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An introduction to symmetry analysis for graduate students in science, engineering and applied mathematics.

Differential equations --- Symmetry (Physics) --- Invariance principles (Physics) --- Symmetry (Chemistry) --- Conservation laws (Physics) --- Physics --- 517.91 Differential equations --- Numerical solutions. --- Differential equations - Numerical solutions --- Lie groups. --- Symmetry (Physics).

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This book is a significant update of the first four chapters of Symmetries and Differential Equations (1989; reprinted with corrections, 1996), by George W. Bluman and Sukeyuki Kumei. Since 1989 there have been considerable developments in symmetry methods (group methods) for differential equations as evidenced by the number of research papers, books, and new symbolic manipulation software devoted to the subject. This is, no doubt, due to the inherent applicability of the methods to nonlinear differential equations. Symmetry methods for differential equations, originally developed by Sophus Lie in the latter half of the nineteenth century, are highly algorithmic and hence amenable to symbolic computation. These methods systematically unify and extend well-known ad hoc techniques to construct explicit solutions for differential equations, especially for nonlinear differential equations. Often ingenious tricks for solving particular differential equations arise transparently from the symmetry point of view, and thus it remains somewhat surprising that symmetry methods are not more widely known. Nowadays it is essential to learn the methods presented in this book to understand existing symbolic manipulation software for obtaining analytical results for differential equations. For ordinary differential equations (ODEs), these include reduction of order through group invariance or integrating factors. For partial differential equations (PDEs), these include the construction of special solutions such as similarity solutions or nonclassical solutions, finding conservation laws, equivalence mappings, and linearizations.

Differential equations --- Differential equations, Partial --- Lie groups. --- Numerical solutions. --- Mathematics. --- Global analysis (Mathematics). --- Applications of Mathematics. --- Theoretical, Mathematical and Computational Physics. --- Analysis. --- Applied mathematics. --- Engineering mathematics. --- Mathematical physics. --- Mathematical analysis. --- Analysis (Mathematics). --- Groups, Lie --- Lie algebras --- Symmetric spaces --- Topological groups --- 517.91 Differential equations --- Numerical analysis

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The subject of this book is numerical methods that preserve geometric properties of the flow of a differential equation: symplectic integrators for Hamiltonian systems, symmetric integrators for reversible systems, methods preserving first integrals and numerical methods on manifolds, including Lie group methods and integrators for constrained mechanical systems, and methods for problems with highly oscillatory solutions. A complete theory of symplectic and symmetric Runge-Kutta, composition, splitting, multistep and various specially designed integrators is presented, and their construction and practical merits are discussed. The long-time behaviour of the numerical solutions is studied using a backward error analysis (modified equations) combined with KAM theory and related perturbation theories. The book is illustrated by many figures, it treats applications from physics and astronomy and contains many numerical experiments and comparisons of different approaches.

519.62 --- 681.3*G17 --- Numerical methods for solution of ordinary differential equations --- Ordinary differential equations: boundary value problems; convergence and stability; error analysis; initial value problems; multistep methods; single step methods; stiff equations (Numerical analysis) --- Differential equations --- Hamiltonian systems. --- Numerical integration. --- Numerical solutions. --- 681.3*G17 Ordinary differential equations: boundary value problems; convergence and stability; error analysis; initial value problems; multistep methods; single step methods; stiff equations (Numerical analysis) --- 519.62 Numerical methods for solution of ordinary differential equations --- Hamiltonian systems --- Numerical integration --- Integration, Numerical --- Mechanical quadrature --- Quadrature, Mechanical --- Definite integrals --- Interpolation --- Numerical analysis --- Hamiltonian dynamical systems --- Systems, Hamiltonian --- Differentiable dynamical systems --- 517.91 Differential equations --- Numerical solutions --- 517.91 --- Dynamique différentiable. --- Systèmes hamiltoniens. --- Differentiable dynamical systems. --- Numerical analysis. --- Mathematical analysis. --- Analysis (Mathematics). --- Mathematical physics. --- Physics. --- Biomathematics. --- Numerical Analysis. --- Analysis. --- Theoretical, Mathematical and Computational Physics. --- Mathematical Methods in Physics. --- Numerical and Computational Physics, Simulation. --- Mathematical and Computational Biology. --- Biology --- Mathematics --- Natural philosophy --- Philosophy, Natural --- Physical sciences --- Dynamics --- Physical mathematics --- Physics --- 517.1 Mathematical analysis --- Mathematical analysis --- Numerical solutions&delete& --- Systèmes hamiltoniens --- Integration numerique --- Analyse numerique --- Equations differentielles