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Saul'yev V.K. — Integration of Equations of Parabolic Type By the Method of Nets
Saul'yev V.K. — Integration of Equations of Parabolic Type By the Method of Nets

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Название: Integration of Equations of Parabolic Type By the Method of Nets

Автор: Saul'yev V.K.

Аннотация:

This book is concerned with the solution of parabolic partial
differential equations by the method of nets (otherwise known
as the method of finite differences). The book is in two parts,
the first of which is devoted to the construction of net equations,
paying particular attention to the stability and the accuracy of
the approximating net equations, and the second part to the
solution of such equations. Together with classical net methods,
this book contains many recent developments due to the author
and his colleagues, which have not previously been published
in the West.
The book can profitably be read by scientists and engineers
without specialized mathematical knowledge, but is of most
value to numerical analysts. Most of the methods described
demand electronic computation, but some of the novel methods
are suitable for hand computation.
The translator has in a few instances added, in a footnote,
a clarification of the original text.
I wish to thank Mr. E. L. Albasiny of the National Physical
Laboratory for checking the accuracy of the translation and
for many helpful suggestions concerning the style. Our grateful
thanks are also due to the author for providing a list of corrections
to the Russian text.


Язык: en

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1964

Количество страниц: 365

Добавлена в каталог: 22.03.2013

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
Abramov A. A.      205 258 279
Absolutely stable net equations      16 23
Absolutely unstable net equations      16 22ff
Accuracy      see. “Error”
Albasiny E. L.      (vii) 186
Albrecht J.      131 153 172
Alekseyenko L. G.      50n
Allen R. D.      90
Alternate use of explicit & implicit net equations      23ff 109 196
Alternating direction method      122 205 308ff
Alternating method      43ff
Analogue computers      (xvii)
Analytic integration      (xiii)
Artificial boundary condition      90
Asymmetric net equations      29ff
Axial node, effect on stability      77 78
Axial symmetry      see “Cylindrical symmetry”
Bauer W. F.      206
Bi-parabolic equation      184
Bilateral approximations      108ff
Birkhoff, G.      313
Birman M. Sh.      205
Blanc C.      (xvii)
Block iteration      299ff
Boundary conditions for fourth-order equation      171
Browder F.      174
Budak B. M.      (xvi)
Buleyev N. I.      223n
Bunyakovskii — Schwartz inequality      8 12
Carr$\acute{e}$ B. A.      291n
Cauchy’s problem      (xvi) 37n 73 75
Central differences      17
Characteristically elliptic net equations      203n 283
Chebyshev polynomials      205 250ff 278 295 315ff
Chessboard order      128 298
Choleski factorization      204
Classification of methods of solving elliptic net equations      208
Classification of one-dimensional net equations      197-199
Coefficient of stability      17
Collatz L.      (xiii) (xvii) 18 152 168 171 257
Complete equations      133
Computational labour      (xvii) 4 250
Computing machines      (11) &
Conditioning of explicit net equations      224ff
Conditioning of implicit net equations      227ff
Conjugate gradients      205 208 247ff
Conte S. D.      171
Convergence      (x) 5 &
Convergence in the mean      14
Convergence rate of S.O.R.      288 293
Convergence, uniform      (vii) (xiii)n 6 12 13 83 270
Cramer’s Rule      204
Crandall S. H.      171 172
Crank J.      (xv) (xvii) 3n 23 90
Crater of functional      245
Critical nodes      87:
Curved boundary      see “Shape of boundary”
Cylindrical symmetry      73ff
D$\ddot{u}$ck W.      281n
Direct methods for solving linear equation      204 208 212ff
Dirichlet problem      85n 219 228 242
Discontinuity of coefficients      162ff 187
Discontinuity of parameters      102
Douglas J.      (xvi) 14 23 94 122 129 147 185 187 193 205 211
du Fort E.C.      123 157 186
Ehrlich L. W.      209
Eigenvalues      (xiv)n 9 &
Eigenvectors      10 11 158 252 263 303
Elliptic direction      154
Elliptic equations      (xiii) (xv) (xvi) 73 242
Ellipticity      204 313
Error of approximation      5 13 16 83ff 130ff 180
Error of solution      4 13 16 83 85
Euler’s constant      83
Explicit application of implicit formulae      67ff 80
Explicit method for general equation      163ff
Explicit methods      15 &
Explicit methods for non-linear equation      188ff
Explicit multi-step equation      156ff
External nodes      88
Extrapolated Liebmann method      see “Successive Over –Relaxation (S.O.R.)”
Extreme accuracy for net equations      85ff
Factorization of differences, method of      208
Factorization, method of      207ff 300
Faddeyev D. K.      205 277
Faddeyeva V. N.      (xvii) 8 11 205 233 256 287
Fictitious nodes      101ff 174ff
Filippov A. F.      (xiii) (xvi) 6n 9 14 15 37n 68n 166 167
Finite differences, method of      (xiii)n
Flanders D. A.      205 257 258
Forsythe G.E.      277 291
Fourier coefficient      10 292
Fourth-order parabolic equation      170ff
Fox L.      (xvii) 91
Frankel S. P.      123 157 186 205 262 287
Franklin J. N.      318
Functional of vector      243
Gallie T.      (xvi) 147
Garabedian P.      291
Garwick J. V.      73
Gauss K.F.      204 205 212 281 299
Gavurin M. K.      205 258
General boundary conditions      (xvi)
Gershgorin S.      187
Goldstine H.      225
Golub G. H.      278n 290n
Gradient of functional      243 245
Gradshtein R. S.      181
graphical methods      (xvii)
Green’s function      219
Groschaftov A. Z.      13
HabetlerG. J.      313n
Hand computation      (vii) (ix) 19 60 195
HarmuthH. F.      (xvii) 19
Heat conduction      (xvi) &
Heinrich H.      (xvii)
Hestenes M. R.      205
Hexagonal nets      (xvii) 153
High-order parabolic equations      170ff 208
Hilbert norm      17
Hildebrand F. B.      181
Hochstrasser U.      262
Hollingsworth B. J.      216
Hurwitz H.      159 200 209
Hyman M.A.      23 137 215
Hyperbolic equations      (xiii) (xiv) 22 154
Ill-conditioning      214ff 294
Implicit method for general equation      167ff
Implicit methods      15 &
Implicit multi-step equation      156ff
Implicitly explicit method      64ff
Increase of time-step      147
Increased accuracy for net equation      83ff 130ff
Influence coefficients      219
Initial conditions      3 &
Instability      6 7 18ff
Instability of recurrence method      210 214
Interfaces      102
Interpolation at boundary      152 153
Inversion of matrix      208
Irregular equation      149ff
Ishizaki H.      215 216
Iterative method for general nonlinear equation      191 192
Iterative methods      (v) 27 28 204
Iterative methods of $n$th degree      268 278
Jacobian matrix      7 11 308
Jacobi’s iterative method      205 233 281 299
Juncosa M. L.      23
Kamynin L. I.      (xvi) 15
Kantorovich L. V.      (xiii) 73 132 145 205 244
Kaplan S.      23 137
Karplus W.J.      11
Keller H. B.      313n
Koval’ P.I.      15
Krylov V. I.      (xiii) 73 132 145
KuroshA. G.      159
Kuz’mina A. N.      29n
Ladyzhenskaya O. A.      (xiii)
Lagrangian interpolation polynomial      155
Laplace equation      (xv)n 108 109 203 205 228 291
Lebedev V. I.      (xvi) 50n 94
Liebmann’s method      208 281 287 289 313
Line methods      (xvi) 313
Linear iterative methods      208
Linearity      9 21
Lipschitz condition      182
Local error      80
Local iterations      282
Loginov L. I.      132
Lotkin M.      187
Lyusternik L. A.      (xi) (xiii)n 9 153 239
Malova M. N.      141n
Mann W. R.      98
Marchuk G. I.      207 214 219
Markov A. A.      80n
Matrix factorization      214
Matrix formulation of general parabolic equation      162ff
Mean arithmetic, method of      52ff
Meiman N.N.      15
Meyer H. I.      216
Mikeladze Sh. E.      15 16n 15 95
Milne W. E.      (xiii)
Minimax properties of Chebyshev polynomials      253
Mitchell A. R.      186
Mixed problem      173
Mnemonic representation of net eqn.      16 197 198 199
Monte Carlo method      (xvi) 206 208
Multi-dimensional equation      27 39 116ff 130ff
Multi-nodal symmetric method      52ff
Multi-step equation      95 153ff
Nets, method of      4 &
Nicolescu M.      184
Nicolson P.      (xv) 23
Nishimura Toru      171
Nodes of net      3 &
Non-central differences      17 88
Non-linear equation      184ff
Non-uniform convergence      50
Non-uniform nets      147ff
Norm      8 17
Number of arithmetic operations      127 131 137 140 141 161 162 204 250
Oleinik O. A.      41
One-dimensional elliptic net equation      206ff
Optimal (standard) accuracy of net equation      85
Optimization of S.O.R.      291 293
Order of magnitude of error      5 &
Over-relaxation      283
O’Brien G. G.      23
PanowD. Ju.      (xiii) 15 84 94 131
Parabolic equation      3 &
Partial factorization, method of      208 220ff
Partitioning of matrix      218
Peaceman D. W.      205 308
Peck J.      215
Petrovskii I. G.      (xiii) 6n 9n 15 18 21
Poisson’s equation      203 215
Polar nets      153
Polynomial of a matrix      240 258
Positive definite matrix      242 250
Quasi-boundary nodes      87 151
Rachford H. H.      129 205 308
Rayleigh quotient      246 257
Recursion, method of      209
Reduced accuracy of net equation      85
Refinement of solution, by the Fox — Volkov method      (xvii) 91
Relaxation methods      282
Residual      241 243
Rhomboidal net      153
Richardson L. F.      (ix) 18 19 208 232 281 284 287
Riesel H.      219
Ritz method      108 244
Rose M.      187
Rounding error      6 8 268 281
Royster W. C.      171
Runge — Kutta method      186
Ryabenkii V. S.      (xiii) (xvi) 6n 9 14 15 157 166 167 262 267
Ryzhik I. M.      181
Saito Sh.      (xvi)
Samarskii A. A.      (xiii) (xvi) 3tt 101 114 149n
Saul’yev V. K.      (xi) (xiv)n 15 109 122
Schmidt’s method      232
Schr$\ddot{o}$dinger equation      19
Seidel’s method      32 205 281 291 299
Self-adjoint elliptic operator      174 226 242 268 315
Separation of variables      10 21 157
Several independent variables      3 116ff 130ff
Severn R. T.      9
Shape of boundary      74 185 109 151ff 200
Sheldon J. W.      205 295
Shortley G.      205 257 258
Simplest iterative method      208 230ff 246
Simultaneous displacement methods      208
Singularity in initial conditions      75 83 193
Six-point symmetric equation      15 22ff
Size of step, its significance      122
Smirnov V. I.      193
Smoothness of initial conditions      13 16 21 75 200
Sobolev S. L.      (xvii) 3n 268
Solution of net equation      Part II passim
Southwell R. V.      241 243
Spherical symmetry      74 78
Stability      (ix) (xiv) 6 &
Steepest descent, method of      205 208 245
Stefan Problem      (xvi)
Stein P.      215
Step (along $t$-axis)      (xiii) 4 &
Step (along $x$-axis)      3 &
Step-wise solution      6
Stiefel E.      205 232 247
Sturm — Liouville problem      114
Successive displacement methods      208 280ff 312
Successive over-relaxation      (S.O.R.) 208 280ff
Sweep over net      222 295
Symmetric S.O.R.      205 208
Szab$\acute{o}$ J.      216
Taylor series expansion      21 135 149
Thomson J. V.      186
Three-dimensional equation      116ff 130ff
Tikhonov A. N.      (xiii) (xvi) 3n
Timlake W. P.      98
Todd J.      18
Triangular net      (xvii) 153
Two-dimensional equation      73ff 116ff 130ff Part
Two-step equation      95ff 125 133 153ff
Under-relaxation      283
Varga R. S.      278n 290n 313n
Variable coefficients      (xiii)n 3 11 38 73ff 162ff
Variational methods of solution      205 208 241ff
Volkov E. A.      (xiv)n (xvii) 91
Von Mises      186
von Neumann J.      21 137 225 278
Wachspress E. L.      313n
Wasow W. R.      291
Yermakov S. M.      223n
Young D. M.      23 242 251 284 285 291 294
Yuan’ Chzhao-Din      11 15 70 71 72 262 272 317 319 320
Yushkov P.P.      (xvii) 15 95 132 153
“Parasitic” harmonics      (x) 11 72 89 319
“Strela” computer      29n 50n 51 80 141
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