Numerical Verification Methods and Computer-Assisted Proofs for Partial Differential Equations

Category: Study

Tag: Mathematics


Posted on 2019-11-11, by selverine.

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Recently, various mathematical problems have been solved by computer-assisted proofs, among them the Kepler conjecture, the existence of chaos, the existence of the Lorenz attractor, the famous four-color problem, and more. In many cases, computer-assisted proofs have the remarkable advantage (compared with a “theoretical” proof) of providing accurate quantitative information.
The authors have been working more than a quarter century to establish the verified computations of solutions for partial differential equations, mainly to the nonlinear elliptic problems of the form -∆u=f(x,u,∇u) with Dirichlet boundary conditions. Here, by "verified computation" is meant a computer-assisted numerical approach to proving the existence of a solution in a close and explicit neighborhood of an approximate solution. Therefore, the quantitative information by the technique shown here should also be significant from the viewpoint of the a posteriori error estimates for approximate solution of concerned partial differential equations with mathematically rigorous sense.

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