Choose an application

• Reference Manager
• EndNote
• RefWorks (Direct export to RefWorks)

The purpose of this book is to give a quick and elementary, yet rigorous, presentation of the rudiments of the so-called theory of Viscosity Solutions which applies to fully nonlinear 1st and 2nd order Partial Differential Equations (PDE). For such equations, particularly for 2nd order ones, solutions generally are non-smooth and standard approaches in order to define a "weak solution" do not apply: classical, strong almost everywhere, weak, measure-valued and distributional solutions either do not exist or may not even be defined. The main reason for the latter failure is that, the standard idea of using "integration-by-parts" in order to pass derivatives to smooth test functions by duality, is not available for non-divergence structure PDE.

Mathematics. --- Partial Differential Equations. --- Calculus of Variations and Optimal Control; Optimization. --- Differential equations, partial. --- Mathematical optimization. --- Mathématiques --- Optimisation mathématique --- Mathematics --- Physical Sciences & Mathematics --- Calculus --- Viscosity solutions. --- Differential equations, Nonlinear. --- Calculus of variations. --- Nonlinear differential equations --- Isoperimetrical problems --- Variations, Calculus of --- Partial differential equations. --- Maxima and minima --- Nonlinear theories --- Hamilton-Jacobi equations --- Optimization (Mathematics) --- Optimization techniques --- Optimization theory --- Systems optimization --- Mathematical analysis --- Operations research --- Simulation methods --- System analysis --- Partial differential equations

Choose an application

• Reference Manager
• EndNote
• RefWorks (Direct export to RefWorks)

This book emphasizes in detail the applicability of the Optimal Homotopy Asymptotic Method to various engineering problems. It is a continuation of the book “Nonlinear Dynamical Systems in Engineering: Some Approximate Approaches”, published at Springer in 2011, and it contains a great amount of practical models from various fields of engineering such as classical and fluid mechanics, thermodynamics, nonlinear oscillations, electrical machines, and so on. The main structure of the book consists of 5 chapters. The first chapter is introductory while the second chapter is devoted to a short history of the development of homotopy methods, including the basic ideas of the Optimal Homotopy Asymptotic Method. The last three chapters, from Chapter 3 to Chapter 5, are introducing three distinct alternatives of the Optimal Homotopy Asymptotic Method with illustrative applications to nonlinear dynamical systems. The third chapter deals with the first alternative of our approach with two iterations. Five applications are presented from fluid mechanics and nonlinear oscillations.  The Chapter 4 presents the Optimal Homotopy Asymptotic Method with a single iteration and solving the linear equation on the first approximation. Here are treated 32 models from different fields of engineering such as fluid mechanics, thermodynamics, nonlinear damped and undamped oscillations, electrical machines and even from physics and biology. The last chapter is devoted to the Optimal Homotopy Asymptotic Method with a single iteration but without solving the equation in the first approximation.

Engineering. --- Theoretical and Applied Mechanics. --- Computational Mathematics and Numerical Analysis. --- Socio- and Econophysics, Population and Evolutionary Models. --- Computer science --- Mechanics, applied. --- Ingénierie --- Informatique --- Mathematics. --- Mathématiques --- Computer science_xMathematics. --- Mathematics --- Physical Sciences & Mathematics --- Geometry --- Homotopy theory. --- Differential equations, Nonlinear --- Asymptotic theory. --- Asymptotic theory in nonlinear differential equations --- Deformations, Continuous --- Computer mathematics. --- Sociophysics. --- Econophysics. --- Mechanics. --- Mechanics, Applied. --- Asymptotic expansions --- Topology --- Data-driven Science, Modeling and Theory Building. --- Computer mathematics --- Discrete mathematics --- Electronic data processing --- Applied mechanics --- Engineering, Mechanical --- Engineering mathematics --- Economics --- Statistical physics --- Mathematical sociology --- Classical mechanics --- Newtonian mechanics --- Physics --- Dynamics --- Quantum theory --- Statistical methods