Approximate methods of higher analysis

Table of Contents......Page all_6230_to_00695.cpc0003.djvu
Preface to the Third Edition......Page all_6230_to_00695.cpc0009.djvu
From the Preface to the Second Edition......Page all_6230_to_00695.cpc0011.djvu
Translator\'s Preface......Page all_6230_to_00695.cpc0013.djvu
1. The Dirichlet problem for the rectangle......Page all_6230_to_00695.cpc0015.djvu
2. The Dirichlet and Neumann problems for an annulus in the case of the Laplace equation......Page all_6230_to_00695.cpc0027.djvu
3. An example of the biharmonic problem......Page all_6230_to_00695.cpc0031.djvu
1. Fundamental definitions......Page all_6230_to_00695.cpc0034.djvu
2. Theorems on the comparison of systems......Page all_6230_to_00695.cpc0035.djvu
3. Regular and fully regular systems......Page all_6230_to_00695.cpc0040.djvu
4. The approximate solution of regular systems......Page all_6230_to_00695.cpc0047.djvu
5. Limitants. Diverse generalizations of regular systems......Page all_6230_to_00695.cpc0052.djvu
6. Brief survey of other investigations relating to infinite systems......Page all_6230_to_00695.cpc0056.djvu
1. General principles......Page all_6230_to_00695.cpc0058.djvu
2. Solution of the problem of the expansion of an arbitrary function in terms of preassigned functions. by means of orthogonalization......Page all_6230_to_00695.cpc0059.djvu
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 all_6230_to_00695.cpc0068.djvu
4 Example 1: A mixed boundary- value problem for the Laplace equation......Page all_6230_to_00695.cpc0070.djvu
5. Example 2: The clamped plate......Page all_6230_to_00695.cpc0075.djvu
1. Statement of the problem. The foundations of the method......Page all_6230_to_00695.cpc0082.djvu
2; Poisson\'s equation for the rectangle......Page all_6230_to_00695.cpc0084.djvu
3. Application to equations of the fourth order......Page all_6230_to_00695.cpc0086.djvu
1. The general principles on which methods for improving the convergence.are founded......Page all_6230_to_00695.cpc0091.djvu
2. The method of Acad. A. N. Krylov for improving the convergence of trigonometric series......Page all_6230_to_00695.cpc0093.djvu
3. Fourier series with strengthened convergence (A. S. Maliev)......Page all_6230_to_00695.cpc0100.djvu
4. General methods for improving the convergence of approximate solutions of boundary-value problems......Page all_6230_to_00695.cpc0103.djvu
1. Fundamental definitions......Page all_6230_to_00695.cpc0111.djvu
2. The replacement of an integral equation by a finite system of linear equations......Page all_6230_to_00695.cpc0112.djvu
3. The evaluation of the error incurred by replacing the integral equation by a system of linear equations......Page all_6230_to_00695.cpc0117.djvu
4. Example......Page all_6230_to_00695.cpc0122.djvu
1. The method of successive approximations......Page all_6230_to_00695.cpc0124.djvu
2. The application of analytic continuation for the approximate solution of integral equations......Page all_6230_to_00695.cpc0130.djvu
1. An integral equation of potential theory......Page all_6230_to_00695.cpc0133.djvu
2. Neumann\'s method......Page all_6230_to_00695.cpc0138.djvu
3. The method of N. M. Krylov and N. N. Bogoliubov......Page all_6230_to_00695.cpc0144.djvu
4. Example......Page all_6230_to_00695.cpc0149.djvu
1. The integral equation with degenerate kernel......Page all_6230_to_00695.cpc0155.djvu
2. Replacement of an arbitrary kernel by a degenerate one......Page all_6230_to_00695.cpc0157.djvu
3. Example......Page all_6230_to_00695.cpc0159.djvu
4. Another estimate of the error......Page all_6230_to_00695.cpc0161.djvu
5. The method of moments......Page all_6230_to_00695.cpc0164.djvu
6. Bateman\'s method......Page all_6230_to_00695.cpc0169.djvu
1. Expressions for the derivatives in terms of difference ratios......Page all_6230_to_00695.cpc0176.djvu
2. Relations between the values of a function at the nodes of a net and the Laplace and biharmonic operators......Page all_6230_to_00695.cpc0193.djvu
1. Ordinary differential equations......Page all_6230_to_00695.cpc0205.djvu
2. Partial differential equations of elliptic type......Page all_6230_to_00695.cpc0214.djvu
3. Boundary conditions for finite-difference equations......Page all_6230_to_00695.cpc0226.djvu
4. The equation Delta^2 u = f(x, y)......Page all_6230_to_00695.cpc0216.djvu
1. The existence and uniqueness of the solution-......Page all_6230_to_00695.cpc0231.djvu
2. Two methods of solving finite-difference equations. Examples......Page all_6230_to_00695.cpc0236.djvu
3. Estimate of the error. The convergence of the process......Page all_6230_to_00695.cpc0245.djvu
IV. Variational Methods......Page all_6230_to_00695.cpc0254.djvu
1. Problems leading to an ordinary differential equation......Page all_6230_to_00695.cpc0255.djvu
2. Variational problems leading to *he place and, Poison. equations......Page all_6230_to_00695.cpc0260.djvu
3. Other forms of boundary conditions......Page all_6230_to_00695.cpc0263.djvu
4. Variational problems connected with the biharmonic equation......Page all_6230_to_00695.cpc0266.djvu
5. Variational problems connected with the determination of proper numbers and proper functions......Page all_6230_to_00695.cpc0269.djvu
1. The fundamental idea of Ritz\'s method and of the method of B. G. Galerkin......Page all_6230_to_00695.cpc0272.djvu
2. Application of Ritz\'s method and that of B. G. Galerkin to ordinary differential equations......Page all_6230_to_00695.cpc0276.djvu
3. The application of the methods of Ritz and of B. G. Galerkin to the solution of partial differential equations of the second order......Page all_6230_to_00695.cpc0273.djvu
4. Application to the biharmonic equation......Page all_6230_to_00695.cpc0297.djvu
5. Application to the determination of the proper values and functions......Page all_6230_to_00695.cpc0306.djvu
3. Reduction to ordinary differential equations......Page all_6230_to_00695.cpc0318.djvu
1. Fundamental equations......Page all_6230_to_00695.cpc0320.djvu
2. Examples of the determination of the first approximation......Page all_6230_to_00695.cpc0323.djvu
3. Examples of the refinement of the solution......Page all_6230_to_00695.cpc0330.djvu
4. An example of the application of the method to the biharmonic equation......Page all_6230_to_00695.cpc0336.djvu
5. Application of the method to the determination of proper values and proper functions......Page all_6230_to_00695.cpc0339.djvu
1. The case of ordinary differential equations......Page all_6230_to_00695.cpc0341.djvu
2. The problem of the convergence of the minimal sequences for equations of elliptic type......Page all_6230_to_00695.cpc0351.djvu
3. The convergence of Ritz\'s method and of the method of reduction to ordinary equations......Page all_6230_to_00695.cpc0361.djvu
1. Conformal transformation and the Laplace equation......Page all_6230_to_00695.cpc0372.djvu
2. The transformation of simply connected regions......Page all_6230_to_00695.cpc0374.djvu
3. The transformation of multiply connected regions......Page all_6230_to_00695.cpc0376.djvu
1. An extremal property of the function transforming a region into a circle......Page all_6230_to_00695.cpc0379.djvu
2. Application of Ritz\'s method......Page all_6230_to_00695.cpc0381.djvu
3. Minimization by means of polynomials......Page all_6230_to_00695.cpc0385.djvu
4. The convergence of the successive approximation. The completeness of the system of coordinate functions......Page all_6230_to_00695.cpc0386.djvu
5. Exterior regions......Page all_6230_to_00695.cpc0389.djvu
1. The extremal property of the transforming function......Page all_6230_to_00695.cpc0390.djvu
2. The application of Ritz s method......Page all_6230_to_00695.cpc0391.djvu
3. The transformation of exterior regions......Page all_6230_to_00695.cpc0393.djvu
1. Polynomials orthogonal on the contour......Page all_6230_to_00695.cpc0395.djvu
2. Application to conformal transformation......Page all_6230_to_00695.cpc0397.djvu
3. Polynomials orthogonal in a region......Page all_6230_to_00695.cpc0401.djvu
4. Application to conformal transformation......Page all_6230_to_00695.cpc0403.djvu
1. Statement of the problem. Reduction to a system of equations......Page all_6230_to_00695.cpc0404.djvu
2. The method of successive approximations......Page all_6230_to_00695.cpc0414.djvu
3. The conformal transformation of exterior regions......Page all_6230_to_00695.cpc0419.djvu
1. The normal representation of a contour......Page all_6230_to_00695.cpc0428.djvu
2. The method of infinite systems......Page all_6230_to_00695.cpc0429.djvu
3. Examples......Page all_6230_to_00695.cpc0433.djvu
4. The method of successive approximations for regions close to a circle, the contour of the regions being given by an implicit equation......Page all_6230_to_00695.cpc0438.djvu
5. The method of successive approximations for regions close to. those into which the conformal transformation of a circle is known......Page all_6230_to_00695.cpc0441.djvu
6. The method of successive approximations for. curves given in parametric form......Page all_6230_to_00695.cpc0445.djvu
7. Proof of the convergence of the process. of, successive approximations......Page all_6230_to_00695.cpc0449.djvu
8.Observations on the mapping of a circle onto the exterior of a curve. Examples......Page all_6230_to_00695.cpc0459.djvu
1, The algorithm of successive approximations......Page all_6230_to_00695.cpc0465.djvu
2: Choice of the first approximation. Computational scheme......Page all_6230_to_00695.cpc0473.djvu
3. The mapping of exterior regions......Page all_6230_to_00695.cpc0486.djvu
4. The case of symmetric contours. Examples......Page all_6230_to_00695.cpc0488.djvu
1. Introduction. Green\'s function for the Dirichlet problem......Page all_6230_to_00695.cpc0492.djvu
2. Approximate construction. of Green\'s. function......Page all_6230_to_00695.cpc0499.djvu
3. Green\'s function  for the Neumann problem......Page all_6230_to_00695.cpc0504.djvu
4. Green.\'s function for the mixed problem......Page all_6230_to_00695.cpc0510.djvu
1. The integral equation for the transformation. of interior regions......Page all_6230_to_00695.cpc0515.djvu
2. Observations on the solution of the integral equation and the approximate construction, of the mapping function......Page all_6230_to_00695.cpc0520.djvu
3. The integral equation for the transformation of exterior regions......Page all_6230_to_00695.cpc0522.djvu
4. The transformation of a region into a plane with parallel cuts......Page all_6230_to_00695.cpc0526.djvu
5. The transformation of multiply connected region into a plane with cuts lying on rays originating at one point......Page all_6230_to_00695.cpc0531.djvu
5. The mapping of a half plane on an arbitrary quadrilateral......Page all_6230_to_00695.cpc0535.djvu
2. The value of the parameters in the Christoffel-Schwarz integral......Page all_6230_to_00695.cpc0537.djvu
3. On the Newton-Fourier method for a system of equations, and on the computation of the improper integrals......Page all_6230_to_00695.cpc0540.djvu
4. Examples......Page all_6230_to_00695.cpc0529.djvu
1. The transformation of the Laplace operator......Page all_6230_to_00695.cpc0557.djvu
2. The transformation \'of the biharmonic operator Goursat\'s formula......Page all_6230_to_00695.cpc0558.djvu
3. Transformation of the boundary , conditions......Page all_6230_to_00695.cpc0567.djvu
4. Integrate  of the Cauchy  type:  their computation......Page all_6230_to_00695.cpc0570.djvu
1. The Poisson integral......Page all_6230_to_00695.cpc0575.djvu
2. The Poisson integral for the exterior of a circle......Page all_6230_to_00695.cpc0579.djvu
3. The Dirichlet problem for a half plane......Page all_6230_to_00695.cpc0581.djvu
4. Dirichlet\'s problem for the annulus......Page all_6230_to_00695.cpc0582.djvu
5. Schwarz\'s formula. Determination of the conjugate harmonic function......Page all_6230_to_00695.cpc0583.djvu
6. Solution of the Poisson equation in the circle......Page all_6230_to_00695.cpc0586.djvu
1. Dini\'s formula......Page all_6230_to_00695.cpc0589.djvu
2. The exterior of a circle......Page all_6230_to_00695.cpc0592.djvu
3. Neumann\'s problem for a half plane......Page all_6230_to_00695.cpc0593.djvu
4. Neumann\'s problem for the annulus......Page all_6230_to_00695.cpc0595.djvu
1. The problem stated. The case of constant coefficients in the boundary condition......Page all_6230_to_00695.cpc0596.djvu
2. Hilbert\'s problem......Page all_6230_to_00695.cpc0601.djvu
3. The general boundary value problem......Page all_6230_to_00695.cpc0605.djvu
1. First fundamental problem. Reduction to a system of equations......Page all_6230_to_00695.cpc0609.djvu
2. The second fundamental problem. Reduction to a system of equations......Page all_6230_to_00695.cpc0619.djvu
3. First fundamental problem. Reduction to functional equations......Page all_6230_to_00695.cpc0620.djvu
4. Second fundamental problem. Reduction to functional equations......Page all_6230_to_00695.cpc0628.djvu
1. Schwarz\'s method in the general case. Investigation of the convergence......Page all_6230_to_00695.cpc0630.djvu
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 all_6230_to_00695.cpc0640.djvu
3. Reduction of Schwarz\'s method to the solution of a system of integral equations by successive approximations......Page all_6230_to_00695.cpc0649.djvu
1. Description of the method and investigation of the convergence of the successive approximations......Page all_6230_to_00695.cpc0654.djvu
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 all_6230_to_00695.cpc0666.djvu
3. Reduction of the Schwarz-Neumann method to the solution of a system of integral equations by successive approximations......Page all_6230_to_00695.cpc0669.djvu
3. An example of the application of Schwarz\'s method......Page all_6230_to_00695.cpc0673.djvu
Bibliography......Page all_6230_to_00695.cpc0685.djvu