Linear integral equations of the third kind arising from neutron transport theory
The aim of this paper is to study, in the class of Hölder functions, linear integral equations with coefficients cos x having zeros in an interval under consideration. Using the theory of singular integral equations, necessary and sufficient conditions for the solvability of these equations under so...
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| Vydané v: | Mathematical methods in the applied sciences Ročník 30; číslo 15; s. 1941 - 1964 |
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| Médium: | Journal Article |
| Jazyk: | English |
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Chichester, UK
John Wiley & Sons, Ltd
01.10.2007
Wiley Teubner |
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| ISSN: | 0170-4214, 1099-1476 |
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| Abstract | The aim of this paper is to study, in the class of Hölder functions, linear integral equations with coefficients cos x having zeros in an interval under consideration. Using the theory of singular integral equations, necessary and sufficient conditions for the solvability of these equations under some assumptions on their kernels are given. The kernels of transport equations satisfy the required assumptions. Copyright © 2007 John Wiley & Sons, Ltd. |
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| AbstractList | The aim of this paper is to study, in the class of Holder functions, linear integral equations with coefficients cos x having zeros in an interval under consideration. Using the theory of singular integral equations, necessary and sufficient conditions for the solvability of these equations under some assumptions on their kernels are given. The kernels of transport equations satisfy the required assumptions. The aim of this paper is to study, in the class of Hölder functions, linear integral equations with coefficients cos x having zeros in an interval under consideration. Using the theory of singular integral equations, necessary and sufficient conditions for the solvability of these equations under some assumptions on their kernels are given. The kernels of transport equations satisfy the required assumptions. Copyright © 2007 John Wiley & Sons, Ltd. The aim of this paper is to study, in the class of Hölder functions, linear integral equations with coefficients cos x having zeros in an interval under consideration. Using the theory of singular integral equations, necessary and sufficient conditions for the solvability of these equations under some assumptions on their kernels are given. The kernels of transport equations satisfy the required assumptions. Copyright © 2007 John Wiley & Sons, Ltd. |
| Author | Shulaia, D. |
| Author_xml | – sequence: 1 givenname: D. surname: Shulaia fullname: Shulaia, D. email: dazshul@yahoo.com organization: I.Vekua Institute of Applied Mathematics, Tbilisi State University, University st.2, Tbilisi 0143, Georgia |
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| Cites_doi | 10.1515/GMJ.2002.179 10.1007/s10625-006-0013-4 10.1137/0504053 10.1081/TT-120037806 10.1063/1.3062231 10.1080/00411450008214531 10.1007/BF01448941 10.24033/asens.641 10.1023/A:1022928500444 10.1016/0003-4916(60)90060-9 10.1090/psapm/011/0119519 |
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| Keywords | Transport equation holomorphic function singular operator Linear equation Necessary and sufficient condition Applied mathematics Singular equation limiting values Integral equation Mathematical method |
| Language | English |
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| References | Shulaia DA. Linear equation of the multivelocity transport theory. Jurnal Vichislitelnoi Matematiki i Matematicheskoi Phisiki 1983; 23(5):1125-1140 (Russian). Case KM. Elementary solutions of the transport equation and their applications. Annals of Physics 1960; 9:1-23. Shulaia DA. Completeness theorems in linear multivelocity transport theory. Doklady Akademii Nauk SSSR 1981; 259(2):332-335 (Russian). Picard E. Les équations integrales de troisième espèce. Annales Scientifigues De L Ecole Normale Superieure 1911; 28(3):459-472 (French). Hilbert D. Grundzüge einer Allgemeinen Theorie der Linearen Integralgleichungen. B.G. Teubner: Leipzig/Berlin, 1912 (German). Titchmarsh EC. Eigenfunction Expansions Associated with Second-Order Differential Equations. Clarendon Press: Oxford, 1958. Lovitt WV. Linear Integral Equations. Mc Graw-Hill: New York, 1924. Fadeyev LD. On Friedrichs model in the theory of perturbations of a continuous spectrum. Trudy Matematicheskogo Instituta Imeni V.A. Steklova 1964; 73:292-313 (Russian). Bart G, Warnock R. Linear equations of the third kind. SIAM Journal on Mathematical Analysis 1973; 4(4):609-622. Shulaia DA. Inverse problem of the linear multivelocity transport theory. Doklady Akademii Nauk SSSR 1983; 270(1):82-87 (Russian). Germogenova TA, Shulaia DA. On the characteristic equation of the radiation transport theory. Doklady Akademii Nauk SSSR 1976; 231(4):841-844 (Russian). Feldman IA. On the discrete spectrum of the characteristic equation of the radiation transport theory. Matematicheskie Issledovaniya 1975; 10 N 1(35):236-242 (Russian). Tricomi FG. Integral Equations. Interscience Publishers: New York, 1957. Shulaia DA. On the expansion of solutions of equations of the linear multivelocity transport theory by eigenfunctions of the characteristic equation. Doklady Akademii Nauk SSSR 1990; 310(4):844-849 (Russian). Friedrichs KO. Über die Spektralzerlegung eines Integraloperators. Mathematische Annalen 1938; 115(2):249-300 (German). Shulaia D, Sharashidze N. A spectral representation of the multigroup transport problem. Transport Theory and Statistical Physics 2004; 33(2):183-202. Tamarkin J. On Fredholm integral equations whose kernels are analytical in a parameter. Annals of Mathematical Series 1927; 2(28):127-152. Davison D. Neutron Transport Theory. Oxford University Press: Oxford, 1957. Muskhelishvili N. Singular Integral Equations. P. Noordhooff: Groningen, 1953. Shulaia DA, Gugushvili EI. Inverse problem of spectral analysis of linear multigroup neutron transport theory in plane geometry. Transport Theory and Statistical Physics 2002; 29(6):711-721. Case KM, Zweifel PF. Linear Transport Equations. Addison-Wesley Publishing Company: MA, 1967. Shulaia D. Solution of a linear integral equation of third kind. Georgian Mathematical Journal 2002; 9(1):179-196. Shulaia D. On one Fredholm integral equation of third kind. Georgian Mathematical Journal 1997; 4(5):461-476. Gachov FD. Boundary Value Problems. Nauka: Moscow, 1963 (Russian). Krasnov ML. Integral Equations. Nauka: Moscow, 1975 (Russian). Gabbasov NS. A special version of the collocation method for integral equations of the third kind. Differential Equations 2005; 41(12):1768-1774. Corduneanu C. Principles of Differential and Integral Equations (2nd edn). Chelsea Publishing Company:New York, 1977. 2002; 9 1983; 270 2005; 41 1953 1975 1960; 9 1911; 28 1997; 4 1958 1957 1912 1977 2004; 33 1976; 231 2002; 29 1927; 2 1963 1961 1981; 259 1990; 310 1938; 115 1969 1964; 73 1924 1967 1973; 4 1983; 23 Shulaia DA (e_1_2_1_18_2) 1983; 270 Fadeyev LD (e_1_2_1_8_2) 1964; 73 Gachov FD (e_1_2_1_23_2) 1963 Shulaia DA (e_1_2_1_17_2) 1983; 23 Titchmarsh EC (e_1_2_1_10_2) 1958 Shulaia DA (e_1_2_1_16_2) 1981; 259 e_1_2_1_20_2 Tamarkin J (e_1_2_1_24_2) 1927; 2 e_1_2_1_21_2 Hilbert D (e_1_2_1_4_2) 1912 Shulaia DA (e_1_2_1_19_2) 1990; 310 e_1_2_1_25_2 Tricomi FG (e_1_2_1_27_2) 1957 Germogenova TA (e_1_2_1_15_2) 1976; 231 Davison D (e_1_2_1_28_2) 1957 Case KM (e_1_2_1_3_2) 1967 e_1_2_1_29_2 Feldman IA (e_1_2_1_31_2) 1975 Krasnov ML (e_1_2_1_26_2) 1975 Shulaia D (e_1_2_1_22_2) 2002; 9 Picard E (e_1_2_1_5_2) 1911; 28 e_1_2_1_6_2 e_1_2_1_30_2 e_1_2_1_7_2 Lovitt WV (e_1_2_1_12_2) 1924 e_1_2_1_2_2 Corduneanu C (e_1_2_1_11_2) 1977 e_1_2_1_14_2 Muskhelishvili N (e_1_2_1_13_2) 1953 e_1_2_1_9_2 |
| References_xml | – reference: Tricomi FG. Integral Equations. Interscience Publishers: New York, 1957. – reference: Gabbasov NS. A special version of the collocation method for integral equations of the third kind. Differential Equations 2005; 41(12):1768-1774. – reference: Friedrichs KO. Über die Spektralzerlegung eines Integraloperators. Mathematische Annalen 1938; 115(2):249-300 (German). – reference: Shulaia DA, Gugushvili EI. Inverse problem of spectral analysis of linear multigroup neutron transport theory in plane geometry. Transport Theory and Statistical Physics 2002; 29(6):711-721. – reference: Shulaia D. On one Fredholm integral equation of third kind. Georgian Mathematical Journal 1997; 4(5):461-476. – reference: Fadeyev LD. On Friedrichs model in the theory of perturbations of a continuous spectrum. Trudy Matematicheskogo Instituta Imeni V.A. Steklova 1964; 73:292-313 (Russian). – reference: Shulaia DA. On the expansion of solutions of equations of the linear multivelocity transport theory by eigenfunctions of the characteristic equation. Doklady Akademii Nauk SSSR 1990; 310(4):844-849 (Russian). – reference: Case KM. Elementary solutions of the transport equation and their applications. Annals of Physics 1960; 9:1-23. – reference: Titchmarsh EC. Eigenfunction Expansions Associated with Second-Order Differential Equations. Clarendon Press: Oxford, 1958. – reference: Germogenova TA, Shulaia DA. On the characteristic equation of the radiation transport theory. Doklady Akademii Nauk SSSR 1976; 231(4):841-844 (Russian). – reference: Shulaia D. Solution of a linear integral equation of third kind. Georgian Mathematical Journal 2002; 9(1):179-196. – reference: Case KM, Zweifel PF. Linear Transport Equations. Addison-Wesley Publishing Company: MA, 1967. – reference: Feldman IA. On the discrete spectrum of the characteristic equation of the radiation transport theory. Matematicheskie Issledovaniya 1975; 10 N 1(35):236-242 (Russian). – reference: Gachov FD. Boundary Value Problems. Nauka: Moscow, 1963 (Russian). – reference: Shulaia DA. Linear equation of the multivelocity transport theory. Jurnal Vichislitelnoi Matematiki i Matematicheskoi Phisiki 1983; 23(5):1125-1140 (Russian). – reference: Hilbert D. Grundzüge einer Allgemeinen Theorie der Linearen Integralgleichungen. B.G. Teubner: Leipzig/Berlin, 1912 (German). – reference: Krasnov ML. Integral Equations. Nauka: Moscow, 1975 (Russian). – reference: Corduneanu C. Principles of Differential and Integral Equations (2nd edn). Chelsea Publishing Company:New York, 1977. – reference: Picard E. Les équations integrales de troisième espèce. Annales Scientifigues De L Ecole Normale Superieure 1911; 28(3):459-472 (French). – reference: Shulaia DA. Inverse problem of the linear multivelocity transport theory. Doklady Akademii Nauk SSSR 1983; 270(1):82-87 (Russian). – reference: Shulaia D, Sharashidze N. A spectral representation of the multigroup transport problem. Transport Theory and Statistical Physics 2004; 33(2):183-202. – reference: Lovitt WV. Linear Integral Equations. Mc Graw-Hill: New York, 1924. – reference: Shulaia DA. Completeness theorems in linear multivelocity transport theory. Doklady Akademii Nauk SSSR 1981; 259(2):332-335 (Russian). – reference: Tamarkin J. On Fredholm integral equations whose kernels are analytical in a parameter. Annals of Mathematical Series 1927; 2(28):127-152. – reference: Davison D. Neutron Transport Theory. Oxford University Press: Oxford, 1957. – reference: Muskhelishvili N. Singular Integral Equations. P. Noordhooff: Groningen, 1953. – reference: Bart G, Warnock R. Linear equations of the third kind. SIAM Journal on Mathematical Analysis 1973; 4(4):609-622. – year: 1912 – year: 1958 – volume: 231 start-page: 841 issue: 4 year: 1976 end-page: 844 article-title: On the characteristic equation of the radiation transport theory publication-title: Doklady Akademii Nauk SSSR – start-page: 236 issue: 35 year: 1975 end-page: 242 article-title: On the discrete spectrum of the characteristic equation of the radiation transport theory publication-title: Matematicheskie Issledovaniya – volume: 23 start-page: 1125 issue: 5 year: 1983 end-page: 1140 article-title: Linear equation of the multivelocity transport theory publication-title: Jurnal Vichislitelnoi Matematiki i Matematicheskoi Phisiki – volume: 4 start-page: 461 issue: 5 year: 1997 end-page: 476 article-title: On one Fredholm integral equation of third kind publication-title: Georgian Mathematical Journal – volume: 2 start-page: 127 issue: 28 year: 1927 end-page: 152 article-title: On Fredholm integral equations whose kernels are analytical in a parameter publication-title: Annals of Mathematical Series – year: 1975 – start-page: 89 year: 1961 end-page: 105 – volume: 9 start-page: 1 year: 1960 end-page: 23 article-title: Elementary solutions of the transport equation and their applications publication-title: Annals of Physics – year: 1977 – volume: 115 start-page: 249 issue: 2 year: 1938 end-page: 300 article-title: Über die Spektralzerlegung eines Integraloperators publication-title: Mathematische Annalen – volume: 29 start-page: 711 issue: 6 year: 2002 end-page: 721 article-title: Inverse problem of spectral analysis of linear multigroup neutron transport theory in plane geometry publication-title: Transport Theory and Statistical Physics – volume: 259 start-page: 332 issue: 2 year: 1981 end-page: 335 article-title: Completeness theorems in linear multivelocity transport theory publication-title: Doklady Akademii Nauk SSSR – volume: 28 start-page: 459 issue: 3 year: 1911 end-page: 472 article-title: Les équations integrales de troisième espèce publication-title: Annales Scientifigues De L Ecole Normale Superieure – year: 1957 – year: 1963 – start-page: 37 year: 1969 end-page: 78 – volume: 270 start-page: 82 issue: 1 year: 1983 end-page: 87 article-title: Inverse problem of the linear multivelocity transport theory publication-title: Doklady Akademii Nauk SSSR – volume: 4 start-page: 609 issue: 4 year: 1973 end-page: 622 article-title: Linear equations of the third kind publication-title: SIAM Journal on Mathematical Analysis – year: 1967 – year: 1969 – volume: 33 start-page: 183 issue: 2 year: 2004 end-page: 202 article-title: A spectral representation of the multigroup transport problem publication-title: Transport Theory and Statistical Physics – year: 1924 – volume: 41 start-page: 1768 issue: 12 year: 2005 end-page: 1774 article-title: A special version of the collocation method for integral equations of the third kind publication-title: Differential Equations – volume: 310 start-page: 844 issue: 4 year: 1990 end-page: 849 article-title: On the expansion of solutions of equations of the linear multivelocity transport theory by eigenfunctions of the characteristic equation publication-title: Doklady Akademii Nauk SSSR – year: 1953 – volume: 73 start-page: 292 year: 1964 end-page: 313 article-title: On Friedrichs model in the theory of perturbations of a continuous spectrum publication-title: Trudy Matematicheskogo Instituta Imeni V.A. Steklova – volume: 9 start-page: 179 issue: 1 year: 2002 end-page: 196 article-title: Solution of a linear integral equation of third kind publication-title: Georgian Mathematical Journal – volume-title: Singular Integral Equations year: 1953 ident: e_1_2_1_13_2 – volume: 231 start-page: 841 issue: 4 year: 1976 ident: e_1_2_1_15_2 article-title: On the characteristic equation of the radiation transport theory publication-title: Doklady Akademii Nauk SSSR – volume-title: Boundary Value Problems year: 1963 ident: e_1_2_1_23_2 – volume-title: Integral Equations year: 1975 ident: e_1_2_1_26_2 – volume: 259 start-page: 332 issue: 2 year: 1981 ident: e_1_2_1_16_2 article-title: Completeness theorems in linear multivelocity transport theory publication-title: Doklady Akademii Nauk SSSR – volume: 9 start-page: 179 issue: 1 year: 2002 ident: e_1_2_1_22_2 article-title: Solution of a linear integral equation of third kind publication-title: Georgian Mathematical Journal doi: 10.1515/GMJ.2002.179 – volume: 23 start-page: 1125 issue: 5 year: 1983 ident: e_1_2_1_17_2 article-title: Linear equation of the multivelocity transport theory publication-title: Jurnal Vichislitelnoi Matematiki i Matematicheskoi Phisiki – volume: 310 start-page: 844 issue: 4 year: 1990 ident: e_1_2_1_19_2 article-title: On the expansion of solutions of equations of the linear multivelocity transport theory by eigenfunctions of the characteristic equation publication-title: Doklady Akademii Nauk SSSR – ident: e_1_2_1_7_2 doi: 10.1007/s10625-006-0013-4 – volume-title: Linear Integral Equations year: 1924 ident: e_1_2_1_12_2 – volume: 2 start-page: 127 issue: 28 year: 1927 ident: e_1_2_1_24_2 article-title: On Fredholm integral equations whose kernels are analytical in a parameter publication-title: Annals of Mathematical Series – volume-title: Principles of Differential and Integral Equations year: 1977 ident: e_1_2_1_11_2 – volume: 270 start-page: 82 issue: 1 year: 1983 ident: e_1_2_1_18_2 article-title: Inverse problem of the linear multivelocity transport theory publication-title: Doklady Akademii Nauk SSSR – start-page: 236 issue: 35 year: 1975 ident: e_1_2_1_31_2 article-title: On the discrete spectrum of the characteristic equation of the radiation transport theory publication-title: Matematicheskie Issledovaniya – ident: e_1_2_1_6_2 doi: 10.1137/0504053 – ident: e_1_2_1_2_2 – ident: e_1_2_1_21_2 doi: 10.1081/TT-120037806 – volume-title: Eigenfunction Expansions Associated with Second‐Order Differential Equations year: 1958 ident: e_1_2_1_10_2 doi: 10.1063/1.3062231 – ident: e_1_2_1_20_2 doi: 10.1080/00411450008214531 – volume-title: Grundzüge einer Allgemeinen Theorie der Linearen Integralgleichungen year: 1912 ident: e_1_2_1_4_2 – ident: e_1_2_1_9_2 doi: 10.1007/BF01448941 – volume: 73 start-page: 292 year: 1964 ident: e_1_2_1_8_2 article-title: On Friedrichs model in the theory of perturbations of a continuous spectrum publication-title: Trudy Matematicheskogo Instituta Imeni V.A. Steklova – volume: 28 start-page: 459 issue: 3 year: 1911 ident: e_1_2_1_5_2 article-title: Les équations integrales de troisième espèce publication-title: Annales Scientifigues De L Ecole Normale Superieure doi: 10.24033/asens.641 – volume-title: Neutron Transport Theory year: 1957 ident: e_1_2_1_28_2 – ident: e_1_2_1_25_2 doi: 10.1023/A:1022928500444 – volume-title: Linear Transport Equations year: 1967 ident: e_1_2_1_3_2 – ident: e_1_2_1_14_2 doi: 10.1016/0003-4916(60)90060-9 – ident: e_1_2_1_29_2 – ident: e_1_2_1_30_2 doi: 10.1090/psapm/011/0119519 – volume-title: Integral Equations year: 1957 ident: e_1_2_1_27_2 |
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| SubjectTerms | Exact sciences and technology holomorphic function Integral equations limiting values Mathematical analysis Mathematics Sciences and techniques of general use Several complex variables and analytic spaces singular operator |
| Title | Linear integral equations of the third kind arising from neutron transport theory |
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