20:
803:. A boundary value problem has conditions specified at the extremes ("boundaries") of the independent variable in the equation whereas an initial value problem has all of the conditions specified at the same value of the independent variable (and that value is at the lower boundary of the domain, thus the term "initial" value). A
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If the problem is dependent on both space and time, one could specify the value of the problem at a given point for all time or at a given time for all space.
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Aside from the boundary condition, boundary value problems are also classified according to the type of differential operator involved. For an
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Boundary value problems arise in several branches of physics as any physical differential equation will have them. Problems involving the
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If the boundary has the form of a curve or surface that gives a value to the normal derivative and the variable itself then it is a
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is devoted to proving that boundary value problems arising from scientific and engineering applications are in fact well-posed.
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is a data value that corresponds to a minimum or maximum input, internal, or output value specified for a system or component.
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For example, if the independent variable is time over the domain , a boundary value problem would specify values for
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One sees that imposing boundary conditions allowed one to determine a unique solution, which in this case is
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Finding a function to describe the temperature of this idealised 2D rod is a boundary value problem with
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and the other end at the freezing point of water would be a boundary value problem.
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Linear
Partial Differential Equations: Exact Solutions and Boundary Value Problems
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Concretely, an example of a boundary value problem (in one spatial dimension) is
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Handbook of Linear
Partial Differential Equations for Engineers and Scientists
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Handbook of Exact
Solutions for Ordinary Differential Equations (2nd edition)
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A boundary condition which specifies the value of the function itself is a
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Without the boundary conditions, the general solution to this equation is
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Finding the temperature at all points of an iron bar with one end kept at
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ISO/IEC/IEEE International
Standard - Systems and software engineering
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753:. The analysis of these problems, in the linear case, involves the
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specified by the boundary conditions, and known scalar functions
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To be useful in applications, a boundary value problem should be
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Among the earliest boundary value problems to be studied is the
2093:, a common problem is to find a function which describes the
891:, whereas an initial value problem would specify a value of
2292:. ISO/IEC/IEEE 24765:2010(E). pp. vol., no., pp.1-418.
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1955:{\displaystyle c_{0}y+c_{1}{\partial y \over \partial n}=g}
1851:{\displaystyle c_{0}y+c_{1}{\partial y \over \partial n}=f}
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Summary of boundary conditions for the unknown function,
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Computation of radiowave attenuation in the atmosphere
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A boundary condition which specifies the value of the
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2363:"Boundary value problem, complex-variable methods"
2325:, Chapman & Hall/CRC Press, Boca Raton, 2002.
2310:, Chapman & Hall/CRC Press, Boca Raton, 2003.
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2382:at EqWorld: The World of Mathematical Equations.
2147:Stochastic processes and boundary value problems
2105:). The boundary conditions in this case are the
2113:in the region, it is also possible to define a
2107:Interface conditions for electromagnetic fields
2066:. These categories are further subdivided into
2033:{\displaystyle {\partial y \over \partial n}=g}
1774:{\displaystyle {\partial y \over \partial n}=f}
2345:"Boundary value problems in potential theory"
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2085:Laplace's equation § Boundary conditions
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1531:. Any solution function will both solve the
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27:is valid and the associated boundary values
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18:
1226:{\displaystyle y(x)=A\sin(x)+B\cos(x).}
799:Boundary value problems are similar to
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1683:specified by the boundary conditions.
1054:to be solved for the unknown function
16:Type of problem involving ODEs or PDEs
1146:{\displaystyle y(0)=0,\ y(\pi /2)=2.}
2125:
158:List of named differential equations
2268:. Elsevier Science. pp. 536–.
1311:{\displaystyle 0=A\cdot 0+B\cdot 1}
231:Dependent and independent variables
13:
2306:A. D. Polyanin and V. F. Zaitsev,
2265:Handbook of Differential Equations
2064:hyperbolic boundary value problems
2015:
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2262:Daniel Zwillinger (12 May 2014).
2162:Perfect thermal contact condition
787:); the solution was given by the
2056:elliptic boundary value problems
1514:Types of boundary value problems
734:subjected to constraints called
366:Carathéodory's existence theorem
2415:Ordinary differential equations
2220:Direct multiple shooting method
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745:, such as the determination of
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2157:Sommerfeld radiation condition
1503:{\displaystyle y(x)=2\sin(x).}
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770:partial differential equations
453: / Integral solutions
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2070:and various nonlinear types.
1695:Form on 2nd part of boundary
1692:Form on 1st part of boundary
1529:Dirichlet boundary conditions
1083:with the boundary conditions
1044:{\displaystyle y''(x)+y(x)=0}
2152:Sturm–Liouville theory
1541:Dirichlet boundary condition
1347:From the boundary condition
1236:From the boundary condition
497:Exponential response formula
243:Coupled / Decoupled
7:
2368:Encyclopedia of Mathematics
2350:Encyclopedia of Mathematics
2120:
2117:using a similar procedure.
1564:
1383:{\displaystyle y(\pi /2)=2}
10:
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1552:Neumann boundary condition
1418:{\displaystyle 2=A\cdot 1}
2115:magnetic scalar potential
2079:Electromagnetic potential
1970:
1789:
1738:
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1559:Cauchy boundary condition
1519:Boundary value conditions
631:Józef Maria Hoene-Wroński
577:Undetermined coefficients
486:Method of characteristics
371:Cauchy–Kowalevski theorem
2387:"Boundary value problem"
2248:
2230:Finite difference method
751:Sturm–Liouville problems
356:Picard–Lindelöf theorem
350:Existence and uniqueness
2410:Boundary value problems
2235:Boundary element method
582:Variation of parameters
572:Separation of variables
361:Peano existence theorem
23:Shows a region where a
2225:Walk-on-spheres method
2170:Physical applications:
2046:Differential operators
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1264:{\displaystyle y(0)=0}
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801:initial value problems
728:boundary-value problem
724:differential equations
651:Carl David Tolmé Runge
194:Differential-algebraic
35:Differential equations
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2209:Numerical algorithms:
2137:Initial value problem
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789:Dirichlet's principle
759:differential operator
732:differential equation
641:Augustin-Louis Cauchy
626:Joseph-Louis Lagrange
458:Numerical integration
440:Exponential stability
303:Relation to processes
25:differential equation
22:
2131:Related mathematics:
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843:
832:{\displaystyle y(t)}
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463:Dirac delta function
199:Integro-differential
2420:Boundary conditions
2060:hyperbolic operator
1990:{\displaystyle y=f}
1885:{\displaystyle y=f}
1723:{\displaystyle y=f}
1321:which implies that
973:{\displaystyle t=0}
884:{\displaystyle t=1}
858:{\displaystyle t=0}
736:boundary conditions
559:Perturbation theory
554:Integral transforms
445:Rate of convergence
311:(discrete analogue)
148:Population dynamics
115:Continuum mechanics
106:Applied mathematics
2099:Laplace's equation
2095:electric potential
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549:Integrating factor
390:Initial conditions
325:Stochastic partial
29:
2275:978-1-4832-2096-3
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2109:. If there is no
2103:harmonic function
2052:elliptic operator
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1944:
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1676:{\displaystyle g}
1656:{\displaystyle f}
1582:{\displaystyle y}
1548:normal derivative
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779:, of finding the
777:Dirichlet problem
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611:Gottfried Leibniz
502:Finite difference
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125:Dynamical systems
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2321:A. D. Polyanin,
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2191:Potential theory
2142:Green's function
2126:
2062:, one discusses
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602:
601:
598:
597:
594:
593:
589:
588:
585:
584:
579:
574:
567:
566:
561:
556:
551:
546:
541:
540:
539:
529:
524:
523:
522:
515:Finite element
512:
508:Crank–Nicolson
499:
494:
488:
483:
479:
478:
475:
474:
471:
470:
466:
465:
460:
455:
447:
442:
429:
427:Phase portrait
424:
419:
418:
417:
415:Cauchy problem
412:
407:
402:
392:
386:
385:
383:General topics
382:
381:
378:
377:
374:
373:
368:
363:
358:
352:
349:
348:
345:
344:
340:
339:
335:
334:
329:
328:
327:
316:
315:
314:
305:
302:
301:
298:
297:
292:
291:
290:
289:
282:
281:
276:
270:
267:
266:
262:
261:
259:
258:
256:Nonhomogeneous
249:
244:
241:
235:
234:
233:
225:
224:
220:
219:
217:
216:
211:
206:
201:
196:
191:
186:
180:
175:
172:
171:
168:
167:
166:Classification
163:
162:
153:
152:
151:
150:
145:
137:
136:
130:
129:
128:
127:
122:
117:
109:
108:
102:
101:
100:
99:
94:
88:
83:
78:
70:
69:
67:
66:
61:
55:
50:
47:
46:
43:
42:
38:
37:
15:
9:
6:
4:
3:
2:
2437:
2426:
2423:
2421:
2418:
2416:
2413:
2411:
2408:
2407:
2405:
2394:
2393:
2388:
2384:
2381:
2378:
2374:
2370:
2369:
2364:
2360:
2356:
2352:
2351:
2346:
2342:
2341:
2332:
2331:1-58488-299-9
2328:
2324:
2320:
2317:
2316:1-58488-297-2
2313:
2309:
2305:
2304:
2291:
2285:
2277:
2271:
2267:
2266:
2258:
2254:
2241:
2236:
2233:
2231:
2228:
2226:
2223:
2221:
2218:
2216:
2213:
2212:
2211:
2210:
2206:
2202:
2199:
2197:
2194:
2192:
2189:
2187:
2184:
2182:
2179:
2177:
2174:
2173:
2172:
2171:
2167:
2163:
2160:
2158:
2155:
2153:
2150:
2148:
2145:
2143:
2140:
2138:
2135:
2134:
2133:
2132:
2128:
2127:
2118:
2116:
2112:
2108:
2104:
2101:(a so-called
2100:
2096:
2092:
2086:
2071:
2069:
2065:
2061:
2057:
2053:
2027:
2024:
2018:
2010:
1984:
1981:
1978:
1968:
1965:
1964:
1949:
1946:
1940:
1932:
1921:
1917:
1913:
1910:
1905:
1901:
1893:
1879:
1876:
1873:
1866:
1864:
1861:
1860:
1845:
1842:
1836:
1828:
1817:
1813:
1809:
1806:
1801:
1797:
1787:
1784:
1783:
1768:
1765:
1759:
1751:
1736:
1733:
1732:
1717:
1714:
1711:
1702:
1699:
1698:
1694:
1691:
1688:
1687:
1684:
1670:
1650:
1628:
1624:
1601:
1597:
1576:
1562:
1560:
1555:
1553:
1549:
1544:
1542:
1534:
1533:heat equation
1530:
1525:
1497:
1491:
1485:
1482:
1479:
1476:
1470:
1464:
1457:
1456:
1455:
1441:
1438:
1435:
1412:
1409:
1406:
1403:
1400:
1393:
1392:
1391:
1377:
1374:
1368:
1364:
1360:
1354:
1334:
1331:
1328:
1305:
1302:
1299:
1296:
1293:
1290:
1287:
1284:
1281:
1274:
1273:
1272:
1258:
1255:
1249:
1243:
1220:
1214:
1208:
1205:
1202:
1199:
1193:
1187:
1184:
1181:
1178:
1172:
1166:
1159:
1158:
1157:
1140:
1137:
1131:
1127:
1123:
1117:
1111:
1108:
1105:
1099:
1093:
1086:
1085:
1084:
1067:
1061:
1038:
1035:
1029:
1023:
1020:
1014:
1007:
1004:
996:
995:
994:
991:
988:
986:
985:absolute zero
981:
967:
964:
961:
938:
931:
928:
904:
898:
878:
875:
872:
852:
849:
846:
823:
817:
808:
806:
802:
792:
790:
786:
782:
778:
773:
771:
767:
762:
760:
756:
752:
748:
744:
743:wave equation
739:
737:
733:
729:
725:
713:
708:
706:
701:
699:
694:
693:
691:
690:
682:
679:
677:
674:
672:
669:
667:
664:
662:
659:
657:
654:
652:
649:
647:
644:
642:
639:
637:
634:
632:
629:
627:
624:
622:
619:
617:
614:
612:
609:
607:
604:
603:
596:
595:
591:
590:
583:
580:
578:
575:
573:
570:
569:
565:
562:
560:
557:
555:
552:
550:
547:
545:
542:
538:
535:
534:
533:
530:
528:
527:Finite volume
525:
521:
518:
517:
516:
513:
509:
503:
500:
498:
495:
493:
489:
487:
484:
481:
480:
473:
472:
464:
461:
459:
456:
452:
448:
446:
443:
441:
437:
433:
430:
428:
425:
423:
420:
416:
413:
411:
408:
406:
403:
401:
398:
397:
396:
393:
391:
388:
387:
380:
379:
372:
369:
367:
364:
362:
359:
357:
354:
353:
347:
346:
342:
341:
333:
330:
326:
323:
322:
321:
318:
317:
313:
307:
306:
300:
299:
288:
285:
284:
280:
277:
275:
272:
271:
269:
268:
264:
263:
257:
253:
250:
248:
245:
242:
240:
237:
236:
232:
229:
228:
227:
226:
222:
221:
215:
212:
210:
207:
205:
202:
200:
197:
195:
192:
190:
187:
185:
182:
181:
179:
178:
170:
169:
165:
164:
159:
149:
146:
144:
141:
140:
139:
138:
135:
132:
131:
126:
123:
121:
118:
116:
113:
112:
111:
110:
107:
104:
103:
98:
95:
93:
89:
87:
84:
82:
79:
77:
74:
73:
72:
71:
65:
62:
60:
57:
56:
54:
53:
45:
44:
40:
39:
36:
33:
32:
26:
21:
2392:Scholarpedia
2390:
2366:
2348:
2322:
2307:
2289:
2284:
2264:
2257:
2239:
2208:
2207:
2169:
2168:
2130:
2129:
2088:
2074:Applications
2049:
1589:, constants
1568:
1556:
1545:
1538:
1427:
1320:
1271:one obtains
1235:
1155:
1053:
992:
989:
982:
809:
804:
798:
774:
763:
747:normal modes
740:
735:
727:
721:
671:Émile Picard
656:Martin Kutta
646:George Green
606:Isaac Newton
438: /
434: /
394:
254: /
120:Chaos theory
2181:Normal mode
795:Explanation
564:Runge–Kutta
309:Difference
252:Homogeneous
64:Engineering
2404:Categories
2301:References
2201:Black hole
2083:See also:
1390:one finds
766:well posed
681:John Crank
482:Inspection
436:Asymptotic
320:Stochastic
239:Autonomous
214:Non-linear
204:Fractional
2373:EMS Press
2355:EMS Press
2016:∂
2008:∂
1938:∂
1930:∂
1834:∂
1826:∂
1757:∂
1749:∂
1701:Dirichlet
1486:
1410:⋅
1361:π
1303:⋅
1291:⋅
1209:
1188:
1124:π
422:Wronskian
400:Dirichlet
143:Economics
86:Chemistry
76:Astronomy
2121:See also
2058:. For a
1565:Examples
1008:″
954:at time
932:′
839:at both
532:Galerkin
432:Lyapunov
343:Solution
287:Notation
279:Operator
265:Features
184:Ordinary
2375:, 2001
2357:, 2001
1735:Neumann
1428:and so
405:Neumann
189:Partial
97:Geology
92:Biology
81:Physics
2329:
2314:
2272:
2240:
2068:linear
1967:Cauchy
1115:
592:People
504:
451:Series
209:Linear
48:Fields
2249:Notes
2176:Waves
1971:both
1863:Mixed
1786:Robin
1689:Name
757:of a
730:is a
492:Euler
410:Robin
332:Delay
274:Order
247:Exact
173:Types
41:Scope
2327:ISBN
2312:ISBN
2270:ISBN
1997:and
1663:and
1616:and
920:and
865:and
726:, a
599:List
2089:In
1483:sin
1206:cos
1185:sin
2406::
2389:.
2371:,
2365:,
2353:,
2347:,
1561:.
1442:2.
1335:0.
1141:2.
980:.
791:.
761:.
2395:.
2333:.
2318:.
2278:.
2028:g
2025:=
2019:n
2011:y
1985:f
1982:=
1979:y
1950:g
1947:=
1941:n
1933:y
1922:1
1918:c
1914:+
1911:y
1906:0
1902:c
1880:f
1877:=
1874:y
1846:f
1843:=
1837:n
1829:y
1818:1
1814:c
1810:+
1807:y
1802:0
1798:c
1769:f
1766:=
1760:n
1752:y
1718:f
1715:=
1712:y
1671:g
1651:f
1629:1
1625:c
1602:0
1598:c
1577:y
1498:.
1495:)
1492:x
1489:(
1480:2
1477:=
1474:)
1471:x
1468:(
1465:y
1439:=
1436:A
1413:1
1407:A
1404:=
1401:2
1378:2
1375:=
1372:)
1369:2
1365:/
1358:(
1355:y
1332:=
1329:B
1306:1
1300:B
1297:+
1294:0
1288:A
1285:=
1282:0
1259:0
1256:=
1253:)
1250:0
1247:(
1244:y
1221:.
1218:)
1215:x
1212:(
1203:B
1200:+
1197:)
1194:x
1191:(
1182:A
1179:=
1176:)
1173:x
1170:(
1167:y
1138:=
1135:)
1132:2
1128:/
1121:(
1118:y
1112:,
1109:0
1106:=
1103:)
1100:0
1097:(
1094:y
1071:)
1068:x
1065:(
1062:y
1039:0
1036:=
1033:)
1030:x
1027:(
1024:y
1021:+
1018:)
1015:x
1012:(
1005:y
968:0
965:=
962:t
942:)
939:t
936:(
929:y
908:)
905:t
902:(
899:y
879:1
876:=
873:t
853:0
850:=
847:t
827:)
824:t
821:(
818:y
711:e
704:t
697:v
510:)
506:(
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