3188:
3869:
2932:
2686:
2113:
1506:
3183:{\displaystyle {\frac {1}{r}}{\frac {\partial }{\partial r}}\left(r{\frac {\partial G}{\partial r}}\right)+{\frac {1}{r^{2}}}{\frac {\partial ^{2}G}{\partial \phi ^{2}}}+{\frac {1}{\alpha }}\delta (t-\tau ){\frac {\delta (r-r')}{2\pi r'}}\delta (\phi -\phi ')={\frac {1}{\alpha }}{\frac {\partial G}{\partial t}}}
3674:
976:
474:
272:
Some of the designations for the Greens function number system are given next. Coordinate system designations include: X, Y, and Z for
Cartesian coordinates; R, Z, φ for cylindrical coordinates; and, RS, φ, θ for spherical coordinates. Designations for several boundary conditions are given in Table
2507:
1934:
40:
Numbers have long been used to identify types of boundary conditions. The Green's function number system was proposed by Beck and
Litkouhi in 1988 and has seen increasing use since then. The number system has been used to catalog a large collection of Green's functions and related solutions.
1313:
3544:
3864:{\displaystyle {\frac {1}{r^{2}}}{\frac {\partial }{\partial r}}\left(r^{2}{\frac {\partial G}{\partial r}}\right)+{\frac {1}{\alpha }}\delta (t-\tau ){\frac {\delta (r-r')}{4\pi r^{2}}}={\frac {1}{\alpha }}{\frac {\partial G}{\partial t}}}
88:
satisfies. The Green's function number has two parts, a letter designation followed by a number designation. The letter(s) designate the coordinate system, while the numbers designate the type of boundary conditions that are satisfied.
2327:
273:
1. The zeroth boundary condition is important for identifying the presence of a coordinate boundary where no physical boundary exists, for example, far away in a semi-infinite body or at the center of a cylindrical or spherical body.
2681:{\displaystyle {\frac {1}{r}}{\frac {\partial }{\partial r}}\left(r{\frac {\partial G}{\partial r}}\right)+{\frac {1}{\alpha }}\delta (t-\tau ){\frac {\delta (r-r')}{2\pi r'}}={\frac {1}{\alpha }}{\frac {\partial G}{\partial t}}}
2108:{\displaystyle {\frac {1}{r}}{\frac {\partial }{\partial r}}\left(r{\frac {\partial G}{\partial r}}\right)+{\frac {1}{\alpha }}\delta (t-\tau ){\frac {\delta (r-r')}{2\pi r'}}={\frac {1}{\alpha }}{\frac {\partial G}{\partial t}}}
843:
341:
1501:{\displaystyle {\frac {\partial ^{2}G}{\partial x^{2}}}+{\frac {\partial ^{2}G}{\partial y^{2}}}+{\frac {1}{\alpha }}\delta (t-\tau )\delta (x-x')\delta (y-y')={\frac {1}{\alpha }}{\frac {\partial G}{\partial t}}}
4047:
1765:
1119:
3446:
262:
198:
2921:
denote the type zero boundaries for angle; here no physical boundary takes the form of the periodic boundary condition. The boundary value problem for the R01φ00 Green's function is given by
1700:
1159:
3441:
4185:
Nowak, A.; Białecki, R.; Kurpisz, K. (February 1987). "Evaluating eigenvalues for boundary value problems of heat conduction in rectangular and cylindrical co-ordinate systems".
3265:
2845:
2727:
1803:
1580:
1547:
1282:
1250:
1017:
774:
4090:
3586:
2888:
2370:
1846:
1202:
678:
3910:
3229:
2154:
515:
826:
3380:
2807:
1660:
636:
595:
2906:
As a two dimensional example, number R01φ00 denotes the Green's function in a solid cylinder with angular dependence, with a type 1 (Dirichlet) boundary condition at
2462:
730:
706:
3984:
3937:
3339:
3292:
2754:
2228:
2181:
1607:
1044:
542:
2233:
146:
2422:
2398:
971:{\displaystyle {\frac {\partial ^{2}G}{\partial x^{2}}}+{\frac {1}{\alpha }}\delta (t-\tau )\delta (x-x')={\frac {1}{\alpha }}{\frac {\partial G}{\partial t}}}
469:{\displaystyle {\frac {\partial ^{2}G}{\partial x^{2}}}+{\frac {1}{\alpha }}\delta (t-\tau )\delta (x-x')={\frac {1}{\alpha }}{\frac {\partial G}{\partial t}}}
4328:
Toptan, A.; Porter, N. W.; Hales, J. D. (2020). "Construction of a code verification matrix for heat conduction with finite element code applications".
1869:
As an example in the cylindrical coordinate system, number R03 denotes the Green's function that satisfies the heat equation in the solid cylinder (
4301:
Lefebvre, G. (December 2010). "A general modal-based numerical simulation of transient heat conduction in a one-dimensional homogeneous slab".
3989:
3539:{\displaystyle \left.{\frac {\partial G}{\partial \phi }}\right|_{\phi =0}=\left.{\frac {\partial G}{\partial \phi }}\right|_{\phi =2\pi }}
1707:
1061:
2496:
denotes the type zero boundary (boundedness) at large values of r. The boundary value problem for the R10 Green's function is given by
4616:
4589:
4552:
4515:
4655:
4665:
4635:
4169:
4456:
212:
2444:
As another example, number R10 denotes the Green's function in a large body containing a cylindrical void (a < r <
4670:
4114:
298:
160:
4119:
3621:
1665:
1124:
2425:
4124:
3609:
As an example in the spherical coordinate system, number RS02 denotes the Green's function for a solid sphere (
3387:
1910:
4660:
4274:
de Monte, Filippo (September 2006). "Multi-layer transient heat conduction using transition time scales".
1299:
829:
327:
3235:
4212:
Beck, James V.; Litkouhi, Bahman (March 1988). "Heat conduction numbering system for basic geometries".
49:
2812:
2700:
1770:
1553:
1520:
1255:
1223:
1220:
As a two-dimensional example, number X10Y20 denotes the Green's function in the quarter-infinite body (
990:
747:
286:
As an example, number X11 denotes the Green's function that satisfies the heat equation in the domain (
4109:
4054:
3550:
2852:
2401:
2334:
1810:
1166:
733:
642:
25:
709:
85:
4476:
4256:
3883:
3202:
2127:
744:
As another
Cartesian example, number X20 denotes the Green's function in the semi-infinite body (
488:
81:
17:
805:
4631:
4612:
4585:
4548:
4511:
4239:
Al-Nimr, M. A.; Alkam, M. K. (19 September 1997). "A generalized thermal boundary condition".
4165:
3344:
2771:
1624:
600:
559:
77:
4472:
2447:
715:
691:
4604:
4468:
4337:
4310:
4283:
4248:
4221:
4194:
3954:
3916:
3309:
3271:
2733:
2322:{\displaystyle \left.k{\frac {\partial G}{\partial n}}\right|_{r=a}+\left.hG\right|_{r=a}=0}
2198:
2160:
1586:
1023:
521:
61:
125:
4287:
2407:
2383:
65:
4649:
4366:
4260:
4225:
4129:
45:
29:
4480:
1898:
denotes the zeroth boundary condition (boundedness) at the center of the cylinder (
4314:
93:
Table 1. Boundary conditions designations for Green's function number system.
57:
53:
4198:
4608:
4252:
3663:. The boundary value problem for the RS02 Green's function is given by
4599:
Cole, Kevin D.; Beck, James; Haji-Sheikh, A.; Litkouhi, Bahman (2011).
4457:"Conduction in rectangular plates with boundary temperatures specified"
4455:
Beck, J. V.; Wright, N.; Haji-Sheikh, A.; Cole, K. D; Amos. D. (2008).
1856:
Applications of related half-space and quarter-space GF are available.
4341:
4042:{\displaystyle \left.{\frac {\partial G}{\partial n}}\right|_{r=a}=0}
1760:{\displaystyle \left.{\frac {\partial G}{\partial n}}\right|_{y=0}=0}
1114:{\displaystyle \left.{\frac {\partial G}{\partial n}}\right|_{x=0}=0}
326:
denotes the type 1 boundary condition at both sides of the body. The
32:
to make existing solutions easier to identify, store, and retrieve.
4330:
Journal of
Verification, Validation and Uncertainty Quantification
1923:. The boundary value problem for R03 Green's function is given by
802:
denotes the zeroth type boundary condition (boundedness) at
4567:
4530:
4419:
4354:
3995:
3596:
Both a transient and steady form of this GF are available.
3495:
3452:
2433:
2287:
2239:
1713:
1671:
1130:
1067:
48:, this number system applies to any phenomena described by
4187:
International
Journal for Numerical Methods in Engineering
4387:
2917:
denotes the angular (azimuthal) coordinate, and numbers
3642:
denotes the zeroth boundary condition (boundedness) at
3638:
denote the radial-spherical coordinate system, number
257:{\displaystyle k{\frac {\partial G}{\partial n}}+hG=0}
4057:
3992:
3957:
3919:
3886:
3677:
3553:
3449:
3390:
3347:
3312:
3274:
3238:
3205:
2935:
2855:
2815:
2774:
2736:
2703:
2510:
2450:
2410:
2386:
2337:
2236:
2201:
2163:
2130:
1937:
1813:
1773:
1710:
1668:
1627:
1589:
1556:
1523:
1316:
1258:
1226:
1169:
1127:
1064:
1026:
993:
846:
808:
750:
718:
694:
645:
603:
562:
524:
491:
344:
215:
163:
128:
4084:
4041:
3978:
3931:
3904:
3863:
3580:
3538:
3435:
3374:
3333:
3286:
3259:
3223:
3182:
2882:
2839:
2801:
2748:
2721:
2680:
2478:denotes the cylindrical coordinate system, number
2464:) with a type 1 (Dirichlet) boundary condition at
2456:
2416:
2392:
2364:
2321:
2222:
2175:
2148:
2107:
1894:denotes the cylindrical coordinate system, number
1840:
1797:
1759:
1694:
1654:
1601:
1574:
1541:
1500:
1276:
1244:
1196:
1153:
1113:
1038:
1011:
970:
820:
768:
724:
700:
672:
630:
589:
536:
509:
468:
256:
193:{\displaystyle {\frac {\partial G}{\partial n}}=0}
192:
140:
1880:) with a boundary condition of type 3 (Robin) at
4461:International Journal of Heat and Mass Transfer
4214:International Journal of Heat and Mass Transfer
44:Although the examples given below are for the
4493:
4442:
2429:
1695:{\displaystyle \left.G\right|_{x\to \infty }}
1154:{\displaystyle \left.G\right|_{x\to \infty }}
8:
1302:for the X10Y20 Green's function is given by
76:The Green's function number specifies the
4276:International Journal of Thermal Sciences
4056:
4021:
3997:
3991:
3956:
3918:
3885:
3841:
3831:
3819:
3781:
3753:
3725:
3719:
3695:
3687:
3678:
3676:
3552:
3521:
3497:
3478:
3454:
3448:
3436:{\displaystyle G(\phi =0)=G(\phi =2\pi )}
3389:
3346:
3311:
3273:
3237:
3204:
3160:
3150:
3079:
3051:
3039:
3021:
3014:
3006:
2997:
2969:
2946:
2936:
2934:
2854:
2825:
2820:
2814:
2773:
2735:
2702:
2658:
2648:
2600:
2572:
2544:
2521:
2511:
2509:
2449:
2409:
2385:
2336:
2301:
2270:
2245:
2235:
2200:
2162:
2129:
2085:
2075:
2027:
1999:
1971:
1948:
1938:
1936:
1812:
1783:
1778:
1772:
1739:
1715:
1709:
1680:
1667:
1626:
1588:
1555:
1522:
1478:
1468:
1391:
1379:
1361:
1354:
1342:
1324:
1317:
1315:
1257:
1225:
1168:
1139:
1126:
1093:
1069:
1063:
1025:
992:
948:
938:
884:
872:
854:
847:
845:
832:for the X20 Green's function is given by
807:
791:denotes the type 2 boundary condition at
749:
717:
693:
644:
602:
561:
523:
490:
446:
436:
382:
370:
352:
345:
343:
330:for the X11 Green's function is given by
219:
214:
164:
162:
127:
4547:. New York: Marcel Dekker. p. 223.
4473:10.1016/j.ijheatmasstransfer.2008.02.020
3665:
2923:
2498:
1925:
1304:
1284:) with a Dirichlet (type 1) boundary at
834:
332:
91:
4141:
24:is used to uniquely categorize certain
4601:Heat Conduction Using Greens Functions
4580:Carslaw, H. S.; Jaeger, J. C. (1959).
4430:
4407:
4148:
776:) with a Neumann (type 2) boundary at
4388:"Exact Analytical Conduction Toolbox"
322:denotes the Cartesian coordinate and
297:) for boundary conditions of type 1 (
7:
3260:{\displaystyle 0<\phi <2\pi }
1860:Examples in cylindrical coordinates
1291:and a Neumann (type 2) boundary at
4288:10.1016/j.ijthermalsci.2005.11.006
4008:
4000:
3852:
3844:
3736:
3728:
3701:
3697:
3508:
3500:
3465:
3457:
3171:
3163:
3032:
3018:
2980:
2972:
2952:
2948:
2832:
2716:
2669:
2661:
2555:
2547:
2527:
2523:
2451:
2256:
2248:
2096:
2088:
1982:
1974:
1954:
1950:
1790:
1726:
1718:
1687:
1569:
1536:
1489:
1481:
1372:
1358:
1335:
1321:
1271:
1239:
1146:
1080:
1072:
1006:
959:
951:
865:
851:
815:
787:denotes the Cartesian coordinate,
763:
736:. This GF is developed elsewhere.
457:
449:
363:
349:
230:
222:
175:
167:
14:
4164:(1st ed.). New York: Wiley.
2898:This GF is available elsewhere.
2840:{\displaystyle G|_{r\to \infty }}
2722:{\displaystyle a<r<\infty }
1798:{\displaystyle G|_{y\to \infty }}
1575:{\displaystyle 0<y<\infty }
1542:{\displaystyle 0<x<\infty }
1277:{\displaystyle 0<y<\infty }
1245:{\displaystyle 0<x<\infty }
1012:{\displaystyle 0<x<\infty }
769:{\displaystyle 0<x<\infty }
277:Examples in Cartesian coordinates
4628:Analytical Heat Diffusion Theory
4545:Influence Functions and Matrices
4506:Thambynayagam, R. K. M. (2011).
4100:This GF is available elsewhere.
3600:Example in spherical coordinates
1212:This GF is published elsewhere.
4630:. Academic Press. p. 388.
4085:{\displaystyle G(t<\tau )=0}
3653:denotes the type 2 boundary at
3581:{\displaystyle G(t<\tau )=0}
2883:{\displaystyle G(t<\tau )=0}
2482:denotes the type 1 boundary at
2365:{\displaystyle G(t<\tau )=0}
1841:{\displaystyle G(t<\tau )=0}
1197:{\displaystyle G(t<\tau )=0}
673:{\displaystyle G(t<\tau )=0}
4073:
4061:
3973:
3961:
3804:
3787:
3778:
3766:
3569:
3557:
3430:
3415:
3406:
3394:
3363:
3351:
3328:
3316:
3144:
3127:
3102:
3085:
3076:
3064:
2871:
2859:
2829:
2821:
2790:
2778:
2623:
2606:
2597:
2585:
2353:
2341:
2217:
2205:
2050:
2033:
2024:
2012:
1829:
1817:
1787:
1779:
1684:
1643:
1631:
1462:
1445:
1439:
1422:
1416:
1404:
1185:
1173:
1143:
932:
915:
909:
897:
661:
649:
619:
607:
578:
566:
430:
413:
407:
395:
1:
4315:10.1016/j.enbuild.2010.07.024
4582:Conduction of Heat in Solids
4510:. McGraw-Hill. p. 432.
4226:10.1016/0017-9310(88)90032-4
4115:Dirichlet boundary condition
2436:, p. 543) for this GF.
4584:. Oxford University Press.
3905:{\displaystyle 0<r<b}
3224:{\displaystyle 0<r<a}
2149:{\displaystyle 0<r<a}
510:{\displaystyle 0<x<L}
4687:
4367:"Green's Function Library"
4160:Özışık, M. Necati (1980).
4120:Neumann boundary condition
2430:Carslaw & Jaeger (1959
4494:Carslaw & Jaeger 1959
4443:Carslaw & Jaeger 1959
3945:
3874:
3668:
3300:
3193:
2926:
2762:
2691:
2501:
2426:heat transfer coefficient
2189:
2118:
1928:
1615:
1511:
1307:
1052:
981:
837:
821:{\displaystyle x=\infty }
550:
479:
335:
4543:Melnikov, Y. A. (1999).
4125:Robin boundary condition
3624:) boundary condition at
3375:{\displaystyle G(r=a)=0}
2802:{\displaystyle G(r=a)=0}
1913:) boundary condition at
1655:{\displaystyle G(x=0)=0}
631:{\displaystyle G(x=L)=0}
590:{\displaystyle G(x=0)=0}
2457:{\displaystyle \infty }
725:{\displaystyle \delta }
701:{\displaystyle \alpha }
22:Green's function number
4656:Differential equations
4626:Luikov, A. V. (1968).
4508:The Diffusion Handbook
4241:Heat and Mass Transfer
4199:10.1002/nme.1620240210
4086:
4043:
3980:
3979:{\displaystyle G(r=0)}
3933:
3932:{\displaystyle t>0}
3906:
3865:
3582:
3540:
3437:
3376:
3335:
3334:{\displaystyle G(r=0)}
3288:
3287:{\displaystyle t>0}
3261:
3225:
3184:
2884:
2841:
2803:
2750:
2749:{\displaystyle t>0}
2723:
2682:
2458:
2418:
2394:
2366:
2323:
2224:
2223:{\displaystyle G(r=0)}
2177:
2176:{\displaystyle t>0}
2150:
2109:
1842:
1799:
1761:
1696:
1656:
1603:
1602:{\displaystyle t>0}
1576:
1543:
1502:
1300:boundary value problem
1278:
1246:
1198:
1155:
1115:
1040:
1039:{\displaystyle t>0}
1013:
972:
830:boundary value problem
822:
770:
726:
702:
674:
632:
591:
538:
537:{\displaystyle t>0}
511:
470:
328:boundary value problem
258:
194:
142:
50:differential equations
4666:Generalized functions
4609:10.1201/9781439895214
4253:10.1007/s002310050173
4087:
4044:
3981:
3934:
3907:
3866:
3583:
3541:
3438:
3377:
3336:
3289:
3262:
3226:
3185:
2885:
2842:
2804:
2751:
2724:
2683:
2459:
2419:
2395:
2367:
2324:
2225:
2178:
2151:
2110:
1843:
1800:
1762:
1697:
1657:
1604:
1577:
1544:
1503:
1279:
1247:
1199:
1156:
1116:
1041:
1014:
973:
823:
771:
727:
703:
675:
633:
592:
539:
512:
471:
301:) at both boundaries
259:
195:
143:
26:fundamental solutions
4467:(19–20): 4676–4690.
4303:Energy and Buildings
4110:Fundamental solution
4055:
3990:
3955:
3946:Boundary conditions
3917:
3884:
3675:
3551:
3447:
3388:
3345:
3310:
3301:Boundary conditions
3272:
3236:
3203:
2933:
2853:
2813:
2772:
2763:Boundary conditions
2734:
2701:
2508:
2448:
2408:
2402:thermal conductivity
2384:
2335:
2234:
2199:
2190:Boundary conditions
2161:
2128:
1935:
1909:denotes the type 3 (
1811:
1771:
1708:
1666:
1625:
1616:Boundary conditions
1587:
1554:
1521:
1314:
1256:
1224:
1167:
1125:
1062:
1053:Boundary conditions
1024:
991:
844:
806:
748:
734:Dirac delta function
716:
692:
643:
601:
560:
551:Boundary conditions
522:
489:
342:
213:
161:
126:
109:No physical boundary
4671:Physical quantities
710:thermal diffusivity
141:{\displaystyle G=0}
94:
82:boundary conditions
4082:
4039:
3976:
3929:
3902:
3861:
3578:
3536:
3433:
3372:
3331:
3284:
3257:
3221:
3180:
2880:
2837:
2799:
2746:
2719:
2678:
2454:
2414:
2390:
2362:
2319:
2220:
2173:
2146:
2105:
1838:
1795:
1757:
1692:
1652:
1599:
1572:
1539:
1498:
1274:
1242:
1194:
1151:
1111:
1036:
1009:
968:
818:
766:
722:
698:
670:
628:
587:
534:
507:
466:
254:
190:
138:
101:Boundary condition
92:
4342:10.1115/1.4049037
4309:(12): 2309–2322.
4098:
4097:
4015:
3859:
3839:
3826:
3761:
3743:
3708:
3693:
3620:) with a type 2 (
3594:
3593:
3515:
3472:
3178:
3158:
3122:
3059:
3046:
3012:
2987:
2959:
2944:
2896:
2895:
2676:
2656:
2643:
2580:
2562:
2534:
2519:
2434:Cole et al. (2011
2417:{\displaystyle h}
2393:{\displaystyle k}
2378:
2377:
2263:
2103:
2083:
2070:
2007:
1989:
1961:
1946:
1854:
1853:
1733:
1496:
1476:
1399:
1386:
1349:
1210:
1209:
1087:
966:
946:
892:
879:
686:
685:
464:
444:
390:
377:
270:
269:
237:
182:
78:coordinate system
4678:
4641:
4622:
4603:(2nd ed.).
4595:
4571:
4568:Cole et al. 2011
4565:
4559:
4558:
4540:
4534:
4531:Cole et al. 2011
4528:
4522:
4521:
4503:
4497:
4491:
4485:
4484:
4452:
4446:
4440:
4434:
4428:
4422:
4420:Cole et al. 2011
4417:
4411:
4405:
4399:
4398:
4396:
4394:
4384:
4378:
4377:
4375:
4373:
4363:
4357:
4355:Cole et al. 2011
4352:
4346:
4345:
4325:
4319:
4318:
4298:
4292:
4291:
4271:
4265:
4264:
4247:(1–2): 157–161.
4236:
4230:
4229:
4209:
4203:
4202:
4182:
4176:
4175:
4157:
4151:
4146:
4091:
4089:
4088:
4083:
4048:
4046:
4045:
4040:
4032:
4031:
4020:
4016:
4014:
4006:
3998:
3985:
3983:
3982:
3977:
3938:
3936:
3935:
3930:
3911:
3909:
3908:
3903:
3870:
3868:
3867:
3862:
3860:
3858:
3850:
3842:
3840:
3832:
3827:
3825:
3824:
3823:
3807:
3803:
3782:
3762:
3754:
3749:
3745:
3744:
3742:
3734:
3726:
3724:
3723:
3709:
3707:
3696:
3694:
3692:
3691:
3679:
3666:
3662:
3648:
3634:. Here letters
3633:
3619:
3587:
3585:
3584:
3579:
3545:
3543:
3542:
3537:
3535:
3534:
3520:
3516:
3514:
3506:
3498:
3489:
3488:
3477:
3473:
3471:
3463:
3455:
3442:
3440:
3439:
3434:
3381:
3379:
3378:
3373:
3340:
3338:
3337:
3332:
3293:
3291:
3290:
3285:
3266:
3264:
3263:
3258:
3230:
3228:
3227:
3222:
3189:
3187:
3186:
3181:
3179:
3177:
3169:
3161:
3159:
3151:
3143:
3123:
3121:
3120:
3105:
3101:
3080:
3060:
3052:
3047:
3045:
3044:
3043:
3030:
3026:
3025:
3015:
3013:
3011:
3010:
2998:
2993:
2989:
2988:
2986:
2978:
2970:
2960:
2958:
2947:
2945:
2937:
2924:
2912:
2889:
2887:
2886:
2881:
2846:
2844:
2843:
2838:
2836:
2835:
2824:
2808:
2806:
2805:
2800:
2755:
2753:
2752:
2747:
2728:
2726:
2725:
2720:
2687:
2685:
2684:
2679:
2677:
2675:
2667:
2659:
2657:
2649:
2644:
2642:
2641:
2626:
2622:
2601:
2581:
2573:
2568:
2564:
2563:
2561:
2553:
2545:
2535:
2533:
2522:
2520:
2512:
2499:
2491:
2474:. Again letter
2473:
2463:
2461:
2460:
2455:
2432:, p. 369),
2423:
2421:
2420:
2415:
2399:
2397:
2396:
2391:
2371:
2369:
2368:
2363:
2328:
2326:
2325:
2320:
2312:
2311:
2300:
2296:
2281:
2280:
2269:
2265:
2264:
2262:
2254:
2246:
2229:
2227:
2226:
2221:
2182:
2180:
2179:
2174:
2155:
2153:
2152:
2147:
2114:
2112:
2111:
2106:
2104:
2102:
2094:
2086:
2084:
2076:
2071:
2069:
2068:
2053:
2049:
2028:
2008:
2000:
1995:
1991:
1990:
1988:
1980:
1972:
1962:
1960:
1949:
1947:
1939:
1926:
1922:
1904:
1889:
1879:
1847:
1845:
1844:
1839:
1804:
1802:
1801:
1796:
1794:
1793:
1782:
1766:
1764:
1763:
1758:
1750:
1749:
1738:
1734:
1732:
1724:
1716:
1701:
1699:
1698:
1693:
1691:
1690:
1679:
1661:
1659:
1658:
1653:
1608:
1606:
1605:
1600:
1581:
1579:
1578:
1573:
1548:
1546:
1545:
1540:
1507:
1505:
1504:
1499:
1497:
1495:
1487:
1479:
1477:
1469:
1461:
1438:
1400:
1392:
1387:
1385:
1384:
1383:
1370:
1366:
1365:
1355:
1350:
1348:
1347:
1346:
1333:
1329:
1328:
1318:
1305:
1297:
1290:
1283:
1281:
1280:
1275:
1251:
1249:
1248:
1243:
1203:
1201:
1200:
1195:
1160:
1158:
1157:
1152:
1150:
1149:
1138:
1120:
1118:
1117:
1112:
1104:
1103:
1092:
1088:
1086:
1078:
1070:
1045:
1043:
1042:
1037:
1018:
1016:
1015:
1010:
977:
975:
974:
969:
967:
965:
957:
949:
947:
939:
931:
893:
885:
880:
878:
877:
876:
863:
859:
858:
848:
835:
827:
825:
824:
819:
797:
782:
775:
773:
772:
767:
731:
729:
728:
723:
707:
705:
704:
699:
679:
677:
676:
671:
637:
635:
634:
629:
596:
594:
593:
588:
543:
541:
540:
535:
516:
514:
513:
508:
475:
473:
472:
467:
465:
463:
455:
447:
445:
437:
429:
391:
383:
378:
376:
375:
374:
361:
357:
356:
346:
333:
317:
307:
296:
263:
261:
260:
255:
238:
236:
228:
220:
199:
197:
196:
191:
183:
181:
173:
165:
147:
145:
144:
139:
95:
86:Green's function
80:and the type of
62:electromagnetics
16:In mathematical
4686:
4685:
4681:
4680:
4679:
4677:
4676:
4675:
4646:
4645:
4644:
4638:
4625:
4619:
4598:
4592:
4579:
4575:
4574:
4566:
4562:
4555:
4542:
4541:
4537:
4529:
4525:
4518:
4505:
4504:
4500:
4492:
4488:
4454:
4453:
4449:
4441:
4437:
4429:
4425:
4418:
4414:
4406:
4402:
4392:
4390:
4386:
4385:
4381:
4371:
4369:
4365:
4364:
4360:
4353:
4349:
4327:
4326:
4322:
4300:
4299:
4295:
4273:
4272:
4268:
4238:
4237:
4233:
4211:
4210:
4206:
4184:
4183:
4179:
4172:
4162:Heat conduction
4159:
4158:
4154:
4147:
4143:
4138:
4106:
4094:
4053:
4052:
4007:
3999:
3994:
3993:
3988:
3987:
3953:
3952:
3941:
3915:
3914:
3882:
3881:
3851:
3843:
3815:
3808:
3796:
3783:
3735:
3727:
3715:
3714:
3710:
3700:
3683:
3673:
3672:
3654:
3643:
3625:
3610:
3607:
3602:
3590:
3549:
3548:
3507:
3499:
3494:
3493:
3464:
3456:
3451:
3450:
3445:
3444:
3386:
3385:
3343:
3342:
3308:
3307:
3296:
3270:
3269:
3234:
3233:
3201:
3200:
3170:
3162:
3136:
3113:
3106:
3094:
3081:
3035:
3031:
3017:
3016:
3002:
2979:
2971:
2965:
2961:
2951:
2931:
2930:
2907:
2904:
2892:
2851:
2850:
2819:
2811:
2810:
2770:
2769:
2758:
2732:
2731:
2699:
2698:
2668:
2660:
2634:
2627:
2615:
2602:
2554:
2546:
2540:
2536:
2526:
2506:
2505:
2483:
2465:
2446:
2445:
2442:
2428:(W/(m K)). See
2406:
2405:
2382:
2381:
2374:
2333:
2332:
2289:
2286:
2285:
2255:
2247:
2241:
2238:
2237:
2232:
2231:
2197:
2196:
2185:
2159:
2158:
2126:
2125:
2095:
2087:
2061:
2054:
2042:
2029:
1981:
1973:
1967:
1963:
1953:
1933:
1932:
1914:
1899:
1890:. Here letter
1881:
1870:
1867:
1862:
1850:
1809:
1808:
1777:
1769:
1768:
1725:
1717:
1712:
1711:
1706:
1705:
1670:
1669:
1664:
1663:
1623:
1622:
1611:
1585:
1584:
1552:
1551:
1519:
1518:
1488:
1480:
1454:
1431:
1375:
1371:
1357:
1356:
1338:
1334:
1320:
1319:
1312:
1311:
1292:
1285:
1254:
1253:
1222:
1221:
1218:
1206:
1165:
1164:
1129:
1128:
1123:
1122:
1079:
1071:
1066:
1065:
1060:
1059:
1048:
1022:
1021:
989:
988:
958:
950:
924:
868:
864:
850:
849:
842:
841:
804:
803:
792:
777:
746:
745:
742:
714:
713:
690:
689:
682:
641:
640:
599:
598:
558:
557:
546:
520:
519:
487:
486:
456:
448:
422:
366:
362:
348:
347:
340:
339:
309:
302:
287:
284:
279:
229:
221:
211:
210:
174:
166:
159:
158:
124:
123:
74:
38:
18:heat conduction
12:
11:
5:
4684:
4682:
4674:
4673:
4668:
4663:
4658:
4648:
4647:
4643:
4642:
4636:
4623:
4617:
4596:
4590:
4576:
4573:
4572:
4560:
4553:
4535:
4523:
4516:
4498:
4486:
4447:
4435:
4423:
4412:
4400:
4379:
4358:
4347:
4320:
4293:
4282:(9): 882–892.
4266:
4231:
4220:(3): 505–515.
4204:
4193:(2): 419–445.
4177:
4170:
4152:
4140:
4139:
4137:
4134:
4133:
4132:
4127:
4122:
4117:
4112:
4105:
4102:
4096:
4095:
4093:
4092:
4081:
4078:
4075:
4072:
4069:
4066:
4063:
4060:
4050:
4038:
4035:
4030:
4027:
4024:
4019:
4013:
4010:
4005:
4002:
3996:
3975:
3972:
3969:
3966:
3963:
3960:
3949:
3947:
3943:
3942:
3940:
3939:
3928:
3925:
3922:
3912:
3901:
3898:
3895:
3892:
3889:
3878:
3876:
3872:
3871:
3857:
3854:
3849:
3846:
3838:
3835:
3830:
3822:
3818:
3814:
3811:
3806:
3802:
3799:
3795:
3792:
3789:
3786:
3780:
3777:
3774:
3771:
3768:
3765:
3760:
3757:
3752:
3748:
3741:
3738:
3733:
3730:
3722:
3718:
3713:
3706:
3703:
3699:
3690:
3686:
3682:
3670:
3606:
3603:
3601:
3598:
3592:
3591:
3589:
3588:
3577:
3574:
3571:
3568:
3565:
3562:
3559:
3556:
3546:
3533:
3530:
3527:
3524:
3519:
3513:
3510:
3505:
3502:
3496:
3492:
3487:
3484:
3481:
3476:
3470:
3467:
3462:
3459:
3453:
3432:
3429:
3426:
3423:
3420:
3417:
3414:
3411:
3408:
3405:
3402:
3399:
3396:
3393:
3383:
3371:
3368:
3365:
3362:
3359:
3356:
3353:
3350:
3330:
3327:
3324:
3321:
3318:
3315:
3304:
3302:
3298:
3297:
3295:
3294:
3283:
3280:
3277:
3267:
3256:
3253:
3250:
3247:
3244:
3241:
3231:
3220:
3217:
3214:
3211:
3208:
3197:
3195:
3191:
3190:
3176:
3173:
3168:
3165:
3157:
3154:
3149:
3146:
3142:
3139:
3135:
3132:
3129:
3126:
3119:
3116:
3112:
3109:
3104:
3100:
3097:
3093:
3090:
3087:
3084:
3078:
3075:
3072:
3069:
3066:
3063:
3058:
3055:
3050:
3042:
3038:
3034:
3029:
3024:
3020:
3009:
3005:
3001:
2996:
2992:
2985:
2982:
2977:
2974:
2968:
2964:
2957:
2954:
2950:
2943:
2940:
2928:
2913:. Here letter
2903:
2900:
2894:
2893:
2891:
2890:
2879:
2876:
2873:
2870:
2867:
2864:
2861:
2858:
2848:
2834:
2831:
2828:
2823:
2818:
2798:
2795:
2792:
2789:
2786:
2783:
2780:
2777:
2766:
2764:
2760:
2759:
2757:
2756:
2745:
2742:
2739:
2729:
2718:
2715:
2712:
2709:
2706:
2695:
2693:
2689:
2688:
2674:
2671:
2666:
2663:
2655:
2652:
2647:
2640:
2637:
2633:
2630:
2625:
2621:
2618:
2614:
2611:
2608:
2605:
2599:
2596:
2593:
2590:
2587:
2584:
2579:
2576:
2571:
2567:
2560:
2557:
2552:
2549:
2543:
2539:
2532:
2529:
2525:
2518:
2515:
2503:
2453:
2441:
2438:
2413:
2404:(W/(m K)) and
2389:
2376:
2375:
2373:
2372:
2361:
2358:
2355:
2352:
2349:
2346:
2343:
2340:
2330:
2318:
2315:
2310:
2307:
2304:
2299:
2295:
2292:
2288:
2284:
2279:
2276:
2273:
2268:
2261:
2258:
2253:
2250:
2244:
2240:
2219:
2216:
2213:
2210:
2207:
2204:
2193:
2191:
2187:
2186:
2184:
2183:
2172:
2169:
2166:
2156:
2145:
2142:
2139:
2136:
2133:
2122:
2120:
2116:
2115:
2101:
2098:
2093:
2090:
2082:
2079:
2074:
2067:
2064:
2060:
2057:
2052:
2048:
2045:
2041:
2038:
2035:
2032:
2026:
2023:
2020:
2017:
2014:
2011:
2006:
2003:
1998:
1994:
1987:
1984:
1979:
1976:
1970:
1966:
1959:
1956:
1952:
1945:
1942:
1930:
1905:), and number
1866:
1863:
1861:
1858:
1852:
1851:
1849:
1848:
1837:
1834:
1831:
1828:
1825:
1822:
1819:
1816:
1806:
1792:
1789:
1786:
1781:
1776:
1756:
1753:
1748:
1745:
1742:
1737:
1731:
1728:
1723:
1720:
1714:
1703:
1689:
1686:
1683:
1678:
1675:
1672:
1651:
1648:
1645:
1642:
1639:
1636:
1633:
1630:
1619:
1617:
1613:
1612:
1610:
1609:
1598:
1595:
1592:
1582:
1571:
1568:
1565:
1562:
1559:
1549:
1538:
1535:
1532:
1529:
1526:
1515:
1513:
1509:
1508:
1494:
1491:
1486:
1483:
1475:
1472:
1467:
1464:
1460:
1457:
1453:
1450:
1447:
1444:
1441:
1437:
1434:
1430:
1427:
1424:
1421:
1418:
1415:
1412:
1409:
1406:
1403:
1398:
1395:
1390:
1382:
1378:
1374:
1369:
1364:
1360:
1353:
1345:
1341:
1337:
1332:
1327:
1323:
1309:
1273:
1270:
1267:
1264:
1261:
1241:
1238:
1235:
1232:
1229:
1217:
1214:
1208:
1207:
1205:
1204:
1193:
1190:
1187:
1184:
1181:
1178:
1175:
1172:
1162:
1148:
1145:
1142:
1137:
1134:
1131:
1110:
1107:
1102:
1099:
1096:
1091:
1085:
1082:
1077:
1074:
1068:
1056:
1054:
1050:
1049:
1047:
1046:
1035:
1032:
1029:
1019:
1008:
1005:
1002:
999:
996:
985:
983:
979:
978:
964:
961:
956:
953:
945:
942:
937:
934:
930:
927:
923:
920:
917:
914:
911:
908:
905:
902:
899:
896:
891:
888:
883:
875:
871:
867:
862:
857:
853:
839:
817:
814:
811:
765:
762:
759:
756:
753:
741:
738:
721:
697:
684:
683:
681:
680:
669:
666:
663:
660:
657:
654:
651:
648:
638:
627:
624:
621:
618:
615:
612:
609:
606:
586:
583:
580:
577:
574:
571:
568:
565:
554:
552:
548:
547:
545:
544:
533:
530:
527:
517:
506:
503:
500:
497:
494:
483:
481:
477:
476:
462:
459:
454:
451:
443:
440:
435:
432:
428:
425:
421:
418:
415:
412:
409:
406:
403:
400:
397:
394:
389:
386:
381:
373:
369:
365:
360:
355:
351:
337:
283:
280:
278:
275:
268:
267:
264:
253:
250:
247:
244:
241:
235:
232:
227:
224:
218:
208:
204:
203:
200:
189:
186:
180:
177:
172:
169:
156:
152:
151:
148:
137:
134:
131:
121:
117:
116:
113:
110:
106:
105:
102:
99:
73:
70:
66:fluid dynamics
37:
34:
13:
10:
9:
6:
4:
3:
2:
4683:
4672:
4669:
4667:
4664:
4662:
4661:Heat transfer
4659:
4657:
4654:
4653:
4651:
4639:
4633:
4629:
4624:
4620:
4618:9781439813546
4614:
4610:
4606:
4602:
4597:
4593:
4591:9780198533689
4587:
4583:
4578:
4577:
4570:, p. 309
4569:
4564:
4561:
4556:
4554:9780824719418
4550:
4546:
4539:
4536:
4533:, p. 554
4532:
4527:
4524:
4519:
4517:9780071751841
4513:
4509:
4502:
4499:
4496:, p. 378
4495:
4490:
4487:
4482:
4478:
4474:
4470:
4466:
4462:
4458:
4451:
4448:
4445:, p. 276
4444:
4439:
4436:
4433:, p. 387
4432:
4427:
4424:
4421:
4416:
4413:
4410:, p. 388
4409:
4404:
4401:
4389:
4383:
4380:
4368:
4362:
4359:
4356:
4351:
4348:
4343:
4339:
4336:(4): 041002.
4335:
4331:
4324:
4321:
4316:
4312:
4308:
4304:
4297:
4294:
4289:
4285:
4281:
4277:
4270:
4267:
4262:
4258:
4254:
4250:
4246:
4242:
4235:
4232:
4227:
4223:
4219:
4215:
4208:
4205:
4200:
4196:
4192:
4188:
4181:
4178:
4173:
4167:
4163:
4156:
4153:
4150:
4145:
4142:
4135:
4131:
4130:Heat equation
4128:
4126:
4123:
4121:
4118:
4116:
4113:
4111:
4108:
4107:
4103:
4101:
4079:
4076:
4070:
4067:
4064:
4058:
4051:
4036:
4033:
4028:
4025:
4022:
4017:
4011:
4003:
3970:
3967:
3964:
3958:
3951:
3950:
3948:
3944:
3926:
3923:
3920:
3913:
3899:
3896:
3893:
3890:
3887:
3880:
3879:
3877:
3873:
3855:
3847:
3836:
3833:
3828:
3820:
3816:
3812:
3809:
3800:
3797:
3793:
3790:
3784:
3775:
3772:
3769:
3763:
3758:
3755:
3750:
3746:
3739:
3731:
3720:
3716:
3711:
3704:
3688:
3684:
3680:
3671:
3667:
3664:
3661:
3657:
3652:
3649:, and number
3646:
3641:
3637:
3632:
3628:
3623:
3618:
3614:
3604:
3599:
3597:
3575:
3572:
3566:
3563:
3560:
3554:
3547:
3531:
3528:
3525:
3522:
3517:
3511:
3503:
3490:
3485:
3482:
3479:
3474:
3468:
3460:
3427:
3424:
3421:
3418:
3412:
3409:
3403:
3400:
3397:
3391:
3384:
3369:
3366:
3360:
3357:
3354:
3348:
3325:
3322:
3319:
3313:
3306:
3305:
3303:
3299:
3281:
3278:
3275:
3268:
3254:
3251:
3248:
3245:
3242:
3239:
3232:
3218:
3215:
3212:
3209:
3206:
3199:
3198:
3196:
3192:
3174:
3166:
3155:
3152:
3147:
3140:
3137:
3133:
3130:
3124:
3117:
3114:
3110:
3107:
3098:
3095:
3091:
3088:
3082:
3073:
3070:
3067:
3061:
3056:
3053:
3048:
3040:
3036:
3027:
3022:
3007:
3003:
2999:
2994:
2990:
2983:
2975:
2966:
2962:
2955:
2941:
2938:
2929:
2925:
2922:
2920:
2916:
2910:
2901:
2899:
2877:
2874:
2868:
2865:
2862:
2856:
2849:
2826:
2816:
2796:
2793:
2787:
2784:
2781:
2775:
2768:
2767:
2765:
2761:
2743:
2740:
2737:
2730:
2713:
2710:
2707:
2704:
2697:
2696:
2694:
2690:
2672:
2664:
2653:
2650:
2645:
2638:
2635:
2631:
2628:
2619:
2616:
2612:
2609:
2603:
2594:
2591:
2588:
2582:
2577:
2574:
2569:
2565:
2558:
2550:
2541:
2537:
2530:
2516:
2513:
2504:
2500:
2497:
2495:
2492:, and number
2490:
2486:
2481:
2477:
2472:
2468:
2439:
2437:
2435:
2431:
2427:
2411:
2403:
2387:
2359:
2356:
2350:
2347:
2344:
2338:
2331:
2316:
2313:
2308:
2305:
2302:
2297:
2293:
2290:
2282:
2277:
2274:
2271:
2266:
2259:
2251:
2242:
2214:
2211:
2208:
2202:
2195:
2194:
2192:
2188:
2170:
2167:
2164:
2157:
2143:
2140:
2137:
2134:
2131:
2124:
2123:
2121:
2117:
2099:
2091:
2080:
2077:
2072:
2065:
2062:
2058:
2055:
2046:
2043:
2039:
2036:
2030:
2021:
2018:
2015:
2009:
2004:
2001:
1996:
1992:
1985:
1977:
1968:
1964:
1957:
1943:
1940:
1931:
1927:
1924:
1921:
1917:
1912:
1908:
1902:
1897:
1893:
1888:
1884:
1878:
1874:
1864:
1859:
1857:
1835:
1832:
1826:
1823:
1820:
1814:
1807:
1784:
1774:
1754:
1751:
1746:
1743:
1740:
1735:
1729:
1721:
1704:
1681:
1676:
1673:
1649:
1646:
1640:
1637:
1634:
1628:
1621:
1620:
1618:
1614:
1596:
1593:
1590:
1583:
1566:
1563:
1560:
1557:
1550:
1533:
1530:
1527:
1524:
1517:
1516:
1514:
1510:
1492:
1484:
1473:
1470:
1465:
1458:
1455:
1451:
1448:
1442:
1435:
1432:
1428:
1425:
1419:
1413:
1410:
1407:
1401:
1396:
1393:
1388:
1380:
1376:
1367:
1362:
1351:
1343:
1339:
1330:
1325:
1310:
1306:
1303:
1301:
1295:
1288:
1268:
1265:
1262:
1259:
1236:
1233:
1230:
1227:
1215:
1213:
1191:
1188:
1182:
1179:
1176:
1170:
1163:
1140:
1135:
1132:
1108:
1105:
1100:
1097:
1094:
1089:
1083:
1075:
1058:
1057:
1055:
1051:
1033:
1030:
1027:
1020:
1003:
1000:
997:
994:
987:
986:
984:
980:
962:
954:
943:
940:
935:
928:
925:
921:
918:
912:
906:
903:
900:
894:
889:
886:
881:
873:
869:
860:
855:
840:
836:
833:
831:
812:
809:
801:
795:
790:
786:
780:
760:
757:
754:
751:
739:
737:
735:
719:
711:
695:
667:
664:
658:
655:
652:
646:
639:
625:
622:
616:
613:
610:
604:
584:
581:
575:
572:
569:
563:
556:
555:
553:
549:
531:
528:
525:
518:
504:
501:
498:
495:
492:
485:
484:
482:
478:
460:
452:
441:
438:
433:
426:
423:
419:
416:
410:
404:
401:
398:
392:
387:
384:
379:
371:
367:
358:
353:
338:
334:
331:
329:
325:
321:
316:
312:
305:
300:
295:
291:
281:
276:
274:
265:
251:
248:
245:
242:
239:
233:
225:
216:
209:
206:
205:
201:
187:
184:
178:
170:
157:
154:
153:
149:
135:
132:
129:
122:
119:
118:
114:
111:
108:
107:
103:
100:
97:
96:
90:
87:
83:
79:
71:
69:
67:
63:
59:
55:
51:
47:
46:heat equation
42:
35:
33:
31:
30:heat equation
27:
23:
19:
4627:
4600:
4581:
4563:
4544:
4538:
4526:
4507:
4501:
4489:
4464:
4460:
4450:
4438:
4426:
4415:
4403:
4391:. Retrieved
4382:
4372:November 19,
4370:. Retrieved
4361:
4350:
4333:
4329:
4323:
4306:
4302:
4296:
4279:
4275:
4269:
4244:
4240:
4234:
4217:
4213:
4207:
4190:
4186:
4180:
4161:
4155:
4144:
4099:
3986:is bounded,
3659:
3655:
3650:
3644:
3639:
3635:
3630:
3626:
3616:
3612:
3608:
3595:
3341:is bounded,
2918:
2914:
2908:
2905:
2897:
2493:
2488:
2484:
2479:
2475:
2470:
2466:
2443:
2379:
2230:is bounded,
1919:
1915:
1906:
1900:
1895:
1891:
1886:
1882:
1876:
1872:
1868:
1855:
1293:
1286:
1219:
1211:
799:
793:
788:
784:
778:
743:
687:
323:
319:
314:
310:
303:
293:
289:
285:
271:
112:G is bounded
75:
43:
39:
21:
15:
4431:Luikov 1968
4408:Luikov 1968
4149:Luikov 1968
2847:is bounded,
1805:is bounded,
1702:is bounded,
1161:is bounded,
4650:Categories
4637:0124597564
4171:047105481X
4136:References
712:(m/s) and
36:Background
4261:119549322
4071:τ
4009:∂
4001:∂
3853:∂
3845:∂
3837:α
3813:π
3794:−
3785:δ
3776:τ
3773:−
3764:δ
3759:α
3737:∂
3729:∂
3702:∂
3698:∂
3669:Equation
3567:τ
3532:π
3523:ϕ
3512:ϕ
3509:∂
3501:∂
3480:ϕ
3469:ϕ
3466:∂
3458:∂
3428:π
3419:ϕ
3398:ϕ
3255:π
3246:ϕ
3172:∂
3164:∂
3156:α
3138:ϕ
3134:−
3131:ϕ
3125:δ
3111:π
3092:−
3083:δ
3074:τ
3071:−
3062:δ
3057:α
3037:ϕ
3033:∂
3019:∂
2981:∂
2973:∂
2953:∂
2949:∂
2927:Equation
2869:τ
2833:∞
2830:→
2717:∞
2670:∂
2662:∂
2654:α
2632:π
2613:−
2604:δ
2595:τ
2592:−
2583:δ
2578:α
2556:∂
2548:∂
2528:∂
2524:∂
2502:Equation
2452:∞
2351:τ
2257:∂
2249:∂
2097:∂
2089:∂
2081:α
2059:π
2040:−
2031:δ
2022:τ
2019:−
2010:δ
2005:α
1983:∂
1975:∂
1955:∂
1951:∂
1929:Equation
1827:τ
1791:∞
1788:→
1727:∂
1719:∂
1688:∞
1685:→
1570:∞
1537:∞
1490:∂
1482:∂
1474:α
1452:−
1443:δ
1429:−
1420:δ
1414:τ
1411:−
1402:δ
1397:α
1373:∂
1359:∂
1336:∂
1322:∂
1308:Equation
1272:∞
1240:∞
1183:τ
1147:∞
1144:→
1081:∂
1073:∂
1007:∞
960:∂
952:∂
944:α
922:−
913:δ
907:τ
904:−
895:δ
890:α
866:∂
852:∂
838:Equation
816:∞
764:∞
720:δ
696:α
659:τ
458:∂
450:∂
442:α
420:−
411:δ
405:τ
402:−
393:δ
388:α
364:∂
350:∂
336:Equation
299:Dirichlet
231:∂
223:∂
176:∂
168:∂
120:Dirichlet
58:acoustics
54:diffusion
4481:12677235
4393:March 4,
4104:See also
3801:′
3141:′
3118:′
3099:′
2639:′
2620:′
2066:′
2047:′
1459:′
1436:′
929:′
783:. Here
427:′
318:. Here
72:Notation
52:such as
3875:Domain
3622:Neumann
3611:0 <
3194:Domain
2692:Domain
2424:is the
2119:Domain
1871:0 <
1512:Domain
1298:. The
982:Domain
732:is the
708:is the
480:Domain
288:0 <
155:Neumann
104:Number
84:that a
68:, etc.
28:of the
4634:
4615:
4588:
4551:
4514:
4479:
4259:
4168:
2902:R01φ00
1216:X10Y20
828:. The
20:, the
4477:S2CID
4257:S2CID
3615:<
2380:Here
1911:Robin
1875:<
688:Here
292:<
207:Robin
4632:ISBN
4613:ISBN
4586:ISBN
4549:ISBN
4512:ISBN
4395:2021
4374:2020
4166:ISBN
4068:<
3924:>
3897:<
3891:<
3605:RS02
3564:<
3279:>
3249:<
3243:<
3216:<
3210:<
2866:<
2741:>
2714:<
2708:<
2348:<
2168:>
2141:<
2135:<
1824:<
1594:>
1567:<
1561:<
1534:<
1528:<
1269:<
1263:<
1237:<
1231:<
1180:<
1031:>
1004:<
998:<
798:and
761:<
755:<
656:<
529:>
502:<
496:<
308:and
98:Name
4605:doi
4469:doi
4338:doi
4311:doi
4284:doi
4249:doi
4222:doi
4195:doi
3647:= 0
2911:= a
2440:R10
2400:is
1903:= 0
1865:R03
1296:= 0
1289:= 0
796:= 0
781:= 0
740:X20
306:= 0
282:X11
4652::
4611:.
4475:.
4465:52
4463:.
4459:.
4332:.
4307:42
4305:.
4280:45
4278:.
4255:.
4245:33
4243:.
4218:31
4216:.
4191:24
4189:.
3658:=
3636:RS
3629:=
3443:,
2919:00
2809:,
2487:=
2469:=
1918:=
1885:=
1767:,
1662:,
1252:,
1121:,
597:,
324:11
313:=
266:3
202:2
150:1
115:0
64:,
60:,
56:,
4640:.
4621:.
4607::
4594:.
4557:.
4520:.
4483:.
4471::
4397:.
4376:.
4344:.
4340::
4334:5
4317:.
4313::
4290:.
4286::
4263:.
4251::
4228:.
4224::
4201:.
4197::
4174:.
4080:0
4077:=
4074:)
4065:t
4062:(
4059:G
4049:,
4037:0
4034:=
4029:a
4026:=
4023:r
4018:|
4012:n
4004:G
3974:)
3971:0
3968:=
3965:r
3962:(
3959:G
3927:0
3921:t
3900:b
3894:r
3888:0
3856:t
3848:G
3834:1
3829:=
3821:2
3817:r
3810:4
3805:)
3798:r
3791:r
3788:(
3779:)
3770:t
3767:(
3756:1
3751:+
3747:)
3740:r
3732:G
3721:2
3717:r
3712:(
3705:r
3689:2
3685:r
3681:1
3660:b
3656:r
3651:2
3645:r
3640:0
3631:b
3627:r
3617:b
3613:r
3576:0
3573:=
3570:)
3561:t
3558:(
3555:G
3529:2
3526:=
3518:|
3504:G
3491:=
3486:0
3483:=
3475:|
3461:G
3431:)
3425:2
3422:=
3416:(
3413:G
3410:=
3407:)
3404:0
3401:=
3395:(
3392:G
3382:,
3370:0
3367:=
3364:)
3361:a
3358:=
3355:r
3352:(
3349:G
3329:)
3326:0
3323:=
3320:r
3317:(
3314:G
3282:0
3276:t
3252:2
3240:0
3219:a
3213:r
3207:0
3175:t
3167:G
3153:1
3148:=
3145:)
3128:(
3115:r
3108:2
3103:)
3096:r
3089:r
3086:(
3077:)
3068:t
3065:(
3054:1
3049:+
3041:2
3028:G
3023:2
3008:2
3004:r
3000:1
2995:+
2991:)
2984:r
2976:G
2967:r
2963:(
2956:r
2942:r
2939:1
2915:φ
2909:r
2878:0
2875:=
2872:)
2863:t
2860:(
2857:G
2827:r
2822:|
2817:G
2797:0
2794:=
2791:)
2788:a
2785:=
2782:r
2779:(
2776:G
2744:0
2738:t
2711:r
2705:a
2673:t
2665:G
2651:1
2646:=
2636:r
2629:2
2624:)
2617:r
2610:r
2607:(
2598:)
2589:t
2586:(
2575:1
2570:+
2566:)
2559:r
2551:G
2542:r
2538:(
2531:r
2517:r
2514:1
2494:0
2489:a
2485:r
2480:1
2476:R
2471:a
2467:r
2412:h
2388:k
2360:0
2357:=
2354:)
2345:t
2342:(
2339:G
2329:,
2317:0
2314:=
2309:a
2306:=
2303:r
2298:|
2294:G
2291:h
2283:+
2278:a
2275:=
2272:r
2267:|
2260:n
2252:G
2243:k
2218:)
2215:0
2212:=
2209:r
2206:(
2203:G
2171:0
2165:t
2144:a
2138:r
2132:0
2100:t
2092:G
2078:1
2073:=
2063:r
2056:2
2051:)
2044:r
2037:r
2034:(
2025:)
2016:t
2013:(
2002:1
1997:+
1993:)
1986:r
1978:G
1969:r
1965:(
1958:r
1944:r
1941:1
1920:a
1916:r
1907:3
1901:r
1896:0
1892:R
1887:a
1883:r
1877:a
1873:r
1836:0
1833:=
1830:)
1821:t
1818:(
1815:G
1785:y
1780:|
1775:G
1755:0
1752:=
1747:0
1744:=
1741:y
1736:|
1730:n
1722:G
1682:x
1677:|
1674:G
1650:0
1647:=
1644:)
1641:0
1638:=
1635:x
1632:(
1629:G
1597:0
1591:t
1564:y
1558:0
1531:x
1525:0
1493:t
1485:G
1471:1
1466:=
1463:)
1456:y
1449:y
1446:(
1440:)
1433:x
1426:x
1423:(
1417:)
1408:t
1405:(
1394:1
1389:+
1381:2
1377:y
1368:G
1363:2
1352:+
1344:2
1340:x
1331:G
1326:2
1294:y
1287:x
1266:y
1260:0
1234:x
1228:0
1192:0
1189:=
1186:)
1177:t
1174:(
1171:G
1141:x
1136:|
1133:G
1109:0
1106:=
1101:0
1098:=
1095:x
1090:|
1084:n
1076:G
1034:0
1028:t
1001:x
995:0
963:t
955:G
941:1
936:=
933:)
926:x
919:x
916:(
910:)
901:t
898:(
887:1
882:+
874:2
870:x
861:G
856:2
813:=
810:x
800:0
794:x
789:2
785:X
779:x
758:x
752:0
668:0
665:=
662:)
653:t
650:(
647:G
626:0
623:=
620:)
617:L
614:=
611:x
608:(
605:G
585:0
582:=
579:)
576:0
573:=
570:x
567:(
564:G
532:0
526:t
505:L
499:x
493:0
461:t
453:G
439:1
434:=
431:)
424:x
417:x
414:(
408:)
399:t
396:(
385:1
380:+
372:2
368:x
359:G
354:2
320:X
315:L
311:x
304:x
294:L
290:x
252:0
249:=
246:G
243:h
240:+
234:n
226:G
217:k
188:0
185:=
179:n
171:G
136:0
133:=
130:G
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