
1210:
558:Perturbation theory
553:Integral transforms
444:Rate of convergence
310:(discrete analogue)
147:Population dynamics
114:Continuum mechanics
105:Applied mathematics
5298:Nonlinear Dynamics
5265:Kuang, Y. (1993).
4783:Halanay Inequality
4728:
4699:
4694:
4581:
4557:
4449:
4374:Lambert W function
4349:
4297:
4252:
4159:
4023:
3856:
3786:
3709:
3686:
3490:
3418:
3337:
3217:
3161:
3129:
3065:
2971:
2849:
2847:
2787:
2692:
2606:
2553:
2512:
2462:
2416:
2367:
2300:
2251:
2216:
2125:
2081:
1986:
1957:
1925:
1867:
1749:
1660:
1593:
1440:
1388:
1250:
1193:
1109:
1077:
1001:
981:
921:
831:
739:time-delay systems
548:Integrating factor
389:Initial conditions
324:Stochastic partial
5033:978-0-89871-632-0
4998:978-0-89871-632-0
4963:978-0-89871-632-0
4907:(1551): 447–468.
4854:978-1-4244-7427-1
4810:(10): 1667–1694.
4486:
4372:th branch of the
4211:
4181:spectral analysis
3902:
3639:
3578:
3520:
3500:is equivalent to
3372:
3305:
3247:
3019:
2935:
2640:
2334:
2019:
1808:
1693:
1472:
1282:
1147:
1127:Continuous delay
1004:{\displaystyle f}
861:
719:
718:
610:Gottfried Leibniz
501:Finite difference
293:
292:
154:
153:
124:Dynamical systems
16:(Redirected from
5521:
5495:
5473:
5444:
5415:
5394:
5382:
5370:
5340:
5339:
5313:
5289:
5283:
5282:
5262:
5256:
5255:
5227:
5221:
5220:
5203:(4): 1802–1826.
5188:
5182:
5181:
5163:
5139:
5133:
5132:
5100:
5094:
5093:
5090:"A288163 - Oeis"
5086:
5080:
5079:
5077:
5065:
5059:
5058:
5056:
5044:
5038:
5037:
5009:
5003:
5002:
4974:
4968:
4967:
4939:
4933:
4932:
4892:
4886:
4885:
4873:
4867:
4866:
4826:
4820:
4819:
4799:
4737:
4735:
4734:
4729:
4708:
4706:
4705:
4700:
4698:
4697:
4685:
4677:
4655:
4647:
4590:
4588:
4587:
4582:
4580:
4566:
4564:
4563:
4558:
4487:
4485:
4474:
4458:
4456:
4455:
4450:
4445:
4444:
4425:
4424:
4358:
4356:
4355:
4350:
4333:
4332:
4306:
4304:
4303:
4298:
4290:
4289:
4261:
4259:
4258:
4253:
4212:
4210:
4199:
4168:
4166:
4165:
4160:
4149:
4148:
4144:
4143:
4126:
4125:
4107:
4106:
4102:
4101:
4084:
4083:
4071:
4070:
4032:
4030:
4029:
4024:
4019:
4018:
3997:
3996:
3975:
3974:
3953:
3952:
3928:
3927:
3903:
3901:
3890:
3865:
3863:
3862:
3857:
3802:
3797:
3725:
3720:
3695:
3693:
3692:
3687:
3640:
3638:
3627:
3579:
3577:
3566:
3521:
3519:
3508:
3499:
3497:
3496:
3491:
3489:
3485:
3434:
3429:
3373:
3371:
3360:
3346:
3344:
3343:
3338:
3306:
3304:
3293:
3248:
3246:
3235:
3226:
3224:
3223:
3218:
3209:
3208:
3177:
3172:
3138:
3136:
3135:
3130:
3125:
3121:
3113:
3112:
3081:
3076:
3020:
3018:
3007:
2984:Reduction to ODE
2980:
2978:
2977:
2972:
2967:
2963:
2956:
2936:
2928:
2858:
2856:
2855:
2850:
2848:
2835:
2831:
2812:
2801:
2756:
2746:
2742:
2711:
2706:
2641:
2639:
2628:
2625:
2620:
2562:
2560:
2559:
2554:
2521:
2519:
2518:
2513:
2471:
2469:
2468:
2463:
2425:
2423:
2422:
2417:
2386:
2381:
2335:
2333:
2322:
2319:
2314:
2260:
2258:
2257:
2252:
2225:
2223:
2222:
2217:
2134:
2132:
2131:
2126:
2090:
2088:
2087:
2082:
2020:
2018:
2007:
1995:
1993:
1992:
1987:
1966:
1964:
1963:
1960:{\displaystyle }
1958:
1934:
1932:
1931:
1926:
1924:
1923:
1918:
1876:
1874:
1873:
1868:
1809:
1807:
1796:
1758:
1756:
1755:
1750:
1694:
1692:
1681:
1669:
1667:
1666:
1661:
1659:
1658:
1647:
1638:
1637:
1619:
1618:
1602:
1600:
1599:
1594:
1589:
1588:
1567:
1566:
1545:
1544:
1523:
1522:
1498:
1497:
1473:
1471:
1460:
1449:
1447:
1446:
1441:
1433:
1432:
1414:
1413:
1397:
1395:
1394:
1389:
1381:
1380:
1347:
1346:
1283:
1281:
1270:
1259:
1257:
1256:
1251:
1249:
1245:
1209:
1204:
1148:
1146:
1135:
1118:
1116:
1115:
1110:
1105:
1104:
1099:
1086:
1084:
1083:
1078:
1073:
1072:
1067:
1058:
1050:
1049:
1037:
1036:
1031:
1022:
1010:
1008:
1007:
1002:
990:
988:
987:
982:
947:
946:
930:
928:
927:
922:
914:
913:
862:
860:
849:
840:
838:
837:
832:
830:
829:
824:
743:functional state
733:) are a type of
711:
704:
697:
675:Phyllis Nicolson
660:Rudolf Lipschitz
543:Green's function
519:Infinite element
510:
475:Solution methods
453:
311:
222:By variable type
176:
58:Natural sciences
51:
30:
21:
5529:
5528:
5524:
5523:
5522:
5520:
5519:
5518:
5499:
5498:
5483:
5480:
5470:
5447:
5441:
5418:
5412:
5397:
5391:
5380:
5373:
5367:
5352:
5349:
5347:Further reading
5344:
5343:
5291:
5290:
5286:
5279:
5264:
5263:
5259:
5252:
5229:
5228:
5224:
5190:
5189:
5185:
5141:
5140:
5136:
5102:
5101:
5097:
5088:
5087:
5083:
5067:
5066:
5062:
5046:
5045:
5041:
5034:
5011:
5010:
5006:
4999:
4976:
4975:
4971:
4964:
4941:
4940:
4936:
4894:
4893:
4889:
4875:
4874:
4870:
4855:
4828:
4827:
4823:
4801:
4800:
4796:
4791:
4774:
4748:
4740:Fabius function
4711:
4710:
4693:
4692:
4663:
4662:
4614:
4593:
4592:
4571:
4570:
4478:
4468:
4467:
4464:
4462:Another example
4416:
4411:
4378:
4377:
4367:
4324:
4313:
4312:
4278:
4264:
4263:
4203:
4193:
4192:
4135:
4127:
4117:
4093:
4085:
4075:
4062:
4039:
4038:
4010:
3988:
3966:
3944:
3919:
3894:
3884:
3883:
3872:
3698:
3697:
3631:
3570:
3512:
3502:
3501:
3396:
3392:
3364:
3354:
3353:
3297:
3239:
3229:
3228:
3197:
3141:
3140:
3101:
3043:
3039:
3011:
3001:
3000:
2986:
2926:
2922:
2863:
2862:
2846:
2845:
2818:
2814:
2754:
2753:
2717:
2713:
2661:
2632:
2567:
2566:
2524:
2523:
2474:
2473:
2430:
2429:
2326:
2265:
2264:
2228:
2227:
2145:
2144:
2141:
2093:
2092:
2011:
2001:
2000:
1969:
1968:
1937:
1936:
1913:
1881:
1880:
1800:
1790:
1789:
1786:
1685:
1675:
1674:
1642:
1629:
1610:
1605:
1604:
1580:
1558:
1536:
1514:
1489:
1464:
1454:
1453:
1424:
1405:
1400:
1399:
1372:
1338:
1274:
1264:
1263:
1262:Discrete delay
1171:
1167:
1139:
1129:
1128:
1124:
1094:
1089:
1088:
1062:
1041:
1026:
1013:
1012:
993:
992:
938:
933:
932:
905:
853:
843:
842:
819:
799:
798:
715:
686:
685:
684:
615:Jacob Bernoulli
599:
586:
585:
567:
536:Petrov–Galerkin
504:
489:
476:
468:
467:
466:
448:
394:Boundary values
383:
375:
374:
350:
337:
336:
335:
309:
303:
295:
294:
282:
259:
217:
173:
160:
159:
155:
133:Social sciences
89:
67:
48:
28:
23:
22:
15:
12:
11:
5:
5527:
5525:
5517:
5516:
5511:
5509:Control theory
5501:
5500:
5497:
5496:
5479:
5478:External links
5476:
5475:
5474:
5469:978-0387743714
5468:
5445:
5440:978-0387902319
5439:
5416:
5411:978-3662440490
5410:
5395:
5390:978-0120848508
5389:
5371:
5366:978-0198506546
5365:
5348:
5345:
5342:
5341:
5304:(1): 621–634.
5284:
5278:978-0080960029
5277:
5257:
5251:978-0792315940
5250:
5222:
5183:
5154:(4): 398–406.
5134:
5115:(3): 559–573.
5095:
5081:
5060:
5039:
5032:
5004:
4997:
4969:
4962:
4934:
4887:
4882:Masters Theses
4868:
4853:
4821:
4793:
4792:
4790:
4787:
4786:
4785:
4780:
4773:
4770:
4769:
4768:
4765:
4760:
4755:
4747:
4744:
4727:
4724:
4721:
4718:
4696:
4691:
4688:
4684:
4680:
4676:
4671:
4668:
4665:
4664:
4661:
4658:
4654:
4650:
4646:
4641:
4638:
4635:
4632:
4629:
4626:
4623:
4620:
4619:
4617:
4612:
4609:
4606:
4603:
4600:
4591:the function:
4579:
4556:
4553:
4550:
4547:
4544:
4541:
4538:
4535:
4532:
4529:
4526:
4523:
4520:
4517:
4514:
4511:
4508:
4505:
4502:
4499:
4496:
4493:
4490:
4484:
4481:
4477:
4463:
4460:
4448:
4443:
4440:
4437:
4434:
4431:
4428:
4423:
4419:
4414:
4409:
4406:
4403:
4400:
4397:
4394:
4391:
4388:
4385:
4363:
4348:
4345:
4342:
4339:
4336:
4331:
4327:
4323:
4320:
4296:
4293:
4288:
4285:
4281:
4277:
4274:
4271:
4251:
4248:
4245:
4242:
4239:
4236:
4233:
4230:
4227:
4224:
4221:
4218:
4215:
4209:
4206:
4202:
4158:
4155:
4152:
4147:
4142:
4138:
4134:
4130:
4124:
4120:
4116:
4113:
4110:
4105:
4100:
4096:
4092:
4088:
4082:
4078:
4074:
4069:
4065:
4061:
4058:
4055:
4052:
4049:
4046:
4022:
4017:
4013:
4009:
4006:
4003:
4000:
3995:
3991:
3987:
3984:
3981:
3978:
3973:
3969:
3965:
3962:
3959:
3956:
3951:
3947:
3943:
3940:
3937:
3934:
3931:
3926:
3922:
3918:
3915:
3912:
3909:
3906:
3900:
3897:
3893:
3871:
3868:
3867:
3866:
3855:
3852:
3849:
3845:
3842:
3839:
3836:
3833:
3830:
3827:
3824:
3821:
3818:
3815:
3812:
3809:
3806:
3801:
3796:
3793:
3789:
3785:
3782:
3778:
3775:
3772:
3768:
3765:
3762:
3759:
3756:
3753:
3750:
3747:
3744:
3741:
3738:
3735:
3732:
3729:
3724:
3719:
3716:
3712:
3708:
3705:
3685:
3682:
3679:
3676:
3673:
3670:
3667:
3664:
3661:
3658:
3655:
3652:
3649:
3646:
3643:
3637:
3634:
3630:
3624:
3621:
3618:
3615:
3612:
3609:
3606:
3603:
3600:
3597:
3594:
3591:
3588:
3585:
3582:
3576:
3573:
3569:
3563:
3560:
3557:
3554:
3551:
3548:
3545:
3542:
3539:
3536:
3533:
3530:
3527:
3524:
3518:
3515:
3511:
3488:
3484:
3481:
3477:
3474:
3471:
3468:
3465:
3462:
3459:
3456:
3453:
3450:
3447:
3444:
3441:
3438:
3433:
3428:
3425:
3421:
3417:
3414:
3411:
3408:
3405:
3402:
3399:
3395:
3391:
3388:
3385:
3382:
3379:
3376:
3370:
3367:
3363:
3347:
3336:
3333:
3330:
3327:
3324:
3321:
3318:
3315:
3312:
3309:
3303:
3300:
3296:
3290:
3287:
3284:
3281:
3278:
3275:
3272:
3269:
3266:
3263:
3260:
3257:
3254:
3251:
3245:
3242:
3238:
3216:
3213:
3207:
3204:
3200:
3196:
3193:
3190:
3187:
3184:
3181:
3176:
3171:
3168:
3164:
3160:
3157:
3154:
3151:
3148:
3128:
3124:
3120:
3117:
3111:
3108:
3104:
3100:
3097:
3094:
3091:
3088:
3085:
3080:
3075:
3072:
3068:
3064:
3061:
3058:
3055:
3052:
3049:
3046:
3042:
3038:
3035:
3032:
3029:
3026:
3023:
3017:
3014:
3010:
2985:
2982:
2970:
2966:
2962:
2959:
2955:
2952:
2949:
2946:
2943:
2940:
2934:
2931:
2925:
2921:
2918:
2915:
2912:
2909:
2906:
2903:
2900:
2897:
2894:
2891:
2888:
2885:
2882:
2879:
2876:
2873:
2870:
2844:
2841:
2838:
2834:
2830:
2827:
2824:
2821:
2817:
2811:
2808:
2805:
2800:
2797:
2794:
2790:
2786:
2783:
2780:
2777:
2774:
2771:
2768:
2765:
2762:
2759:
2757:
2755:
2752:
2749:
2745:
2741:
2738:
2735:
2732:
2729:
2726:
2723:
2720:
2716:
2710:
2705:
2702:
2699:
2695:
2691:
2688:
2685:
2682:
2679:
2676:
2673:
2670:
2667:
2664:
2662:
2660:
2657:
2653:
2650:
2647:
2644:
2638:
2635:
2631:
2624:
2619:
2616:
2613:
2609:
2605:
2602:
2599:
2596:
2593:
2590:
2587:
2584:
2581:
2578:
2575:
2574:
2552:
2549:
2546:
2543:
2540:
2537:
2534:
2531:
2511:
2508:
2505:
2502:
2499:
2496:
2493:
2490:
2487:
2484:
2481:
2461:
2458:
2455:
2452:
2449:
2446:
2443:
2440:
2437:
2415:
2412:
2409:
2405:
2402:
2399:
2396:
2393:
2390:
2385:
2380:
2377:
2374:
2370:
2366:
2363:
2360:
2357:
2354:
2351:
2347:
2344:
2341:
2338:
2332:
2329:
2325:
2318:
2313:
2310:
2307:
2303:
2299:
2296:
2293:
2290:
2287:
2284:
2281:
2278:
2275:
2272:
2250:
2247:
2244:
2241:
2238:
2235:
2215:
2212:
2209:
2206:
2203:
2200:
2197:
2194:
2191:
2188:
2185:
2182:
2179:
2176:
2173:
2170:
2167:
2164:
2161:
2158:
2155:
2152:
2140:
2137:
2124:
2121:
2118:
2115:
2112:
2109:
2106:
2103:
2100:
2080:
2077:
2074:
2071:
2068:
2065:
2062:
2059:
2056:
2053:
2050:
2047:
2044:
2041:
2038:
2035:
2032:
2029:
2026:
2023:
2017:
2014:
2010:
1985:
1982:
1979:
1976:
1956:
1953:
1950:
1947:
1944:
1922:
1917:
1912:
1909:
1906:
1903:
1900:
1897:
1894:
1891:
1888:
1866:
1863:
1860:
1857:
1854:
1851:
1848:
1845:
1842:
1839:
1836:
1833:
1830:
1827:
1824:
1821:
1818:
1815:
1812:
1806:
1803:
1799:
1785:
1782:
1781:
1780:
1748:
1745:
1742:
1739:
1736:
1733:
1730:
1727:
1724:
1721:
1718:
1715:
1712:
1709:
1706:
1703:
1700:
1697:
1691:
1688:
1684:
1671:
1657:
1654:
1651:
1646:
1641:
1636:
1632:
1628:
1625:
1622:
1617:
1613:
1592:
1587:
1583:
1579:
1576:
1573:
1570:
1565:
1561:
1557:
1554:
1551:
1548:
1543:
1539:
1535:
1532:
1529:
1526:
1521:
1517:
1513:
1510:
1507:
1504:
1501:
1496:
1492:
1488:
1485:
1482:
1479:
1476:
1470:
1467:
1463:
1450:
1439:
1436:
1431:
1427:
1423:
1420:
1417:
1412:
1408:
1387:
1384:
1379:
1375:
1371:
1368:
1365:
1362:
1359:
1356:
1353:
1350:
1345:
1341:
1337:
1334:
1331:
1328:
1325:
1322:
1319:
1316:
1313:
1310:
1307:
1304:
1301:
1298:
1295:
1292:
1289:
1286:
1280:
1277:
1273:
1260:
1248:
1244:
1241:
1238:
1235:
1232:
1228:
1225:
1222:
1219:
1216:
1213:
1208:
1203:
1200:
1196:
1192:
1189:
1186:
1183:
1180:
1177:
1174:
1170:
1166:
1163:
1160:
1157:
1154:
1151:
1145:
1142:
1138:
1123:
1120:
1108:
1103:
1098:
1076:
1071:
1066:
1061:
1057:
1053:
1048:
1044:
1040:
1035:
1030:
1025:
1021:
1000:
980:
977:
974:
971:
968:
965:
962:
959:
956:
953:
950:
945:
941:
920:
917:
912:
908:
904:
901:
898:
895:
892:
889:
886:
883:
880:
877:
874:
871:
868:
865:
859:
856:
852:
828:
823:
818:
815:
812:
809:
806:
795:
794:
787:
784:control system
780:
772:
717:
716:
714:
713:
706:
699:
691:
688:
687:
683:
682:
677:
672:
667:
665:Ernst Lindelöf
662:
657:
652:
647:
642:
637:
635:Joseph Fourier
632:
627:
622:
620:Leonhard Euler
617:
612:
607:
601:
600:
597:
596:
593:
592:
588:
587:
584:
583:
578:
573:
566:
565:
560:
555:
550:
545:
540:
539:
538:
528:
523:
522:
521:
514:Finite element
511:
507:Crank–Nicolson
498:
493:
487:
482:
478:
477:
474:
473:
470:
469:
465:
464:
459:
454:
446:
441:
428:
426:Phase portrait
423:
418:
417:
416:
414:Cauchy problem
411:
406:
401:
391:
385:
384:
382:General topics
381:
380:
377:
376:
373:
372:
367:
362:
357:
351:
348:
347:
344:
343:
339:
338:
334:
333:
328:
327:
326:
315:
314:
313:
304:
301:
300:
297:
296:
291:
290:
289:
288:
281:
280:
275:
269:
266:
265:
261:
260:
258:
257:
255:Nonhomogeneous
248:
243:
240:
234:
233:
232:
224:
223:
219:
218:
216:
215:
210:
205:
200:
195:
190:
185:
179:
174:
171:
170:
167:
166:
165:Classification
162:
161:
152:
151:
150:
149:
144:
136:
135:
129:
128:
127:
126:
121:
116:
108:
107:
101:
100:
99:
98:
93:
87:
82:
77:
69:
68:
66:
65:
60:
54:
49:
46:
45:
42:
41:
37:
36:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
5526:
5515:
5512:
5510:
5507:
5506:
5504:
5493:
5492:
5487:
5482:
5481:
5477:
5471:
5465:
5461:
5457:
5453:
5452:
5446:
5442:
5436:
5432:
5428:
5424:
5423:
5417:
5413:
5407:
5403:
5402:
5396:
5392:
5386:
5379:
5378:
5372:
5368:
5362:
5358:
5357:
5351:
5350:
5346:
5337:
5333:
5329:
5325:
5321:
5317:
5312:
5307:
5303:
5299:
5295:
5288:
5285:
5280:
5274:
5270:
5269:
5261:
5258:
5253:
5247:
5243:
5239:
5235:
5234:
5226:
5223:
5218:
5214:
5210:
5206:
5202:
5198:
5194:
5187:
5184:
5179:
5175:
5171:
5167:
5162:
5157:
5153:
5149:
5145:
5138:
5135:
5130:
5126:
5122:
5118:
5114:
5110:
5106:
5099:
5096:
5091:
5085:
5082:
5076:
5071:
5064:
5061:
5055:
5050:
5043:
5040:
5035:
5029:
5025:
5021:
5017:
5016:
5008:
5005:
5000:
4994:
4990:
4986:
4982:
4981:
4973:
4970:
4965:
4959:
4955:
4951:
4947:
4946:
4938:
4935:
4930:
4926:
4922:
4918:
4914:
4910:
4906:
4902:
4898:
4891:
4888:
4883:
4879:
4872:
4869:
4864:
4860:
4856:
4850:
4846:
4842:
4838:
4837:
4832:
4825:
4822:
4817:
4813:
4809:
4805:
4798:
4795:
4788:
4784:
4781:
4779:
4776:
4775:
4771:
4766:
4764:
4761:
4759:
4756:
4754:
4750:
4749:
4745:
4743:
4741:
4722:
4716:
4689:
4686:
4678:
4669:
4666:
4659:
4656:
4648:
4639:
4633:
4630:
4627:
4621:
4615:
4610:
4604:
4598:
4567:
4554:
4548:
4545:
4542:
4539:
4533:
4530:
4527:
4521:
4518:
4515:
4512:
4506:
4503:
4500:
4494:
4488:
4482:
4479:
4475:
4461:
4459:
4446:
4441:
4438:
4432:
4429:
4421:
4417:
4412:
4404:
4398:
4395:
4389:
4383:
4375:
4371:
4366:
4362:
4346:
4340:
4337:
4329:
4325:
4321:
4318:
4310:
4294:
4291:
4286:
4283:
4279:
4275:
4272:
4269:
4249:
4243:
4240:
4237:
4231:
4228:
4225:
4219:
4213:
4207:
4204:
4200:
4190:
4185:
4182:
4178:
4174:
4169:
4156:
4153:
4145:
4140:
4136:
4132:
4128:
4122:
4118:
4114:
4111:
4108:
4103:
4098:
4094:
4090:
4086:
4080:
4076:
4072:
4067:
4063:
4059:
4056:
4053:
4050:
4036:
4015:
4011:
4007:
4004:
3998:
3993:
3989:
3985:
3982:
3979:
3971:
3967:
3963:
3960:
3954:
3949:
3945:
3941:
3935:
3929:
3924:
3920:
3916:
3910:
3904:
3898:
3895:
3891:
3881:
3877:
3869:
3853:
3850:
3847:
3840:
3837:
3834:
3831:
3825:
3822:
3816:
3813:
3810:
3804:
3799:
3791:
3787:
3783:
3780:
3776:
3773:
3770:
3763:
3760:
3757:
3754:
3748:
3745:
3739:
3736:
3733:
3727:
3722:
3714:
3710:
3706:
3703:
3683:
3680:
3677:
3674:
3671:
3665:
3659:
3656:
3653:
3647:
3641:
3635:
3632:
3628:
3622:
3619:
3616:
3613:
3610:
3604:
3598:
3595:
3592:
3586:
3580:
3574:
3571:
3567:
3561:
3555:
3552:
3549:
3546:
3543:
3537:
3534:
3528:
3522:
3516:
3513:
3509:
3486:
3482:
3479:
3472:
3469:
3466:
3463:
3457:
3454:
3448:
3445:
3442:
3436:
3431:
3423:
3419:
3415:
3409:
3403:
3400:
3397:
3393:
3389:
3386:
3380:
3374:
3368:
3365:
3361:
3351:
3348:
3334:
3331:
3328:
3325:
3322:
3319:
3313:
3307:
3301:
3298:
3294:
3288:
3282:
3279:
3276:
3273:
3270:
3264:
3261:
3255:
3249:
3243:
3240:
3236:
3214:
3211:
3205:
3202:
3198:
3191:
3188:
3185:
3179:
3174:
3166:
3162:
3158:
3152:
3146:
3126:
3122:
3118:
3115:
3109:
3106:
3102:
3095:
3092:
3089:
3083:
3078:
3070:
3066:
3062:
3056:
3050:
3047:
3044:
3040:
3036:
3033:
3027:
3021:
3015:
3012:
3008:
2998:
2995:
2994:
2993:
2991:
2983:
2981:
2968:
2964:
2960:
2957:
2950:
2947:
2944:
2938:
2932:
2929:
2923:
2916:
2913:
2910:
2904:
2901:
2895:
2892:
2889:
2886:
2880:
2874:
2868:
2859:
2842:
2839:
2836:
2832:
2828:
2825:
2822:
2819:
2815:
2809:
2806:
2803:
2798:
2795:
2792:
2788:
2784:
2781:
2775:
2772:
2769:
2766:
2760:
2758:
2750:
2747:
2743:
2739:
2736:
2730:
2727:
2724:
2718:
2714:
2708:
2703:
2700:
2697:
2693:
2689:
2686:
2680:
2677:
2674:
2671:
2665:
2663:
2658:
2655:
2648:
2642:
2636:
2633:
2629:
2622:
2617:
2614:
2611:
2607:
2603:
2597:
2591:
2588:
2582:
2576:
2564:
2547:
2544:
2541:
2538:
2532:
2529:
2509:
2506:
2500:
2494:
2491:
2485:
2479:
2459:
2456:
2453:
2450:
2447:
2441:
2435:
2426:
2413:
2410:
2407:
2400:
2397:
2394:
2388:
2383:
2378:
2375:
2372:
2368:
2364:
2361:
2358:
2355:
2352:
2349:
2342:
2336:
2330:
2327:
2323:
2316:
2311:
2308:
2305:
2301:
2297:
2291:
2285:
2282:
2276:
2270:
2262:
2248:
2245:
2239:
2233:
2210:
2207:
2204:
2198:
2195:
2192:
2183:
2180:
2177:
2171:
2168:
2162:
2156:
2150:
2138:
2136:
2119:
2113:
2110:
2104:
2098:
2078:
2069:
2066:
2063:
2057:
2054:
2048:
2042:
2036:
2033:
2027:
2021:
2015:
2012:
2008:
1999:
1980:
1974:
1951:
1948:
1945:
1920:
1904:
1901:
1898:
1895:
1889:
1886:
1877:
1858:
1855:
1852:
1846:
1843:
1837:
1831:
1825:
1822:
1816:
1810:
1804:
1801:
1797:
1783:
1778:
1774:
1770:
1766:
1762:
1746:
1740:
1737:
1731:
1728:
1725:
1719:
1713:
1710:
1707:
1701:
1695:
1689:
1686:
1682:
1672:
1655:
1652:
1649:
1639:
1634:
1630:
1626:
1623:
1620:
1615:
1611:
1585:
1581:
1577:
1574:
1568:
1563:
1559:
1555:
1552:
1549:
1541:
1537:
1533:
1530:
1524:
1519:
1515:
1511:
1505:
1499:
1494:
1490:
1486:
1480:
1474:
1468:
1465:
1461:
1451:
1437:
1434:
1429:
1425:
1421:
1418:
1415:
1410:
1406:
1377:
1373:
1369:
1366:
1360:
1357:
1354:
1351:
1343:
1339:
1335:
1332:
1326:
1323:
1317:
1311:
1308:
1305:
1299:
1296:
1290:
1284:
1278:
1275:
1271:
1261:
1246:
1239:
1233:
1230:
1223:
1220:
1217:
1211:
1206:
1198:
1194:
1190:
1184:
1178:
1175:
1172:
1168:
1164:
1161:
1155:
1149:
1143:
1140:
1136:
1126:
1125:
1121:
1119:
1106:
1101:
1069:
1059:
1046:
1042:
1038:
1033:
1023:
998:
975:
972:
969:
966:
960:
954:
948:
943:
939:
918:
910:
906:
902:
896:
890:
887:
884:
878:
875:
869:
863:
857:
854:
850:
826:
816:
810:
804:
792:
788:
785:
781:
777:
773:
769:
765:
761:
756:
755:
754:
752:
748:
744:
740:
736:
732:
728:
724:
712:
707:
705:
700:
698:
693:
692:
690:
689:
681:
678:
676:
673:
671:
668:
666:
663:
661:
658:
656:
653:
651:
648:
646:
643:
641:
638:
636:
633:
631:
628:
626:
623:
621:
618:
616:
613:
611:
608:
606:
603:
602:
595:
594:
589:
582:
579:
577:
574:
572:
569:
568:
564:
561:
559:
556:
554:
551:
549:
546:
544:
541:
537:
534:
533:
532:
529:
527:
526:Finite volume
524:
520:
517:
516:
515:
512:
508:
502:
499:
497:
494:
492:
488:
486:
483:
480:
479:
472:
471:
463:
460:
458:
455:
451:
447:
445:
442:
440:
436:
432:
429:
427:
424:
422:
419:
415:
412:
410:
407:
405:
402:
400:
397:
396:
395:
392:
390:
387:
386:
379:
378:
371:
368:
366:
363:
361:
358:
356:
353:
352:
346:
345:
340:
332:
329:
325:
322:
321:
320:
317:
316:
312:
306:
305:
299:
298:
287:
284:
283:
279:
276:
274:
271:
270:
268:
267:
262:
256:
252:
249:
247:
244:
241:
239:
236:
235:
231:
228:
227:
226:
225:
220:
214:
211:
209:
206:
204:
201:
199:
196:
194:
191:
189:
186:
184:
181:
180:
178:
177:
169:
168:
163:
158:
148:
145:
143:
140:
139:
138:
137:
134:
130:
125:
122:
120:
117:
115:
112:
111:
110:
109:
106:
102:
97:
94:
92:
88:
86:
83:
81:
78:
76:
73:
72:
71:
70:
64:
61:
59:
56:
55:
52:
44:
43:
38:
35:
31:
19:
5491:Scholarpedia
5489:
5450:
5421:
5400:
5376:
5355:
5301:
5297:
5287:
5267:
5260:
5232:
5225:
5200:
5196:
5186:
5151:
5147:
5137:
5112:
5108:
5098:
5084:
5063:
5042:
5014:
5007:
4979:
4972:
4944:
4937:
4904:
4900:
4890:
4881:
4871:
4835:
4824:
4807:
4803:
4797:
4758:Epidemiology
4751:Dynamics of
4746:Applications
4568:
4465:
4369:
4364:
4360:
4308:
4186:
4172:
4170:
3873:
3352:An equation
3349:
2996:
2987:
2860:
2565:
2427:
2263:
2142:
1967:is given by
1878:
1787:
1784:Solving DDEs
1772:
1768:
1764:
1760:
796:
775:
771:engineering.
738:
730:
726:
720:
670:Émile Picard
655:Martin Kutta
645:George Green
605:Isaac Newton
437: /
433: /
330:
253: /
119:Chaos theory
3874:Similar to
1777:pantographs
723:mathematics
563:Runge–Kutta
308:Difference
251:Homogeneous
63:Engineering
5503:Categories
5311:2001.07392
5075:1702.06487
5054:2305.00341
4804:Automatica
4789:References
4171:The roots
3139:Introduce
1779:on trains.
680:John Crank
481:Inspection
435:Asymptotic
319:Stochastic
238:Autonomous
213:Non-linear
203:Fractional
5336:210838940
5328:1573-269X
5217:1468-1218
5170:0002-9262
5129:0168-9274
4929:110981464
4687:≥
4546:−
4528:−
4439:⋅
4430:−
4338:−
4319:λ
4287:λ
4284:−
4276:−
4273:λ
4270:−
4241:−
4229:−
4146:λ
4137:τ
4133:−
4112:⋯
4104:λ
4095:τ
4091:−
4054:λ
4051:−
4012:τ
4008:−
3983:⋯
3968:τ
3964:−
3851:τ
3841:β
3835:τ
3832:α
3826:
3817:τ
3795:∞
3792:−
3788:∫
3774:τ
3764:β
3758:τ
3755:α
3749:
3740:τ
3718:∞
3715:−
3711:∫
3678:α
3675:−
3666:β
3660:
3617:α
3605:β
3599:
3483:τ
3473:β
3467:τ
3464:α
3458:
3449:τ
3427:∞
3424:−
3420:∫
3350:Example 2
3329:λ
3326:−
3215:τ
3206:τ
3203:λ
3192:τ
3170:∞
3167:−
3163:∫
3119:τ
3110:τ
3107:λ
3096:τ
3074:∞
3071:−
3067:∫
2997:Example 1
2951:τ
2948:−
2917:τ
2914:−
2890:τ
2810:τ
2807:−
2789:∫
2770:τ
2731:τ
2728:−
2704:τ
2694:∫
2675:τ
2618:τ
2608:∫
2598:τ
2548:τ
2539:τ
2533:∈
2495:ϕ
2401:τ
2398:−
2389:ϕ
2369:∫
2302:∫
2234:ϕ
2211:τ
2208:−
2184:τ
2181:−
2114:ϕ
2099:ψ
2070:τ
2067:−
2058:ϕ
2043:ψ
2022:ψ
1975:ψ
1952:τ
1911:→
1899:τ
1896:−
1890::
1887:ϕ
1859:τ
1856:−
1738:λ
1653:×
1640:∈
1624:…
1582:τ
1578:−
1553:⋯
1538:τ
1534:−
1435:≥
1426:τ
1419:⋯
1407:τ
1374:τ
1370:−
1355:…
1340:τ
1336:−
1240:τ
1234:μ
1224:τ
1202:∞
1199:−
1195:∫
1039:×
1024:×
973:≤
970:τ
961:τ
817:∈
776:classical
760:actuators
421:Wronskian
399:Dirichlet
142:Economics
85:Chemistry
75:Astronomy
4772:See also
4753:diabetes
4177:spectrum
4037:given by
2143:Suppose
1122:Examples
531:Galerkin
431:Lyapunov
342:Solution
286:Notation
278:Operator
264:Features
183:Ordinary
5178:9508108
4909:Bibcode
4863:1200900
4368:is the
4033:is the
2139:Example
793:(PDEs).
764:sensors
745:, i.e.
404:Neumann
188:Partial
96:Geology
91:Biology
80:Physics
5466:
5437:
5408:
5387:
5363:
5334:
5326:
5275:
5248:
5215:
5176:
5168:
5127:
5030:
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4960:
4927:
4861:
4851:
4376:, so:
4359:where
3696:where
2861:i.e.,
2428:i.e.,
1759:where
1603:where
931:where
766:, and
591:People
503:
450:Series
208:Linear
47:Fields
5381:(PDF)
5332:S2CID
5306:arXiv
5070:arXiv
5049:arXiv
4925:S2CID
4859:S2CID
4709:with
2091:with
491:Euler
409:Robin
331:Delay
273:Order
246:Exact
172:Types
40:Scope
5464:ISBN
5435:ISBN
5406:ISBN
5385:ISBN
5361:ISBN
5324:ISSN
5273:ISBN
5246:ISBN
5213:ISSN
5174:PMID
5166:ISSN
5125:ISSN
5028:ISBN
4993:ISBN
4958:ISBN
4849:ISBN
4738:the
4657:<
3876:ODEs
2226:and
1767:and
1422:>
1416:>
1398:for
731:DDEs
598:List
5456:doi
5427:doi
5316:doi
5302:102
5238:doi
5205:doi
5156:doi
5152:147
5117:doi
5020:doi
4985:doi
4950:doi
4917:doi
4905:322
4841:doi
4812:doi
4045:det
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2992:.
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841:is
721:In
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