Parametric Continuation and Optimal Parametrization in Applied Mathematics and Mechanics

Preface.1: Nonlinear Equations with a Parameter. 1. Two forms of the method of continuation of the solution with respect to a parameter. 2. The problem of choosing the continuation parameter. Replacement of the parameter. 3. The best continuation parameter. 4. The algorithms using the best continuation parameter and examples of their application. 5. Geometrical visualization of step-by-step processes. 6. The solution continuation in vicinity of essential singularity points.2: The Cauchy Problem for Ordinary Differential Equations. 1. The Cauchy problem as a problem of solution continuation with respect to a parameter. 2. Certain properties of lambda-transformation. 3. Algorithms, softwares, examples.3: Stiff Systems of Ordinary Differential Equations. 1. Characteristic features of numerical integration of stiff system of ordinary differential equations. 2. Sinular perturbed equations. 3. Stiff systems. 4. Stiff equations for partial derivatives.4: Differential-Algebraic Equations. 1. Classification of systems of DAE. 2. The best argument for a system of differential-algebraic equations. 3. Explicit differential-algebraic equations. 4. Implicit ordinary differential equations. 5. Implicit differential-algebraic equations.5: Functional-Differential Equations. 1. The Cauchy problem for equations with a retarded argument. 2. The Cauchy problem for Volterra's integro-differential equations.6: The Parametric Approximation. 1. The parametric interpolation. 2. The parametric approximation. 3. The continuous approximation.7: Nonlinear Boundary Value Problems for Ordinary Differential Equations. 1. The equations of solution continuation for nonlinear one-dimensional boundary value problems. 2. The discrete orthagonal shooting method. 3. The algorithms for continuous and discrete continuation of the solution with respect to a parameter for nonlinear one-dimensional boundary value problems. 4. Example: large deflections of the circle arch.8: Continuation of the Solution Near