The Final Aim Of The Book Is To Construct Effective Discretization Methods To Solve Multidimensional Weakly Singular Integral Equations Of The Second Kind On A Region Of Rn E.g. Equations Arising In The Radiation Transfer Theory. To This End, The Smoothness Of The Solution Is Examined Proposing Sharp Estimates Of The Growth Of The Derivatives Of The Solution Near The Boundary G. The Superconvergence Effect Of Collocation Methods At The Collocation Points Is Established. This Is A Book For Graduate Students And Researchers In The Fields Of Analysis, Integral Equations, Mathematical Physics And Numerical Methods. No Special Knowledge Beyond Standard Undergraduate Courses Is Assumed. 1. Some Problems Leading To Multidimensional Weakly Singular Integral Equations -- 1.1. An Interior-exterior Problem -- 1.2. A Physical Background (n = 2) -- 1.3. Integral Equation Formulation [actual Symbol Not Reproducible] -- 1.4. Integral Equation Formulation (n = 2) -- 1.5. Radiation Transfer Problem -- 1.6. Integral Equation Formulation Of The Radiation Transfer Problem -- 1.7. Peierls Integral Equations -- 2. Preliminaries -- 2.1. Metric D[subscript G] And Space Bc(g*) -- 2.2. Weakly Singular Integral Operators -- 2.3. Compactness Of A Weakly Singular Integral Operator -- 2.4. Weakly Singular Integral Operators With Differentiable Kernels -- 2.5. Weighted Space [actual Symbol Not Reproducible] -- 2.6. Weighted Space [actual Symbol Not Reproducible] -- 3. Smoothness Of The Solution -- 3.1. The Class Of Weakly Singular Kernels -- 3.2. Main Results And Comments -- 3.3. Differentiation Of Weakly Singular Integral -- 3.4. Estimates Of The Derivatives Of Tu --^ 3.5. Proof Of Theorem 3.1 -- 3.6. Tangential Differentiation Of Weakly Singular Integral -- 3.7. Estimates Of The Tangential Derivatives Of Tu -- 3.8. Proof Of Theorem 3.2 -- 3.9. Sharpness Of Theorem 3.1 (general Analysis) -- 3.10. Sharpness Of Theorem 3.1 (analysis Of Model Examples) -- 3.11. Two Dimensional Integral Equation With A Logarithmically Singular Kernel -- 4. Outlines Of Discrete Convergence Theory -- 4.1. Discrete Convergence And Discrete Compactness Of A Family Of Elements -- 4.2. Example -- 4.3. Discrete Convergence Of A Family Of Linear Operators -- 4.4. Discrete Convergence Of Approximate Solutions -- 4.5. Convergence And Convergence Rates In Eigenvalue Problems -- 4.6. Approximation Of Non-linear Equations -- 5. Piecewise Constant Collocation And Related Methods -- 5.1. Assumptions About The Boundary -- 5.2. Partition Of G -- 5.3. Pccm And Related Methods -- 5.4. Convergence Rates Of The Methods -- 5.5. Block Scheme Of The Proof Of Theorem 5.1 --^ 5.6. Compact Convergence Of The Discretized Operators -- 5.7. Approximation Error Of The Basic Method -- 5.8. Estimation Of Prolonged Approximate Solution -- 5.9. Proof Of Assertion (ii) -- 5.10. Proof Of Assertion (iii) -- 5.11. Proof Of Assertion (iv) -- 5.12. Two Grid Iteration Method -- 5.13. Initial Guesses -- 5.14. Amount Of Arithmetical Work -- 5.15. Eigenvalue Problem -- 6. Composite Cubature Algorithms -- 6.1. Partition Of G And Composite Cubature Formulae -- 6.2. Basic Methods -- 6.3. Refined Cubatures For The Evaluation Of The Coefficients -- 6.4. Error Analysis (preliminaries) -- 6.5. Error Analysis Of Algorithms 6.1 And 6.2 -- 6.6. Error Estimates For Approximate Solution -- 6.7. Some Further Algorithms -- 6.8. Two Grid Methods -- 6.9. Case Of Convolution Type Integral Equations -- 7. Higher Order Methods -- 7.1. Space [actual Symbol Not Reproducible] -- 7.2. Piecewise Polynomial Interpolation -- 7.3. Error Estimates Of The Interpolation --^ 7.4. Piecewise Polynomial Collocation Method -- 7.5. Error Estimates At The Collocation Points -- 7.6. Superconvergence At The Collocation Points -- 7.7. Piecewise Constant Collocation (m = 1) -- 7.8. Piecewise Polylinear Collocation (m = 2) -- 7.9. Piecewise Polysquare Collocation (m = 3) -- 8. Nonlinear Integral Equation -- 8.1. Smoothness Of The Solution -- 8.2. Piecewise Constant Collocation And Related Methods -- 8.3. Higher Order Methods. Gennadi Vainikko. Includes Bibliographical References And Index. Some problems leading to multidimensional weakly singular integral equations....Pages 1-9 Preliminaries....Pages 10-23 Smoothness of the solution....Pages 24-50 Outlines of the discrete convergence theory....Pages 51-59 Piecewise constant collocation and related methods....Pages 60-93 Composite cubature algorithms....Pages 94-111 Higher order methods....Pages 112-136 Nonlinear integral equation....Pages 137-144
