Abstract
Let x = (x{sub 1}....,x{sub q}), R{sub q}(x) be the q-dimensional space (q{>=}2), s be the unit vector, and #Greek Capital Letter Omega With Tonos# the surface of the unit sphere. The problem of solving transport equations with a degenerate scattering indicatrix is a multidimensional problem in R{sub q}x#Greek Capital Letter Omega With Tonos#; the nature of these problems calls for rapidly converging iterative methods which do not require all the information on the preceding step. The paper proposes a KP method: the idea consists in solving in R{sub q} simplified problems of error determination using successive iterative steps of decreasing difficulty. Two operations are performed in the KP: the K-operation is a simple iteration in R{sub q}x{Omega} and operation P = {l_brace}P{sub 1}(n{sub 1}),...P{sub 0}(n{sub 0}){r_brace} is for the error with P{sub k} (n{sub k}) representing the solution in R{sub q} of the ultimate problem for a differential equation of order 2n{sub k}. P-operations are found and the convergence of the following methods is studied: KP{sub 1}(n), P{sub 2}(0), K{sup 2}P{sub 1}(n), cyclic KP{sub 1}(1) and KP{sub 1}(0) etc. For 2{pi}T periodic problems the convergence is estimated, P(KP) (KP price) and cheap algorithms are found, and the non-improvability is
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Lebedev, V. I.
[1]
- Institut Atomnoj Ehnergii Im. I.V. Kurchatova Moskva, SSSR (Russian Federation)
Citation Formats
Lebedev, V. I.
Possible Solutions of Various Transport Problems; O nakhozhdenii reshenij nekotorykh zadach perenosa.
IAEA: N. p.,
1968.
Web.
Lebedev, V. I.
Possible Solutions of Various Transport Problems; O nakhozhdenii reshenij nekotorykh zadach perenosa.
IAEA.
Lebedev, V. I.
1968.
"Possible Solutions of Various Transport Problems; O nakhozhdenii reshenij nekotorykh zadach perenosa."
IAEA.
@misc{etde_22108108,
title = {Possible Solutions of Various Transport Problems; O nakhozhdenii reshenij nekotorykh zadach perenosa}
author = {Lebedev, V. I.}
abstractNote = {Let x = (x{sub 1}....,x{sub q}), R{sub q}(x) be the q-dimensional space (q{>=}2), s be the unit vector, and #Greek Capital Letter Omega With Tonos# the surface of the unit sphere. The problem of solving transport equations with a degenerate scattering indicatrix is a multidimensional problem in R{sub q}x#Greek Capital Letter Omega With Tonos#; the nature of these problems calls for rapidly converging iterative methods which do not require all the information on the preceding step. The paper proposes a KP method: the idea consists in solving in R{sub q} simplified problems of error determination using successive iterative steps of decreasing difficulty. Two operations are performed in the KP: the K-operation is a simple iteration in R{sub q}x{Omega} and operation P = {l_brace}P{sub 1}(n{sub 1}),...P{sub 0}(n{sub 0}){r_brace} is for the error with P{sub k} (n{sub k}) representing the solution in R{sub q} of the ultimate problem for a differential equation of order 2n{sub k}. P-operations are found and the convergence of the following methods is studied: KP{sub 1}(n), P{sub 2}(0), K{sup 2}P{sub 1}(n), cyclic KP{sub 1}(1) and KP{sub 1}(0) etc. For 2{pi}T periodic problems the convergence is estimated, P(KP) (KP price) and cheap algorithms are found, and the non-improvability is shown. Numerical calculations indicated that the KP method is very efficient. The Case results are generalized for the q-dimensional case: in Rqx{Omega} a system of solutions is found for a homogeneous single-velocity transport equation with constant coefficients and an isotropic scattering indicatrix {Phi}{sub {omega}}(S)exp({+-}(x, {omega})/v) These are generalized functions with {sub v}#Greek Lunate Epsilon Symbol#{l_brace}(-1, 1], {+-} v{sub 0}{r_brace} and {omega} #Greek Lunate Epsilon Symbol#. Theorems are proved for the completeness of {l_brace}{Phi}{sub v{omega}}{r_brace} in L{sub 2}({Omega}), for the partial orthogonality of and the possibility of representing {psi}#Greek Lunate Epsilon Symbol# L{sub 2}({Omega}) by {l_brace}{Phi}{sub v{omega}}{r_brace}. For q>3 the second component of v can be missing. 2{pi}T periodic solutions of non-homogeneous transport equations are found for cells of different dimensions. (author) [Russian] Pust' x = (x{sub 1}....,x{sub q}), R{sub q}(x) - ja-mernoe prostranstvo (q Greater-Than-Or-Equal-To 2), s - edinichnyj vektor, #Greek Capital Letter Omega With Tonos# - poverhnost' edinich- Zadacha reshenija uravnenij perenosa s vyrozhdennoj indikatrisoj rassejanija javljaetsja mnogomernoj zadachej v R{sub q}x#Greek Capital Letter Omega With Tonos#; specifika takih zadach trebuet razrabotki bystroshodjashhih- sja iteracionnyh metodov, trebujushhih zapominanija ne vsej informacii o provedennom iteracionnom shage. Izlagaetsja KR-metod; ideja ego sostoit v reshenii v Rja na kazhdom iteracionnom shage posledovatel'nosti ubyvajushhej trudnosti uproshhennyh zadach dlja oshibki. V KR vypolnjaetsja dve operacii: operacija K-prostaja iteracija v R{sub q}x Greek-Capital-Letter-Omega , i operacija P = {l_brace}P{sub 1}(n{sub 1}),...P{sub 0}(n{sub 0}){r_brace} dlja oshibki, zdes' P{sub k} (n{sub k}) oznachaet reshenie v R{sub q} kraevoj zadachi dlja differencial'nogo uravnenija 2n{sub k} go porjadka. Najdeny operacii R i issledovana shodimost' metodov: KP{sub 1}(n), P{sub 2}(0), K{sup 2}P{sub 1}(n), cyclic KP{sub 1}(1) ciklicheskogo KP{sub 1}(0) i KPi (0) i dr. Dlja 2{pi}T periodicheskih zadach ocenena shodimost', najdeny C(KR) (cena KR) i deshevye algorifmy, pokazana neuluchshaemost'. Chislennye raschety pokazali vysokuju jeffektivnost' KR-metoda. Obobshheny na ja-mernyj sluchaj rezul'taty Kejsa: v Rqx{Omega} najdena sistema reshenij odnorodnogo odnoskorostnogo uravnenija perenosa s postojannymi kojefficientami i izotropnoj indikatrisoj rassejanija {Phi}{sub {omega}}(S)exp({+-}(x, {omega})/v) javljajushhihsja obobshhennymi funkcijami, u kotoryh {sub v}#Greek Lunate Epsilon Symbol#{l_brace}(-1, 1], {+-} v{sub 0}{r_brace}, i {omega} #Greek Lunate Epsilon Symbol#. Dokazany teoremy o polnote {l_brace}{Phi}{sub v{omega}}{r_brace} B L{sub 2}({Omega}), o chastichnoj ortogonal'nosti {psi}#Greek Lunate Epsilon Symbol# L{sub 2}({Omega}) by {l_brace}{Phi}{sub v{omega}}{r_brace}, o predstavimosti cherez Pri ja>3 vtoraja komponenta mnozhestva v mozhet otsutstvovat'. Najdeny 2{pi}T-periodicheskie reshenija neodnorodnyh uravnenij perenosa, jeto reshenija dlja jacheek,razlichnoj formy. (author)}
place = {IAEA}
year = {1968}
month = {Jan}
}
title = {Possible Solutions of Various Transport Problems; O nakhozhdenii reshenij nekotorykh zadach perenosa}
author = {Lebedev, V. I.}
abstractNote = {Let x = (x{sub 1}....,x{sub q}), R{sub q}(x) be the q-dimensional space (q{>=}2), s be the unit vector, and #Greek Capital Letter Omega With Tonos# the surface of the unit sphere. The problem of solving transport equations with a degenerate scattering indicatrix is a multidimensional problem in R{sub q}x#Greek Capital Letter Omega With Tonos#; the nature of these problems calls for rapidly converging iterative methods which do not require all the information on the preceding step. The paper proposes a KP method: the idea consists in solving in R{sub q} simplified problems of error determination using successive iterative steps of decreasing difficulty. Two operations are performed in the KP: the K-operation is a simple iteration in R{sub q}x{Omega} and operation P = {l_brace}P{sub 1}(n{sub 1}),...P{sub 0}(n{sub 0}){r_brace} is for the error with P{sub k} (n{sub k}) representing the solution in R{sub q} of the ultimate problem for a differential equation of order 2n{sub k}. P-operations are found and the convergence of the following methods is studied: KP{sub 1}(n), P{sub 2}(0), K{sup 2}P{sub 1}(n), cyclic KP{sub 1}(1) and KP{sub 1}(0) etc. For 2{pi}T periodic problems the convergence is estimated, P(KP) (KP price) and cheap algorithms are found, and the non-improvability is shown. Numerical calculations indicated that the KP method is very efficient. The Case results are generalized for the q-dimensional case: in Rqx{Omega} a system of solutions is found for a homogeneous single-velocity transport equation with constant coefficients and an isotropic scattering indicatrix {Phi}{sub {omega}}(S)exp({+-}(x, {omega})/v) These are generalized functions with {sub v}#Greek Lunate Epsilon Symbol#{l_brace}(-1, 1], {+-} v{sub 0}{r_brace} and {omega} #Greek Lunate Epsilon Symbol#. Theorems are proved for the completeness of {l_brace}{Phi}{sub v{omega}}{r_brace} in L{sub 2}({Omega}), for the partial orthogonality of and the possibility of representing {psi}#Greek Lunate Epsilon Symbol# L{sub 2}({Omega}) by {l_brace}{Phi}{sub v{omega}}{r_brace}. For q>3 the second component of v can be missing. 2{pi}T periodic solutions of non-homogeneous transport equations are found for cells of different dimensions. (author) [Russian] Pust' x = (x{sub 1}....,x{sub q}), R{sub q}(x) - ja-mernoe prostranstvo (q Greater-Than-Or-Equal-To 2), s - edinichnyj vektor, #Greek Capital Letter Omega With Tonos# - poverhnost' edinich- Zadacha reshenija uravnenij perenosa s vyrozhdennoj indikatrisoj rassejanija javljaetsja mnogomernoj zadachej v R{sub q}x#Greek Capital Letter Omega With Tonos#; specifika takih zadach trebuet razrabotki bystroshodjashhih- sja iteracionnyh metodov, trebujushhih zapominanija ne vsej informacii o provedennom iteracionnom shage. Izlagaetsja KR-metod; ideja ego sostoit v reshenii v Rja na kazhdom iteracionnom shage posledovatel'nosti ubyvajushhej trudnosti uproshhennyh zadach dlja oshibki. V KR vypolnjaetsja dve operacii: operacija K-prostaja iteracija v R{sub q}x Greek-Capital-Letter-Omega , i operacija P = {l_brace}P{sub 1}(n{sub 1}),...P{sub 0}(n{sub 0}){r_brace} dlja oshibki, zdes' P{sub k} (n{sub k}) oznachaet reshenie v R{sub q} kraevoj zadachi dlja differencial'nogo uravnenija 2n{sub k} go porjadka. Najdeny operacii R i issledovana shodimost' metodov: KP{sub 1}(n), P{sub 2}(0), K{sup 2}P{sub 1}(n), cyclic KP{sub 1}(1) ciklicheskogo KP{sub 1}(0) i KPi (0) i dr. Dlja 2{pi}T periodicheskih zadach ocenena shodimost', najdeny C(KR) (cena KR) i deshevye algorifmy, pokazana neuluchshaemost'. Chislennye raschety pokazali vysokuju jeffektivnost' KR-metoda. Obobshheny na ja-mernyj sluchaj rezul'taty Kejsa: v Rqx{Omega} najdena sistema reshenij odnorodnogo odnoskorostnogo uravnenija perenosa s postojannymi kojefficientami i izotropnoj indikatrisoj rassejanija {Phi}{sub {omega}}(S)exp({+-}(x, {omega})/v) javljajushhihsja obobshhennymi funkcijami, u kotoryh {sub v}#Greek Lunate Epsilon Symbol#{l_brace}(-1, 1], {+-} v{sub 0}{r_brace}, i {omega} #Greek Lunate Epsilon Symbol#. Dokazany teoremy o polnote {l_brace}{Phi}{sub v{omega}}{r_brace} B L{sub 2}({Omega}), o chastichnoj ortogonal'nosti {psi}#Greek Lunate Epsilon Symbol# L{sub 2}({Omega}) by {l_brace}{Phi}{sub v{omega}}{r_brace}, o predstavimosti cherez Pri ja>3 vtoraja komponenta mnozhestva v mozhet otsutstvovat'. Najdeny 2{pi}T-periodicheskie reshenija neodnorodnyh uravnenij perenosa, jeto reshenija dlja jacheek,razlichnoj formy. (author)}
place = {IAEA}
year = {1968}
month = {Jan}
}