Adjoint equations and perturbation algorithms in nonlinear problems

Content: Principles of Construction of Adjoint Operators in Non-Linear ProblemsDual Spaces and Adjoint OperatorsConstruction of Adjoint Operators Based on Using the Lagrange IdentityDefinition of Adjoint Operators Based on Using Taylor's FormulaOperators of the Class D and their Adjoint OperatorsProperties of Adjoint Operators Constructed on the Basis of Various PrinciplesGeneral Properties of Main and Adjoint Operators Corresponding to Non-Linear OperatorsProperties of Operators of the Class DProperties of Adjoint Operators Constructed with the Use of the Taylor FormulaSolvability of Main and Adjoint Equations in Non-Linear ProblemsMain and Adjoint Equations. ProblemsSolvability of the Equation F(u) = ySolvability of the Equation A(u)v = ySolvability of the Equation A(u)v = ySolvability of the Equation A*(u)w = pSolvability of the Equation A*(u)w = pTransformation Groups, Conservation Laws and Construction of the Adjoint Operators in Non-Linear ProblemsDefinitions. Non-Linear Equations and Operators. Conservation LawsTransformation of EquationsAdjoint EquationsRelationship between Different Adjoint OperatorsGeneral Remarks on Constructing the Adjoint Equations with the Use of the Lie Groups and Conservation LawsConstruction of Adjoint Operators with Prescribed PropertiesThe Noether Theorem, Conservation Laws and Adjoint OperatorsOn Some Applications of Adjoint EquationsPerturbation Algorithms in Non-Linear ProblemsPerturbation Algorithms for Original Non-Linear Equations and Equations Involving Adjoint OperatorsPerturbation Algorithms for Non-Linear Functionals Based on Using Main and Adjoint EquationsSpectral Method in Perturbation AlgorithmsJustification of the N-th Order Perturbation AlgorithmsConvergence Rate Estimates for Perturbation Algorithms. Comparison with the Successive Approximation MethodJustification of Perturbation Algorithms in Quasi-Linear Elliptic ProblemsAdjoint Equations and the N-th Order Perturbation Algorithms in Non-Linear Problems of Transport TheorySome Problems of Transport TheoryThe N-th Order Perturbation Algorithms for an Eigenvalue ProblemA Problem of Control and its Approximate Solution with the Use of Perturbation AlgorithmsInvestigation and Approximate Solution of a Non-Linear Problem for the Transport EquationAdjoint Equations and Perturbation Algorithms for a Quasilinear Equation of MotionStatement of the Problem. Basic Assumptions. Operator FormulationTransformation of the Problem. Properties of the Non-Linear OperatorAdjoint EquationAn Algorithm for Computing the FunctionalThe Problem on Chemical Exchange ProcessesAdjoint Equations and Perturbation Algorithms for a Non-Linear Mathematical Model of Mass Transfer in SoilMathematical Models of Mass Transfer in SoilFormulation of a Non-Linear Mathematical ModelTransformation of the Problem. Properties of the Non-Linear OperatorPerturbation Algorithm. Adjoint EquationApproximate Solution of the Problem on Finding an Effective Dispersion CoefficientAn Algorithm for Solving the ProblemApplications of Adjoint Equations in Science and TechnologyAdjoint Equations in Data Assimilation ProblemsApplication of Adjoint Equations for Solving the Problem of Liquid Boundary Conditions in HydrodynamicsShape Optimization Using Adjoint Equation ApproachesGlobal Transport of PollutantsProblems of Climate Change Sensitivity in Various Regions of the World