351:
decreases. The solution to a singularly perturbed problem cannot be approximated in this way: As seen in the examples below, a singular perturbation generally occurs when a problem's small parameter multiplies its highest operator. Thus naively taking the parameter to be zero changes the very nature
662:
An electrically driven robot manipulator can have slower mechanical dynamics and faster electrical dynamics, thus exhibiting two time scales. In such cases, we can divide the system into two subsystems, one corresponding to faster dynamics and other corresponding to slower dynamics, and then design
2717:
582:
937:
1949:
800:
283:
problems, for which a uniform approximation of this form can be obtained. Singularly perturbed problems are generally characterized by dynamics operating on multiple scales. Several classes of singular perturbations are outlined below.
416:
Differential equations that contain a small parameter that premultiplies the highest order term typically exhibit boundary layers, so that the solution evolves in two different scales. For example, consider the boundary value problem
2486:
407:
Each of the examples described below shows how a naive perturbation analysis, which assumes that the problem is regular instead of singular, will fail. Some show how the problem may be solved by more sophisticated singular methods.
123:
1837:
355:
Singular perturbation theory is a rich and ongoing area of exploration for mathematicians, physicists, and other researchers. The methods used to tackle problems in this field are many. The more basic of these include the
2195:
2625:
423:
277:
1467:
1204:
331:
by zero everywhere in the problem statement. This corresponds to taking only the first term of the expansion, yielding an approximation that converges, perhaps slowly, to the true solution as
1026:
2386:
1652:
663:
controllers for each one of them separately. Through a singular perturbation technique, we can make these two subsystems independent of each other, thereby simplifying the control problem.
652:
27:
problem is a problem containing a small parameter that cannot be approximated by setting the parameter value to zero. More precisely, the solution cannot be uniformly approximated by an
208:
1271:
2548:
2028:
1061:
1533:
2326:
to zero by becoming equally dominant to another term, is called significant degeneration; this yields the correct rescaling to make the remaining root visible. This choice yields
806:
1868:
1678:
1325:
152:
611:
637:
2324:
2088:
1972:
1860:
1348:
349:
329:
228:
172:
2224:
352:
of the problem. In the case of differential equations, boundary conditions cannot be satisfied; in algebraic equations, the possible number of solutions is decreased.
2617:
2584:
1747:
1718:
672:
2277:
2304:
2251:
1115:
1088:
2048:
2619:
are the two roots that we've found above that collapse to zero in the limit of an infinite rescaling. Calculating the first few terms of the series then yields
2108:
2068:
1992:
1954:
To find the other root, singular perturbation analysis must be used. We must then deal with the fact that the equation degenerates into a quadratic when we let
1576:
1556:
2397:
36:
1974:
tend to zero, in that limit one of the roots escapes to infinity. To prevent this root from becoming invisible to the perturbative analysis, we must rescale
1755:
373:
2116:
311:. Most often in applications, an acceptable approximation to a regularly perturbed problem is found by simply replacing the small parameter
2712:{\displaystyle x(\varepsilon )={\frac {y(\varepsilon )}{\varepsilon }}={\frac {1}{\varepsilon }}-\varepsilon -2\varepsilon ^{3}+\cdots .}
577:{\displaystyle {\begin{matrix}\varepsilon u^{\prime \prime }(x)+u^{\prime }(x)=-e^{-x},\ \ 0<x<1\\u(0)=0,\ \ u(1)=1.\end{matrix}}}
1121:
states that, with the correct conditions on the system, it will initially and very quickly approximate the solution to the equations
1994:
to keep track with this escaping root so that in terms of the rescaled variables, it doesn't escape. We define a rescaled variable
2861:
Tikhonov, A. N. (1952), "Systems of differential equations containing a small parameter multiplying the derivative" (in
Russian),
644:
357:
233:
2850:
2830:
2810:
2306:
term while they both dominate the remaining term. This point where the highest order term will no longer vanish in the limit
1386:
1127:
303:
A perturbed problem whose solution can be approximated on the whole problem domain, whether space or time, by a single
943:
2883:
2332:
1593:
365:
2914:
177:
1376:
marked by areas where a reagent exists, and areas where it does not, with sharp transitions between them. In
1369:
1210:
2763:"A rational spectral collocation method for solving a class of parameterized singular perturbation problems"
2497:
1997:
1034:
932:{\displaystyle \varepsilon {\dot {x}}_{2}=f_{2}(x_{1},x_{2})+\varepsilon g_{2}(x_{1},x_{2},\varepsilon ),\,}
2924:
1944:{\displaystyle x(\varepsilon )=\pm 1+{\frac {1}{2}}\varepsilon \pm {\frac {5}{8}}\varepsilon ^{2}+\cdots .}
1473:
1118:
613:
is the solid curve shown below. Note that the solution changes rapidly near the origin. If we naively set
2919:
1657:
1277:
131:
2090:
to zero, but such that it doesn't collapse to zero where the other two roots will end up. In terms of
639:, we would get the solution labelled "outer" below which does not model the boundary layer, for which
590:
369:
1362:, the properties of a slightly viscous fluid are dramatically different outside and inside a narrow
616:
2876:
Methods and
Applications of Singular Perturbations: Boundary Layers and Multiple Timescale Dynamics
2309:
2073:
1957:
1845:
1333:
643:
is close to zero. For more details that show how to obtain the uniformly valid approximation, see
334:
314:
213:
157:
1350:
decreases toward zero, the system will approach the solution more closely in that same interval.
795:{\displaystyle {\dot {x}}_{1}=f_{1}(x_{1},x_{2})+\varepsilon g_{1}(x_{1},x_{2},\varepsilon ),\,}
2203:
2050:
will be chosen such that we rescale just fast enough so that the root is at a finite value of
2589:
2556:
1723:
1690:
2256:
2895:
Owen, M. R. and Lewis, M. A. "How
Predation can Slow, Stop, or Reverse a Prey Invasion",
2774:
2282:
2229:
1093:
1066:
304:
288:
28:
2033:
376:. The numerical methods for solving singular perturbation problems are also very popular.
8:
2481:{\displaystyle y(\varepsilon )=y_{0}+\varepsilon ^{2}y_{1}+\varepsilon ^{4}y_{2}+\cdots }
308:
280:
118:{\displaystyle \varphi (x)\approx \sum _{n=0}^{N}\delta _{n}(\varepsilon )\psi _{n}(x)\,}
2778:
2093:
2053:
1977:
1685:
1587:
1561:
1541:
2744:
Mathematics
Research Center, University of Wisconsin-Madison, Technical Summary Report
2879:
2846:
2826:
2806:
1832:{\displaystyle x(\varepsilon )=x_{0}+\varepsilon x_{1}+\varepsilon ^{2}x_{2}+\cdots }
1373:
361:
292:
2782:
1359:
2736:
1681:
1363:
388:
2787:
2762:
2908:
2737:"ON BOUNDARY LAYER PROBLEMS IN THE THEORY OF ORDINARY DIFFERENTIAL EQUATIONS"
392:
379:
For books on singular perturbation in ODE and PDE's, see for example Holmes,
411:
2190:{\displaystyle y^{3}-\varepsilon ^{\nu -1}y^{2}+\varepsilon ^{3\nu -1}=0.}
651:
20:
666:
Consider a class of system described by the following set of equations:
1372:
in which one reagent diffuses much more slowly than another can form
1377:
402:
287:
The term "singular perturbation" was coined in the 1940s by
1578:
is the predator, have been shown to exhibit such patterns.
2843:
397:
272:{\displaystyle \delta _{n}(\varepsilon )=\varepsilon ^{n}}
412:
Vanishing coefficients in ordinary differential equations
1462:{\displaystyle u_{t}=\varepsilon u_{xx}+uf(u)-vg(u),\,}
442:
428:
2628:
2592:
2559:
2500:
2400:
2335:
2312:
2285:
2259:
2232:
2206:
2119:
2096:
2076:
2056:
2036:
2000:
1980:
1960:
1871:
1848:
1758:
1726:
1693:
1660:
1596:
1564:
1544:
1476:
1389:
1336:
1280:
1213:
1130:
1096:
1069:
1063:. The second equation indicates that the dynamics of
1037:
946:
809:
675:
619:
593:
426:
337:
317:
236:
216:
180:
160:
134:
39:
1199:{\displaystyle {\dot {x}}_{1}=f_{1}(x_{1},x_{2}),\,}
1366:. Thus the fluid exhibits multiple spatial scales.
2711:
2611:
2578:
2542:
2480:
2380:
2318:
2298:
2271:
2245:
2218:
2189:
2102:
2082:
2062:
2042:
2022:
1986:
1966:
1943:
1854:
1831:
1741:
1712:
1672:
1646:
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1550:
1527:
1461:
1342:
1319:
1265:
1198:
1109:
1082:
1055:
1020:
931:
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631:
605:
576:
343:
323:
271:
222:
202:
166:
146:
117:
2761:Wang, Yingwei; Chen, Suqin; Wu, Xionghua (2010).
2253:is dominated by the lower degree terms, while at
2906:
2767:Journal of Computational and Applied Mathematics
1021:{\displaystyle x_{1}(0)=a_{1},x_{2}(0)=a_{2},\,}
2381:{\displaystyle y^{3}-y^{2}+\varepsilon ^{2}=0.}
1647:{\displaystyle p(x)=\varepsilon x^{3}-x^{2}+1}
1842:in the equation and equating equal powers of
1749:. Substituting a regular perturbation series
1862:only yields corrections to these two roots:
2760:
403:Examples of singular perturbative problems
174:is the small parameter of the problem and
2786:
1524:
1458:
1316:
1262:
1195:
1017:
928:
791:
203:{\displaystyle \delta _{n}(\varepsilon )}
114:
2868:
2855:
1266:{\displaystyle f_{2}(x_{1},x_{2})=0,\,}
645:method of matched asymptotic expansions
364:for spatial problems, and in time, the
358:method of matched asymptotic expansions
2907:
2841:Bender, Carl M. and Orszag, Steven A.
2553:We are then interested in the root at
2543:{\displaystyle y_{0}^{3}-y_{0}^{2}=0.}
2023:{\displaystyle y=x\varepsilon ^{\nu }}
1581:
1330:on some interval of time and that, as
1056:{\displaystyle 0<\varepsilon \ll 1}
298:
2795:
2734:
2391:Substituting the perturbation series
1528:{\displaystyle v_{t}=v_{xx}+vh(u),\,}
2825:. Cambridge University Press, 1991.
2815:
2803:Introduction to Perturbation Methods
2754:
1586:Consider the problem of finding all
1353:
381:Introduction to Perturbation Methods
657:
13:
464:
439:
14:
2936:
2835:
1673:{\displaystyle \varepsilon \to 0}
1320:{\displaystyle x_{1}(0)=a_{1},\,}
147:{\displaystyle \varepsilon \to 0}
2897:Bulletin of Mathematical Biology
650:
606:{\displaystyle \varepsilon =0.1}
1380:, predator-prey models such as
210:are a sequence of functions of
2889:
2728:
2656:
2650:
2638:
2632:
2410:
2404:
2279:it becomes as dominant as the
1881:
1875:
1768:
1762:
1664:
1606:
1600:
1518:
1512:
1452:
1446:
1434:
1428:
1297:
1291:
1250:
1224:
1189:
1163:
998:
992:
963:
957:
922:
890:
871:
845:
785:
753:
734:
708:
632:{\displaystyle \varepsilon =0}
561:
555:
534:
528:
475:
469:
453:
447:
253:
247:
197:
191:
138:
111:
105:
92:
86:
49:
43:
1:
2722:
230:of increasing order, such as
2319:{\displaystyle \varepsilon }
2083:{\displaystyle \varepsilon }
1967:{\displaystyle \varepsilon }
1855:{\displaystyle \varepsilon }
1343:{\displaystyle \varepsilon }
1090:is much faster than that of
344:{\displaystyle \varepsilon }
324:{\displaystyle \varepsilon }
223:{\displaystyle \varepsilon }
167:{\displaystyle \varepsilon }
7:
2735:Wasow, Wolfgang R. (1981),
10:
2941:
1370:Reaction–diffusion systems
2788:10.1016/j.cam.2009.11.011
2219:{\displaystyle \nu <1}
370:method of multiple scales
366:Poincaré–Lindstedt method
279:. This is in contrast to
2612:{\displaystyle y_{0}=0}
2579:{\displaystyle y_{0}=1}
1742:{\displaystyle x=\pm 1}
1713:{\displaystyle 1-x^{2}}
2915:Differential equations
2713:
2613:
2580:
2544:
2482:
2382:
2320:
2300:
2273:
2272:{\displaystyle \nu =1}
2247:
2220:
2191:
2104:
2084:
2064:
2044:
2024:
1988:
1968:
1945:
1856:
1833:
1743:
1714:
1674:
1648:
1572:
1552:
1529:
1463:
1344:
1321:
1267:
1200:
1111:
1084:
1057:
1022:
933:
796:
633:
607:
578:
345:
325:
273:
224:
204:
168:
148:
119:
75:
16:Concept in mathematics
2874:Verhulst, Ferdinand.
2714:
2614:
2586:; the double root at
2581:
2545:
2483:
2383:
2321:
2301:
2299:{\displaystyle y^{2}}
2274:
2248:
2246:{\displaystyle y^{3}}
2221:
2192:
2105:
2085:
2065:
2045:
2025:
1989:
1969:
1946:
1857:
1834:
1744:
1715:
1684:degenerates into the
1675:
1649:
1573:
1553:
1530:
1464:
1345:
1322:
1268:
1201:
1112:
1110:{\displaystyle x_{1}}
1085:
1083:{\displaystyle x_{2}}
1058:
1023:
934:
797:
634:
608:
579:
346:
326:
274:
225:
205:
169:
149:
120:
55:
25:singular perturbation
2865:31 (73), pp. 575–586
2823:Perturbation methods
2626:
2590:
2557:
2498:
2398:
2333:
2310:
2283:
2257:
2230:
2204:
2200:We can see that for
2117:
2094:
2074:
2054:
2043:{\displaystyle \nu }
2034:
1998:
1978:
1958:
1869:
1846:
1756:
1724:
1691:
1658:
1594:
1562:
1542:
1474:
1387:
1334:
1278:
1211:
1128:
1094:
1067:
1035:
944:
807:
673:
617:
591:
424:
385:Perturbation methods
335:
315:
309:regular perturbation
305:asymptotic expansion
289:Kurt Otto Friedrichs
281:regular perturbation
234:
214:
178:
158:
132:
37:
29:asymptotic expansion
2925:Perturbation theory
2899:(2001) 63, 655-684.
2845:. Springer, 1999.
2779:2010JCoAM.233.2652W
2533:
2515:
2030:where the exponent
1582:Algebraic equations
1117:. A theorem due to
299:Methods of analysis
2878:, Springer, 2005.
2805:. Springer, 1995.
2709:
2609:
2576:
2540:
2519:
2501:
2478:
2378:
2316:
2296:
2269:
2243:
2216:
2187:
2100:
2080:
2060:
2040:
2020:
1984:
1964:
1941:
1852:
1829:
1739:
1710:
1670:
1644:
1590:of the polynomial
1568:
1548:
1525:
1459:
1340:
1317:
1263:
1196:
1107:
1080:
1053:
1018:
929:
792:
629:
603:
587:Its solution when
574:
572:
374:periodic averaging
341:
321:
269:
220:
200:
164:
144:
115:
2920:Nonlinear control
2851:978-0-387-98931-0
2831:978-0-521-37897-0
2811:978-0-387-94203-2
2773:(10): 2652–2660.
2676:
2663:
2103:{\displaystyle y}
2063:{\displaystyle y}
1987:{\displaystyle x}
1920:
1904:
1571:{\displaystyle v}
1551:{\displaystyle u}
1354:Examples in space
1141:
823:
686:
551:
548:
505:
502:
362:WKB approximation
293:Wolfgang R. Wasow
2932:
2900:
2893:
2887:
2872:
2866:
2859:
2853:
2839:
2833:
2819:
2813:
2801:Holmes, Mark H.
2799:
2793:
2792:
2790:
2758:
2752:
2751:
2741:
2732:
2718:
2716:
2715:
2710:
2699:
2698:
2677:
2669:
2664:
2659:
2645:
2618:
2616:
2615:
2610:
2602:
2601:
2585:
2583:
2582:
2577:
2569:
2568:
2549:
2547:
2546:
2541:
2532:
2527:
2514:
2509:
2487:
2485:
2484:
2479:
2471:
2470:
2461:
2460:
2448:
2447:
2438:
2437:
2425:
2424:
2387:
2385:
2384:
2379:
2371:
2370:
2358:
2357:
2345:
2344:
2325:
2323:
2322:
2317:
2305:
2303:
2302:
2297:
2295:
2294:
2278:
2276:
2275:
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2252:
2250:
2249:
2244:
2242:
2241:
2225:
2223:
2222:
2217:
2196:
2194:
2193:
2188:
2180:
2179:
2158:
2157:
2148:
2147:
2129:
2128:
2109:
2107:
2106:
2101:
2089:
2087:
2086:
2081:
2070:in the limit of
2069:
2067:
2066:
2061:
2049:
2047:
2046:
2041:
2029:
2027:
2026:
2021:
2019:
2018:
1993:
1991:
1990:
1985:
1973:
1971:
1970:
1965:
1950:
1948:
1947:
1942:
1931:
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1921:
1913:
1905:
1897:
1861:
1859:
1858:
1853:
1838:
1836:
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1821:
1812:
1811:
1799:
1798:
1783:
1782:
1748:
1746:
1745:
1740:
1719:
1717:
1716:
1711:
1709:
1708:
1679:
1677:
1676:
1671:
1653:
1651:
1650:
1645:
1637:
1636:
1624:
1623:
1577:
1575:
1574:
1569:
1558:is the prey and
1557:
1555:
1554:
1549:
1534:
1532:
1531:
1526:
1502:
1501:
1486:
1485:
1468:
1466:
1465:
1460:
1418:
1417:
1399:
1398:
1374:spatial patterns
1349:
1347:
1346:
1341:
1326:
1324:
1323:
1318:
1312:
1311:
1290:
1289:
1272:
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1205:
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1202:
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1188:
1187:
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1174:
1162:
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1149:
1148:
1143:
1142:
1134:
1116:
1114:
1113:
1108:
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1105:
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1087:
1086:
1081:
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1078:
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1059:
1054:
1027:
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1024:
1019:
1013:
1012:
991:
990:
978:
977:
956:
955:
938:
936:
935:
930:
915:
914:
902:
901:
889:
888:
870:
869:
857:
856:
844:
843:
831:
830:
825:
824:
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801:
799:
798:
793:
778:
777:
765:
764:
752:
751:
733:
732:
720:
719:
707:
706:
694:
693:
688:
687:
679:
658:Examples in time
654:
638:
636:
635:
630:
612:
610:
609:
604:
583:
581:
580:
575:
573:
549:
546:
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468:
467:
446:
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350:
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103:
85:
84:
74:
69:
2940:
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2905:
2904:
2903:
2894:
2890:
2873:
2869:
2860:
2856:
2840:
2836:
2820:
2816:
2800:
2796:
2759:
2755:
2739:
2733:
2729:
2725:
2694:
2690:
2668:
2646:
2644:
2627:
2624:
2623:
2597:
2593:
2591:
2588:
2587:
2564:
2560:
2558:
2555:
2554:
2528:
2523:
2510:
2505:
2499:
2496:
2495:
2466:
2462:
2456:
2452:
2443:
2439:
2433:
2429:
2420:
2416:
2399:
2396:
2395:
2366:
2362:
2353:
2349:
2340:
2336:
2334:
2331:
2330:
2311:
2308:
2307:
2290:
2286:
2284:
2281:
2280:
2258:
2255:
2254:
2237:
2233:
2231:
2228:
2227:
2205:
2202:
2201:
2166:
2162:
2153:
2149:
2137:
2133:
2124:
2120:
2118:
2115:
2114:
2095:
2092:
2091:
2075:
2072:
2071:
2055:
2052:
2051:
2035:
2032:
2031:
2014:
2010:
1999:
1996:
1995:
1979:
1976:
1975:
1959:
1956:
1955:
1926:
1922:
1912:
1896:
1870:
1867:
1866:
1847:
1844:
1843:
1817:
1813:
1807:
1803:
1794:
1790:
1778:
1774:
1757:
1754:
1753:
1725:
1722:
1721:
1704:
1700:
1692:
1689:
1688:
1659:
1656:
1655:
1654:. In the limit
1632:
1628:
1619:
1615:
1595:
1592:
1591:
1584:
1563:
1560:
1559:
1543:
1540:
1539:
1494:
1490:
1481:
1477:
1475:
1472:
1471:
1410:
1406:
1394:
1390:
1388:
1385:
1384:
1360:fluid mechanics
1356:
1335:
1332:
1331:
1307:
1303:
1285:
1281:
1279:
1276:
1275:
1244:
1240:
1231:
1227:
1218:
1214:
1212:
1209:
1208:
1183:
1179:
1170:
1166:
1157:
1153:
1144:
1133:
1132:
1131:
1129:
1126:
1125:
1101:
1097:
1095:
1092:
1091:
1074:
1070:
1068:
1065:
1064:
1036:
1033:
1032:
1008:
1004:
986:
982:
973:
969:
951:
947:
945:
942:
941:
910:
906:
897:
893:
884:
880:
865:
861:
852:
848:
839:
835:
826:
815:
814:
813:
808:
805:
804:
773:
769:
760:
756:
747:
743:
728:
724:
715:
711:
702:
698:
689:
678:
677:
676:
674:
671:
670:
660:
655:
618:
615:
614:
592:
589:
588:
571:
570:
522:
521:
488:
484:
463:
459:
438:
434:
427:
425:
422:
421:
414:
405:
336:
333:
332:
316:
313:
312:
301:
263:
259:
241:
237:
235:
232:
231:
215:
212:
211:
185:
181:
179:
176:
175:
159:
156:
155:
133:
130:
129:
99:
95:
80:
76:
70:
59:
38:
35:
34:
17:
12:
11:
5:
2938:
2928:
2927:
2922:
2917:
2902:
2901:
2888:
2867:
2854:
2834:
2814:
2794:
2753:
2726:
2724:
2721:
2720:
2719:
2708:
2705:
2702:
2697:
2693:
2689:
2686:
2683:
2680:
2675:
2672:
2667:
2662:
2658:
2655:
2652:
2649:
2643:
2640:
2637:
2634:
2631:
2608:
2605:
2600:
2596:
2575:
2572:
2567:
2563:
2551:
2550:
2539:
2536:
2531:
2526:
2522:
2518:
2513:
2508:
2504:
2489:
2488:
2477:
2474:
2469:
2465:
2459:
2455:
2451:
2446:
2442:
2436:
2432:
2428:
2423:
2419:
2415:
2412:
2409:
2406:
2403:
2389:
2388:
2377:
2374:
2369:
2365:
2361:
2356:
2352:
2348:
2343:
2339:
2315:
2293:
2289:
2268:
2265:
2262:
2240:
2236:
2215:
2212:
2209:
2198:
2197:
2186:
2183:
2178:
2175:
2172:
2169:
2165:
2161:
2156:
2152:
2146:
2143:
2140:
2136:
2132:
2127:
2123:
2099:
2079:
2059:
2039:
2017:
2013:
2009:
2006:
2003:
1983:
1963:
1952:
1951:
1940:
1937:
1934:
1929:
1925:
1919:
1916:
1911:
1908:
1903:
1900:
1895:
1892:
1889:
1886:
1883:
1880:
1877:
1874:
1851:
1840:
1839:
1828:
1825:
1820:
1816:
1810:
1806:
1802:
1797:
1793:
1789:
1786:
1781:
1777:
1773:
1770:
1767:
1764:
1761:
1738:
1735:
1732:
1729:
1720:with roots at
1707:
1703:
1699:
1696:
1669:
1666:
1663:
1643:
1640:
1635:
1631:
1627:
1622:
1618:
1614:
1611:
1608:
1605:
1602:
1599:
1583:
1580:
1567:
1547:
1536:
1535:
1523:
1520:
1517:
1514:
1511:
1508:
1505:
1500:
1497:
1493:
1489:
1484:
1480:
1469:
1457:
1454:
1451:
1448:
1445:
1442:
1439:
1436:
1433:
1430:
1427:
1424:
1421:
1416:
1413:
1409:
1405:
1402:
1397:
1393:
1364:boundary layer
1355:
1352:
1339:
1328:
1327:
1315:
1310:
1306:
1302:
1299:
1296:
1293:
1288:
1284:
1273:
1261:
1258:
1255:
1252:
1247:
1243:
1239:
1234:
1230:
1226:
1221:
1217:
1206:
1194:
1191:
1186:
1182:
1178:
1173:
1169:
1165:
1160:
1156:
1152:
1147:
1140:
1137:
1104:
1100:
1077:
1073:
1052:
1049:
1046:
1043:
1040:
1029:
1028:
1016:
1011:
1007:
1003:
1000:
997:
994:
989:
985:
981:
976:
972:
968:
965:
962:
959:
954:
950:
939:
927:
924:
921:
918:
913:
909:
905:
900:
896:
892:
887:
883:
879:
876:
873:
868:
864:
860:
855:
851:
847:
842:
838:
834:
829:
822:
819:
812:
802:
790:
787:
784:
781:
776:
772:
768:
763:
759:
755:
750:
746:
742:
739:
736:
731:
727:
723:
718:
714:
710:
705:
701:
697:
692:
685:
682:
659:
656:
649:
628:
625:
622:
602:
599:
596:
585:
584:
569:
566:
563:
560:
557:
554:
545:
542:
539:
536:
533:
530:
527:
524:
523:
520:
517:
514:
511:
508:
499:
494:
491:
487:
483:
480:
477:
474:
471:
466:
462:
458:
455:
452:
449:
444:
441:
437:
433:
430:
429:
413:
410:
404:
401:
340:
320:
300:
297:
266:
262:
258:
255:
252:
249:
244:
240:
219:
199:
196:
193:
188:
184:
163:
143:
140:
137:
126:
125:
113:
110:
107:
102:
98:
94:
91:
88:
83:
79:
73:
68:
65:
62:
58:
54:
51:
48:
45:
42:
15:
9:
6:
4:
3:
2:
2937:
2926:
2923:
2921:
2918:
2916:
2913:
2912:
2910:
2898:
2892:
2885:
2884:0-387-22966-3
2881:
2877:
2871:
2864:
2858:
2852:
2848:
2844:
2838:
2832:
2828:
2824:
2821:Hinch, E. J.
2818:
2812:
2808:
2804:
2798:
2789:
2784:
2780:
2776:
2772:
2768:
2764:
2757:
2749:
2745:
2738:
2731:
2727:
2706:
2703:
2700:
2695:
2691:
2687:
2684:
2681:
2678:
2673:
2670:
2665:
2660:
2653:
2647:
2641:
2635:
2629:
2622:
2621:
2620:
2606:
2603:
2598:
2594:
2573:
2570:
2565:
2561:
2537:
2534:
2529:
2524:
2520:
2516:
2511:
2506:
2502:
2494:
2493:
2492:
2475:
2472:
2467:
2463:
2457:
2453:
2449:
2444:
2440:
2434:
2430:
2426:
2421:
2417:
2413:
2407:
2401:
2394:
2393:
2392:
2375:
2372:
2367:
2363:
2359:
2354:
2350:
2346:
2341:
2337:
2329:
2328:
2327:
2313:
2291:
2287:
2266:
2263:
2260:
2238:
2234:
2213:
2210:
2207:
2184:
2181:
2176:
2173:
2170:
2167:
2163:
2159:
2154:
2150:
2144:
2141:
2138:
2134:
2130:
2125:
2121:
2113:
2112:
2111:
2097:
2077:
2057:
2037:
2015:
2011:
2007:
2004:
2001:
1981:
1961:
1938:
1935:
1932:
1927:
1923:
1917:
1914:
1909:
1906:
1901:
1898:
1893:
1890:
1887:
1884:
1878:
1872:
1865:
1864:
1863:
1849:
1826:
1823:
1818:
1814:
1808:
1804:
1800:
1795:
1791:
1787:
1784:
1779:
1775:
1771:
1765:
1759:
1752:
1751:
1750:
1736:
1733:
1730:
1727:
1705:
1701:
1697:
1694:
1687:
1683:
1667:
1661:
1641:
1638:
1633:
1629:
1625:
1620:
1616:
1612:
1609:
1603:
1597:
1589:
1579:
1565:
1545:
1521:
1515:
1509:
1506:
1503:
1498:
1495:
1491:
1487:
1482:
1478:
1470:
1455:
1449:
1443:
1440:
1437:
1431:
1425:
1422:
1419:
1414:
1411:
1407:
1403:
1400:
1395:
1391:
1383:
1382:
1381:
1379:
1375:
1371:
1367:
1365:
1361:
1351:
1337:
1313:
1308:
1304:
1300:
1294:
1286:
1282:
1274:
1259:
1256:
1253:
1245:
1241:
1237:
1232:
1228:
1219:
1215:
1207:
1192:
1184:
1180:
1176:
1171:
1167:
1158:
1154:
1150:
1145:
1138:
1135:
1124:
1123:
1122:
1120:
1102:
1098:
1075:
1071:
1050:
1047:
1044:
1041:
1038:
1014:
1009:
1005:
1001:
995:
987:
983:
979:
974:
970:
966:
960:
952:
948:
940:
925:
919:
916:
911:
907:
903:
898:
894:
885:
881:
877:
874:
866:
862:
858:
853:
849:
840:
836:
832:
827:
820:
817:
810:
803:
788:
782:
779:
774:
770:
766:
761:
757:
748:
744:
740:
737:
729:
725:
721:
716:
712:
703:
699:
695:
690:
683:
680:
669:
668:
667:
664:
653:
648:
646:
642:
626:
623:
620:
600:
597:
594:
567:
564:
558:
552:
543:
540:
537:
531:
525:
518:
515:
512:
509:
506:
497:
492:
489:
485:
481:
478:
472:
460:
456:
450:
435:
431:
420:
419:
418:
409:
400:
398:
394:
390:
386:
382:
377:
375:
371:
367:
363:
359:
353:
338:
318:
310:
306:
296:
294:
290:
285:
282:
264:
260:
256:
250:
242:
238:
217:
194:
186:
182:
161:
141:
135:
108:
100:
96:
89:
81:
77:
71:
66:
63:
60:
56:
52:
46:
40:
33:
32:
31:
30:
26:
22:
2896:
2891:
2875:
2870:
2862:
2857:
2842:
2837:
2822:
2817:
2802:
2797:
2770:
2766:
2756:
2750:: PDF page 5
2747:
2743:
2730:
2552:
2490:
2390:
2199:
1953:
1841:
1585:
1537:
1368:
1357:
1329:
1030:
665:
661:
640:
586:
415:
406:
396:
384:
380:
378:
354:
302:
286:
127:
24:
18:
21:mathematics
2909:Categories
2723:References
2704:⋯
2692:ε
2685:−
2682:ε
2679:−
2674:ε
2661:ε
2654:ε
2636:ε
2517:−
2476:⋯
2454:ε
2431:ε
2408:ε
2364:ε
2347:−
2314:ε
2261:ν
2208:ν
2174:−
2171:ν
2164:ε
2142:−
2139:ν
2135:ε
2131:−
2078:ε
2038:ν
2016:ν
2012:ε
1962:ε
1936:⋯
1924:ε
1910:±
1907:ε
1888:±
1879:ε
1850:ε
1827:⋯
1805:ε
1788:ε
1766:ε
1734:±
1698:−
1686:quadratic
1665:→
1662:ε
1626:−
1613:ε
1438:−
1404:ε
1338:ε
1139:˙
1048:≪
1045:ε
920:ε
878:ε
821:˙
811:ε
783:ε
741:ε
684:˙
621:ε
595:ε
490:−
482:−
465:′
443:′
440:′
432:ε
383:, Hinch,
339:ε
319:ε
261:ε
251:ε
239:δ
218:ε
195:ε
183:δ
162:ε
139:→
136:ε
97:ψ
90:ε
78:δ
57:∑
53:≈
41:φ
2863:Mat. Sb.
2110:we have
1119:Tikhonov
2775:Bibcode
2491:yields
1680:, this
1378:ecology
154:. Here
2882:
2849:
2829:
2809:
1538:where
550:
547:
504:
501:
393:Orszag
389:Bender
368:, the
307:has a
2740:(PDF)
1682:cubic
1588:roots
1031:with
2880:ISBN
2847:ISBN
2827:ISBN
2807:ISBN
2748:2244
2226:the
2211:<
1042:<
516:<
510:<
391:and
372:and
360:and
291:and
23:, a
2783:doi
2771:233
1358:In
601:0.1
387:or
128:as
19:In
2911::
2781:.
2769:.
2765:.
2746:,
2742:,
2538:0.
2376:0.
2185:0.
647:.
568:1.
399:.
395:,
295:.
2886:.
2791:.
2785::
2777::
2707:.
2701:+
2696:3
2688:2
2671:1
2666:=
2657:)
2651:(
2648:y
2642:=
2639:)
2633:(
2630:x
2607:0
2604:=
2599:0
2595:y
2574:1
2571:=
2566:0
2562:y
2535:=
2530:2
2525:0
2521:y
2512:3
2507:0
2503:y
2473:+
2468:2
2464:y
2458:4
2450:+
2445:1
2441:y
2435:2
2427:+
2422:0
2418:y
2414:=
2411:)
2405:(
2402:y
2373:=
2368:2
2360:+
2355:2
2351:y
2342:3
2338:y
2292:2
2288:y
2267:1
2264:=
2239:3
2235:y
2214:1
2182:=
2177:1
2168:3
2160:+
2155:2
2151:y
2145:1
2126:3
2122:y
2098:y
2058:y
2008:x
2005:=
2002:y
1982:x
1939:.
1933:+
1928:2
1918:8
1915:5
1902:2
1899:1
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1891:1
1885:=
1882:)
1876:(
1873:x
1824:+
1819:2
1815:x
1809:2
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1792:x
1785:+
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1772:=
1769:)
1763:(
1760:x
1737:1
1731:=
1728:x
1706:2
1702:x
1695:1
1668:0
1642:1
1639:+
1634:2
1630:x
1621:3
1617:x
1610:=
1607:)
1604:x
1601:(
1598:p
1566:v
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1522:,
1519:)
1516:u
1513:(
1510:h
1507:v
1504:+
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1492:v
1488:=
1483:t
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1453:)
1450:u
1447:(
1444:g
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1432:u
1429:(
1426:f
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1412:x
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1301:=
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1295:0
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1287:1
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1185:2
1181:x
1177:,
1172:1
1168:x
1164:(
1159:1
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1151:=
1146:1
1136:x
1103:1
1099:x
1076:2
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1051:1
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1015:,
1010:2
1006:a
1002:=
999:)
996:0
993:(
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980:,
975:1
971:a
967:=
964:)
961:0
958:(
953:1
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926:,
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917:,
912:2
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891:(
886:2
882:g
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867:2
863:x
859:,
854:1
850:x
846:(
841:2
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833:=
828:2
818:x
789:,
786:)
780:,
775:2
771:x
767:,
762:1
758:x
754:(
749:1
745:g
738:+
735:)
730:2
726:x
722:,
717:1
713:x
709:(
704:1
700:f
696:=
691:1
681:x
641:x
627:0
624:=
598:=
565:=
562:)
559:1
556:(
553:u
544:,
541:0
538:=
535:)
532:0
529:(
526:u
519:1
513:x
507:0
498:,
493:x
486:e
479:=
476:)
473:x
470:(
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451:x
448:(
436:u
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257:=
254:)
248:(
243:n
198:)
192:(
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109:x
106:(
101:n
93:)
87:(
82:n
72:N
67:0
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61:n
50:)
47:x
44:(
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