Adjoint Equations and Perturbation Algorithms in Nonlinear Problems

Abstract

Principles of Construction of Adjoint Operators in Non-Linear Problems Dual Spaces and Adjoint Operators Construction of Adjoint Operators Based on Using the Lagrange Identity Definition of Adjoint Operators Based on Using Taylor's Formula Operators of the Class D and their Adjoint Operators Properties of Adjoint Operators Constructed on the Basis of Various Principles General Properties of Main and Adjoint Operators Corresponding to Non-Linear Operators Properties of Operators of the Class D Properties of Adjoint Operators Constructed with the Use of the Taylor Formula Solvability of Main and Adjoint Equations in Non-Linear Problems Main and Adjoint Equations. Problems Solvability of the Equation F(u) = y Solvability of the Equation A(u)v = y Solvability of the Equation A(u)v = y Solvability of the Equation A*(u)w = p Solvability of the Equation A*(u)w = p Transformation Groups, Conservation Laws and Construction of the Adjoint Operators in Non-Linear Problems Definitions. Non-Linear Equations and Operators. Conservation Laws Transformation of Equations Adjoint Equations Relationship between Different Adjoint Operators General Remarks on Constructing the Adjoint Equations with the Use of the Lie Groups and Conservation Laws Construction of Adjoint Operators with Prescribed Properties The Noether Theorem, Conservation Laws and Adjoint Operators On Some Applications of Adjoint Equations Perturbation Algorithms in Non-Linear Problems Perturbation Algorithms for Original Non-Linear Equations and Equations Involving Adjoint Operators Perturbation Algorithms for Non-Linear Functionals Based on Using Main and Adjoint Equations Spectral Method in Perturbation Algorithms Justification of the N-th Order Perturbation Algorithms Convergence Rate Estimates for Perturbation Algorithms. Comparison with the Successive Approximation Method Justification of Perturbation Algorithms in Quasi-Linear Elliptic Problems Adjoint Equations and the N-th Order Perturbation Algorithms in Non-Linear Problems of Transport Theory Some Problems of Transport Theory The N-th Order Perturbation Algorithms for an Eigenvalue Problem A Problem of Control and its Approximate Solution with the Use of Perturbation Algorithms Investigation and Approximate Solution of a Non-Linear Problem for the Transport Equation Adjoint Equations and Perturbation Algorithms for a Quasilinear Equation of Motion Statement of the Problem. Basic Assumptions. Operator Formulation Transformation of the Problem. Properties of the Non-Linear Operator Adjoint Equation An Algorithm for Computing the Functional The Problem on Chemical Exchange Processes Adjoint Equations and Perturbation Algorithms for a Non-Linear Mathematical Model of Mass Transfer in Soil Mathematical Models of Mass Transfer in Soil Formulation of a Non-Linear Mathematical Model Transformation of the Problem. Properties of the Non-Linear Operator Perturbation Algorithm. Adjoint Equation Approximate Solution of the Problem on Finding an Effective Dispersion Coefficient An Algorithm for Solving the Problem Applications of Adjoint Equations in Science and Technology Adjoint Equations in Data Assimilation Problems Application of Adjoint Equations for Solving the Problem of Liquid Boundary Conditions in Hydrodynamics Shape Optimization Using Adjoint Equation Approaches Global Transport of Pollutants Problems of Climate Change Sensitivity in Various Regions of the World