1st Edition

Adjoint Equations and Perturbation Algorithms in Nonlinear Problems

288 Pages
by CRC Press

288 Pages
by CRC Press

288 Pages
by CRC Press

Also available as eBook on:

Sparked by demands inherent to the mathematical study of pollution, intensive industry, global warming, and the biosphere, Adjoint Equations and Perturbation Algorithms in Nonlinear Problems is the first book ever to systematically present the theory of adjoint equations for nonlinear problems, as well as their application to perturbation algorithms. This new approach facilitates analysis of observational data, the application of adjoint equations to retrospective study of processes governed by imitation models, and the study of computer models themselves. Specifically, the book discusses:

• Principles for constructing adjoint operators in nonlinear problems
• Perturbation algorithms using the adjoint equations theory for nonlinear problems in transport theory, quasilinear motion, substance transfer, and nonlinear data assimilation
• Known results on adjoint equations and perturbation algorithms in nonlinear problems

This groundbreaking text contains some results that have no analogs in the scientific literature, opening unbounded possibilities in construction and application of adjoint equations to nonlinear problems of mathematical physics.
• Principles of Construction of Adjoint Operators in Non-Linear Problems
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
Solvability of the Equation F(u) = y
Solvability of the Equation A(u)v = y
Solvability of the Equation Ã(u)v = y
Solvability of the Equation A*(u)w = p
Solvability of the Equation Ã*(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
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
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