632:
711:, respectively. This case would not be typically included under the umbrella of Fredholm integral equations, a name that is usually reserved for when the integral operator defines a compact operator (convolution operators on non-compact groups are non-compact, since, in general, the spectrum of the operator of convolution with
1047:. In physics, the solution of such integral equations allows for experimental spectra to be related to various underlying distributions, for instance the mass distribution of polymers in a polymeric melt, or the distribution of relaxation times in the system. In addition, Fredholm integral equations also arise in
344:
1058:
A specific application of
Fredholm equation is the generation of photo-realistic images in computer graphics, in which the Fredholm equation is used to model light transport from the virtual light sources to the image plane. The Fredholm equation is often called the
627:{\displaystyle f(s)={\mathcal {F}}_{\omega }^{-1}\left(\omega ) \over {\mathcal {F}}_{t}(\omega )}\right]=\int _{-\infty }^{\infty }{{\mathcal {F}}_{t}(\omega ) \over {\mathcal {F}}_{t}(\omega )}e^{2\pi i\omega s}\mathrm {d} \omega }
876:
165:
705:
666:
758:
60:
A Fredholm equation is an integral equation in which the term containing the kernel function (defined below) has constants as integration limits. A closely related form is the
292:
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914:
943:
729:
336:
316:
233:
213:
193:
774:
1221:
Daddi-Moussa-Ider, A. (25 November 2020). "Asymmetric Stokes flow induced by a transverse point force acting near a finite-sized elastic membrane".
1137:
SchÀfer, H.; Sternin, E.; Stannarius, R.; Arndt, M.; Kremer, F. (18 March 1996). "Novel
Approach to the Analysis of Broadband Dielectric Spectra".
238:
An important case of these types of equation is the case when the kernel is a function only of the difference of its arguments, namely
78:
1361:
1373:
1398:
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671:
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Mathews, Jon; Walker, Robert L. (1970), Mathematical methods of physics (2nd ed.), New York: W. A. Benjamin,
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45:
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640:
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294:, and the limits of integration are 屉, then the right hand side of the equation can be rewritten as a
172:
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1283:
760:, which is usually a non-countable set, whereas compact operators have discrete countable spectra).
1425:
1300:
Simons, F. J.; Wieczorek, M. A.; Dahlen, F. A. (2006). "Spatiospectral concentration on a sphere".
241:
948:
68:
1184:(9 April 2019). "Axisymmetric flow due to a Stokeslet near a finite-sized elastic membrane".
884:
1319:
1146:
1111:
1102:
Honerkamp, J.; Weese, J. (1990). "Tikhonovs regularization method for ill-posed problems".
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8:
978:
1323:
1150:
1115:
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871:{\displaystyle \varphi (t)=f(t)+\lambda \int _{a}^{b}K(t,s)\varphi (s)\,\mathrm {d} s.}
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1394:
1369:
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1028:
37:
25:
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Slepian, D. (1983). "Some comments on
Fourier Analysis, uncertainty and modeling".
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41:
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IntEQ: a Python package for numerically solving
Fredholm integral equations
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1410:
977:
A standard approach to solving this is to use iteration, amounting to the
295:
17:
1123:
1016:
1314:
1352:
1235:
1198:
768:
An inhomogeneous
Fredholm equation of the second kind is given as
993:
The general theory underlying the
Fredholm equations is known as
1359:
1360:
Press, WH; Teukolsky, SA; Vetterling, WT; Flannery, BP (2007).
1051:
problems involving hydrodynamic interactions near finite-sized
1136:
160:{\displaystyle g(t)=\int _{a}^{b}K(t,s)f(s)\,\mathrm {d} s~,}
1179:
1043:. They also commonly arise in linear forward modeling and
1015:
that can be understood in terms of a discrete spectrum of
1368:(3rd ed.). New York: Cambridge University Press.
1362:"Section 19.1. Fredholm Equations of the Second Kind"
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Fredholm equations arise naturally in the theory of
997:. One of the principal results is that the kernel
981:; written as a series, the solution is known as the
1281:
338:and therefore, formally, the solution is given by
1366:Numerical Recipes: The Art of Scientific Computing
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71:Fredholm equation of the first kind is written as
1220:
945:, the problem is typically to find the function
1417:
1264:at EqWorld: The World of Mathematical Equations.
44:. A useful method to solve such equations, the
1101:
763:
700:{\displaystyle {\mathcal {F}}_{\omega }^{-1}}
55:
1282:Khvedelidze, B.V.; Litvinov, G.L. (2001) ,
1313:
1234:
1197:
1039:. The operators involved are the same as
856:
171:and the problem is, given the continuous
142:
1223:Journal of the Physical Society of Japan
1186:Journal of the Physical Society of Japan
1007:. Compactness may be shown by invoking
40:. The integral equation was studied by
1338:
1418:
1104:Continuum Mechanics and Thermodynamics
64:which has variable integral limits.
13:
1267:A.D. Polyanin and A.V. Manzhirov,
1255:
1180:Daddi-Moussa-Ider, A.; Kaoui, B.;
858:
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661:{\displaystyle {\mathcal {F}}_{t}}
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553:
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438:
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366:
144:
14:
1442:
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988:
753:{\displaystyle {\mathcal {F}}{K}}
1271:, CRC Press, Boca Raton, 1998.
1022:
1269:Handbook of Integral Equations
1214:
1173:
1130:
1095:
1033:spectral concentration problem
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1:
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287:{\displaystyle K(t,s)=K(t-s)}
28:whose solution gives rise to
1031:, for example as the famous
46:Adomian decomposition method
7:
1289:Encyclopedia of Mathematics
1159:10.1103/PhysRevLett.76.2177
1066:
1011:. As an operator, it has a
967:{\displaystyle \varphi (t)}
764:Equation of the second kind
707:are the direct and inverse
10:
1447:
1078:Volterra integral equation
62:Volterra integral equation
56:Equation of the first kind
22:Fredholm integral equation
1332:10.1137/S0036144504445765
1073:LiouvilleâNeumann series
983:LiouvilleâNeumann series
1139:Physical Review Letters
215:, to find the function
1245:10.7566/JPSJ.89.124401
1208:10.7566/JPSJ.88.054401
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939:
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909:{\displaystyle K(t,s)}
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731:contains the range of
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1324:2006SIAMR..48..504S
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1116:1990CMT.....2...17H
979:resolvent formalism
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1431:Integral equations
1262:Integral Equations
1124:10.1007/BF01170953
1061:rendering equation
1053:elastic interfaces
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38:Fredholm operators
1375:978-0-521-88068-8
1284:"Fredholm kernel"
1145:(12): 2177â2180.
1063:in this context.
1029:signal processing
724:{\displaystyle K}
592:
477:
331:{\displaystyle f}
311:{\displaystyle K}
298:of the functions
228:{\displaystyle f}
208:{\displaystyle g}
195:and the function
188:{\displaystyle K}
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26:integral equation
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1378:. Archived from
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34:Fredholm kernels
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1426:Fredholm theory
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1353:10.1137/1025078
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1256:Further reading
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32:, the study of
30:Fredholm theory
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1406:
1405:External links
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69:inhomogeneous
65:
63:
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43:
42:Ivar Fredholm
39:
35:
31:
27:
23:
19:
1384:. Retrieved
1380:the original
1365:
1344:
1340:
1315:math/0408424
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1110:(1): 17â30.
1107:
1103:
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1026:
1023:Applications
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976:
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237:
170:
66:
59:
48:, is due to
21:
15:
1341:SIAM Review
1302:SIAM Review
1017:eigenvalues
296:convolution
18:mathematics
1420:Categories
1386:2011-08-17
1236:2006.14375
1229:: 124401.
1199:1901.04485
1089:References
1294:EMS Press
1182:Löwen, H.
1003:yields a
953:φ
845:φ
813:∫
809:λ
779:φ
690:−
685:ω
622:ω
609:ω
603:π
586:ω
543:ω
500:∞
495:∞
492:−
488:∫
471:ω
428:ω
378:−
373:ω
276:−
175:function
99:∫
1167:10060625
1067:See also
1320:Bibcode
1147:Bibcode
1112:Bibcode
1397:
1372:
1275:
1165:
637:where
173:kernel
152:
24:is an
20:, the
1310:arXiv
1231:arXiv
1194:arXiv
1395:ISBN
1370:ISBN
1273:ISBN
1163:PMID
668:and
318:and
36:and
1349:doi
1328:doi
1241:doi
1204:doi
1155:doi
1120:doi
67:An
16:In
1422::
1364:.
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1343:.
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1304:.
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1000:K
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956:(
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927:(
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892:(
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863:s
859:d
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848:(
842:)
839:s
836:,
833:t
830:(
827:K
822:b
817:a
806:+
803:)
800:t
797:(
794:f
791:=
788:)
785:t
782:(
747:K
741:F
719:K
693:1
679:F
654:t
648:F
618:d
612:s
606:i
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596:e
589:)
583:(
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571:(
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565:[
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546:)
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531:t
528:(
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522:[
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484:=
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468:(
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459:t
456:(
453:K
450:[
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431:)
425:(
422:]
419:)
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413:(
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387:[
381:1
367:F
361:=
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355:s
352:(
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326:f
306:K
282:)
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273:t
270:(
267:K
264:=
261:)
258:s
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249:(
246:K
223:f
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183:K
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149:s
145:d
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137:s
134:(
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128:)
125:s
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119:t
116:(
113:K
108:b
103:a
95:=
92:)
89:t
86:(
83:g
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