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Analysis of Carrier's problem
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A computational and asymptotic analysis of the solutions of Carrier's problem is presented in this paper, which reveals a striking and beautiful bifurcation diagram, with an infinite sequence of alternating pitchfork and fold bifurbcations as the bifurlcation parameter tends to zero.Abstract:
A computational and asymptotic analysis of the solutions of Carrier's problem is presented. The computations reveal a striking and beautiful bifurcation diagram, with an infinite sequence of alternating pitchfork and fold bifurcations as the bifurcation parameter tends to zero. The method of Kuzmak is then applied to construct asymptotic solutions to the problem. This asymptotic approach explains the bifurcation structure identified numerically, and its predictions of the bifurcation points are in excellent agreement with the numerical results. The analysis yields a novel and complete taxonomy of the solutions to the problem, and demonstrates that a claim of Bender & Orszag is incorrect.read more
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PETSc Users Manual
Satish Balay,Shrirang Abhyankar,Mark F. Adams,Jed Brown,Peter R. Brune,Kristopher R. Buschelman,Lisandro Dalcin,Alp Dener,Eijkhout,William Gropp,Dmitry Karpeyev,Dinesh K. Kaushik,Matthew G. Knepley,Dave A. May,L. Curfman McInnes,Richard T. Mills,Todd Munson,Karl Rupp,Patrick Sanan,Barry Smith,Stefano Zampini,Hong Zhang +21 more
TL;DR: The Portable, Extensible Toolkit for Scientific Computation (PETSc), is a suite of data structures and routines for the scalable (parallel) solution of scientific applications modeled by partial differential equations that supports MPI, and GPUs through CUDA or OpenCL, as well as hybrid MPI-GPU parallelism.
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Automated Solution of Differential Equations by the Finite Element Method: The FEniCS Book
TL;DR: This book is a tutorial written by researchers and developers behind the FEniCS Project and explores an advanced, expressive approach to the development of mathematical software.
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Advanced Mathematical Methods for Scientists and Engineers I: Asymptotic Methods and Perturbation Theory
Carl M. Bender,Steven A. Orszag +1 more
TL;DR: Eventually, you will enormously discover a supplementary experience and execution by Spending more cash by spending more cash.
Journal ArticleDOI
Ordinary Differential Equations.
TL;DR: In this article, the Fundamental Theorem of Calculus gives us an important connection between differential equations and integrals, and modern numerical methods automatically determine the step sizes hn = tn+1 − tn so that the estimated error in the numerical solution is controlled by a specified tolerance.
Journal ArticleDOI
The Calculation of Turning Points of Nonlinear Equations
G. Moore,Alastair Spence +1 more
TL;DR: In this article, an enlarged system is introduced for which the turning point is a nonsingular solution and thus standard methods can be used to compute it, and numerical results for discretizations of a differential equation and an integral equation are given.