Approximate methods of higher analysis

TABLE OF CONTENTS......Page 3
Preface to the Third Edition......Page 9
From the Preface to the Second Edition......Page 11
Translator\'s Preface......Page 13
1. The Dirichlet problem for the rectangle......Page 15
2. The Dirichlet and Neumann problems for an annulus in the case of the Laplace equation......Page 27
3. An example of the biharmonic problem......Page 31
1. Fundamental definitions......Page 34
2. Theorems on the comparison of systems......Page 35
3. Regular and fully regular systems......Page 40
4. The approximate solution of regular systems......Page 47
5. Limitants. Diverse generalizations of regular systems......Page 52
6. Brief survey of other investigations relating to infinite systems......Page 56
1. General principles......Page 58
2. Solution of the problem of the expansion of an arbitrary function in terms of preassigned functions. by means of orthogonalization......Page 59
3. Solution of the problem-of the expansion of an arbitrary function in terms of given functions by, means of infinite systems of equations......Page 68
4 Example 1: A mixed boundary- value problem for the Laplace equation......Page 70
5. Example 2: The clamped plate......Page 75
1. Statement of the problem. The foundations of the method......Page 82
2; Poisson\'s equation for the rectangle......Page 84
3. Application to equations of the fourth order......Page 86
1. The general principles on which methods for improving the convergence.are founded......Page 91
2. The method of Aca41. A. N. Krylov for improving the convergence of trigonometric series......Page 93
3. Fourier series with strengthened convergence (A. S. Maliev)......Page 100
4. General methods for improving the convergence of approximate solutions of lloundary-value problems......Page 103
1. Fundamental definitions......Page 111
2. The replacement of an integral equation by a finite system of linear equations......Page 112
3. The evaluation of the error incurred by replacing the integral equation by a system of linear equations......Page 117
4. Example......Page 122
1. The method of successive approximations......Page 124
2. The application of analytic continuation for the approximate solution of integral equations......Page 130
1. An integral equation of potential theory......Page 133
2. Neumann\'s method......Page 138
3. The method of N. M. Krylov and N. N. Bogoliubov......Page 144
4. Example......Page 149
1. The integral equation with degenerate kernel......Page 155
2. Replacement of an arbitrary kernel by a degenerate one......Page 157
3. Example......Page 159
4. Another estlmite of the error......Page 161
5. The method of moments......Page 164
6. Bateman\'s method......Page 169
1. Expressions for the derivatives in terms of difference ratios......Page 176
2. Relations between the values of a function at the nodes of a net and the Laplace and biharmonic operators......Page 193
1. Ordinary differential equations......Page 205
2. Partial differential equations of elliptic type......Page 214
3. Beundary conditions for finite-difference equations......Page 226
1. The existence and uniqueness of the solution-......Page 231
2. Two methods of solving finite-difference equations. Examples......Page 236
3. Estimate of the error. The convergence of the process......Page 245
1. Problems leading to an ordinary differential equation......Page 255
2. Variational problems leading to *he place and, Poison. equagns......Page 260
3. Other forms of boundary conditions......Page 263
4. Variational problems connected with the biharmonicequation......Page 266
5. Variational problems connected with the determination of proper numbers and proper functions......Page 269
1. The fundamental idea of Ritz\'s method and of the method of B. G. Galerkin......Page 272
2. Application of Ritz\'s method and that of B. G. Galerkin to ordinary differential equations......Page 276
4. Application to the biharmonic equation......Page 297
5. Application to the determination of the proper values and functions......Page 306
3. Reduction to ordinary differential equations......Page 318
1. Fundamental equations......Page 320
2. Examples of the determination of the first approximation......Page 323
3. Examples of the refinement of the solution......Page 330
4. An example of the application of the method to the biharmonic equation......Page 336
5. Application of the method to the determination of proper values and proper functions......Page 339
1. The case of ordinary differential equations......Page 341
2. The problem of the convergence of the minimal sequences for equations of elliptic type......Page 351
3. The convergence of Ritz\'s method and of the method of reduction to ordinary equations......Page 361
1. Conformal transformation and the Laplace equation......Page 372
2. The transformation of simply connected regions......Page 374
3. The transfor- mation of multiply connected regions......Page 376
1. An extremal property of the function transforming a region into a circle......Page 379
2. App icattion of Ritz\'s method......Page 381
3. Minimization by means of polynomials......Page 385
4. The convergence of the successive appximations. The completeness of the system of coordinate functions......Page 386
5. Exterior regions......Page 389
1. The extremal property of the transforming function......Page 390
2. The application of Ritz s method......Page 391
3. The transformation of exterior regions......Page 393
1. Polynomials orthogonal on the contour......Page 395
2. Application to conformal transformation......Page 397
3. Polynomials orthogonal in a region......Page 401
4. Application to conformal transformation......Page 403
1. Statement of the problem. Reduction to a system of equations......Page 404
2. The method of successive approximations......Page 414
3. The conformal transformatioW of eiferior regions......Page 419
1. The normal representation of a contour......Page 428
2. The method of infinite systems......Page 429
3. Exam les......Page 433
4. The method of successive approximations for regions clos to a circle, the contour of the regions being given by an implicit equation......Page 438
5. The method of successive approximations for regions close to. those into which the conformal transformation of a circle is known......Page 441
6. The method of successive approximations for. curves given in parametric form......Page 445
7. Proof of the convergence of the process. of, successive approximations......Page 449
8.Observations on the mapping of a circle onto the exterior of a curve. Examples......Page 459
1, The algorithm of successive approximations......Page 465
2: Choice of the first approximation. Computational scheme......Page 473
3. The mapping of exterior regions......Page 486
4. The case of symmetric contours. Examples......Page 488
1. Introduction. Green\'s function for the Dirichlet problem......Page 492
2. Approximate construction. of Green\'s. function......Page 499
3. Green\'s function  for the Neumann problem......Page 504
4. Green.\'s function for the mixed problem......Page 510
1. The integral equation for the transformation. of interior regions......Page 515
2. Observations on the solution of the integral equation and the approximate construction, of the mapping function......Page 520
3. The inteal equation for the transformation of exterior regions......Page 522
4. T ie transformation of a region into a plane with parallel cuts......Page 526
5. the\' transformation of a\'tnultiply connected region into a plane with cuts lying on rays originating at one point......Page 531
1. Derivation of the- Christoffel-Schwarz formula......Page 535
2. The vahie of the parameters in the Christoffel-Schwarz integral......Page 537
3. On the Newton-Fourier method for a system of equations, and on the computation of the improper integrals......Page 540
1. The trainsformation of the Laplace operator......Page 557
2. The transformation \'of the biharmonic operator ursat\'s formula......Page 558
3. Transformation of the boundary , conditions......Page 567
4. Integrate  of the Cauchy  type:  their computation......Page 570
1. The Poisson integral......Page 575
2. The Poisson integral for the exterior of a circle......Page 579
3. The Dirichlet problem for a half plane......Page 581
4. Dirichlet\'s problem for the annulus......Page 582
5. Schwarz\'s formula. Determination of the conjugate harmonic function......Page 583
6. Solution of the Poisson equation in the circle......Page 586
1. Dini\'s formula......Page 589
2. The exterior of a circle......Page 592
3. Neumann\'s problem for a half plane......Page 593
4. Neumann\'s problem for the annulus......Page 595
1. The problem stated. The case of constant coefficients in the boundary condition......Page 596
2. Hilbert\'s problem......Page 601
3. The general boundary- value problem......Page 605
1. First fundamental problem. Reduction to a system of equations......Page 609
2. The second fundamental problem. Reduction to a system of equations......Page 619
3. First fundamental problem. Reduction to functional equations......Page 620
4. Second fundamental problem. Reduction to functional equations......Page 628
1. Schwarz\'s method in the general case. Investigation of the convergence......Page 630
2. The case of a linear equation of elliptic type. An estimate of the rapidity of convergence of Schwarz\'s process for the Laplace equation......Page 640
3. Reduction of Schwarz\'s method to the solution of a system of integral equations by successive approximations......Page 649
1. Description of the method and investigation of the convergence of the successive approximations......Page 654
2. Example of the investigation of the convergence of the Schwarz-Neumann method. An estimate of the rate of convergence in the case of the Laplace equation......Page 666
3. Reduction of the Schwarz-Neumann method to the solution of a system of integral equations by successive approximations......Page 669
3. An example of the application of Schwarz\'s method......Page 673
BIBLIOGRAPHY......Page 685