Mark J. Friedman

Improved multiquadric method for elliptic partial differential equations via PDE collocation on the boundary

The multiquadric radial basis function (MQ) method is a recent meshless collocation method with global basis functions. It was introduced for discretizing partial differential equations (PDEs) by Kansa in the early 1990s. The MQ method was originally used for interpolation of scattered data, and it was shown to have exponential convergence for interpolation problems. In [1], we have extended the Kansa-MQ method to numerical solution and detection of bifurcations in 1D and 2D parameterized nonlinear elliptic PDEs. We have found there that the modest size nonlinear systems resulting from the MQ discretization can be efficiently continued by a standard continuation software, such as auto. We have observed high accuracy with a small number of unknowns, as compared with most known results from the literature. In this paper, we formulate an improved Kansa-MQ method with PDE collocation on the boundary (MQ PDECB): we add an additional set of nodes (which can lie inside or outside of the domain) adjacent to the boundary and, correspondingly, add an additional set of collocation equations obtained via collocation of the PDE on the boundary. Numerical results are given that show a considerable improvement in accuracy of the MQ PDECB method over the Kansa-MQ method, with both methods having exponential convergence with essentially the same rates.

Numerical computation and continuation of invariant manifolds connecting fixed points

A numerical method for the computation of an invariant manifold that connects two fixed points of a vector field in

Numerical computation of heteroclinic orbits

\mathbb{R}^n

CONTINUATION FOR NONLINEAR ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS DISCRETIZED BY THE MULTIQUADRIC METHOD

is given, extending the results of an earlier paper [Comput. Appl. Math., 26 (1989), pp. 159’170] by the authors. Basically, a boundary value problem on the real line is truncated to a finite interval. The method applies, in particular, to the computation of heteroclinic orbits. The emphasis is on the systematic computation of such orbits by continuation. Using the fact that the linearized operator of our problem is Fredholm in appropriate Banach spaces, the general theory of approximation of nonlinear problems is employed to show that the errors in the approximate solution decay exponentially with the length of the approximating interval. Several applications are considered, including the computation of traveling wave solutions to reaction diffusion problems. Computations were done using the software package AUTO.

Computing Connecting Orbits via an Improved Algorithm for Continuing Invariant Subspaces

We give a numerical method for the computation of heteroclinic orbits connecting two saddle points in ℝ2. These can be computed to very high period due to an integral phase condition and an adaptive discretization. We can also compute entire branches (one-dimensional continua) of such orbits. The method can be extended to compute an invariant manifold that connects two fixed points in ℝ n . As an example we compute branches of traveling wave front solutions to the Huxley equation. Using weighted Sobolev spaces and the general theory of approximation of nonlinear problems we show that the errors in the approximate wave speed and in the approximate wave front decay exponentially with the period.

Continuation of invariant subspaces

The Multiquadric Radial Basis Function (MQ) Method is a meshless collocation method with global basis functions. It is known to have exponentional convergence for interpolation problems. We descretize nonlinear elliptic PDEs by the MQ method. This results in modest-size systems of nonlinear algebraic equations which can be efficiently continued by standard continuation software such as AUTO and CONTENT. Examples are given of detection of bifurcations in 1D and 2D PDEs. These examples show high accuracy with small number of unknowns, as compared with known results from the literature.

Continuation of Invariant Subspaces in Large Bifurcation Problems

A successive continuation method for locating connecting orbits in parametrized systems of autonomous ODEs was considered in [Numer. Algorithms, 14 (1997), pp. 103--124]. In this paper we present an improved algorithm for locating and continuing connecting orbits, which includes a new algorithm for the continuation of invariant subspaces. The latter algorithm is of independent interest and can be used in contexts different than the present one.

Computational methods for global analysis of homoclinic and heteroclinic orbits: A case study

In this work we consider implementation and testing of an algorithm for continuation of invariant subspaces. Copyright

A Preliminary Study of the Safety of Red Light Phototherapy of Tissues Harboring Cancer

We summarize an algorithm for computing a smooth orthonormal basis for an invariant subspace of a parameter-dependent matrix, and describe how to extend it for numerical bifurcation analysis. We adapt the continued subspace to track behavior relevant to bifurcations, and use projection methods to deal with large problems. To test our ideas, we have integrated our code into MATCONT, a program for numerical continuation and bifurcation analysis.

Successive continuation for locating connecting orbits

In earlier paper we have developed a numerical method for the computation of branches of heteroclinic orbits for a system of autonomous ordinary differential equations in ℝn. The idea of the method is to reduce a boundary value problem on the real line to a boundary value problem on a finite interval by using linear approximation of the unstable and stable manifolds. In this paper we extend our algorithm to incorporate higher-order approximations of the unstable and stable manifolds. This approximation is especially useful if we want to compute center manifolds accurately. A procedure for switching between the periodic approximation of homoclinic orbits and the higher-order approximation of homoclinic orbits provides additional flexibility to the method. The algorithm is applied to a model problem: the DC Josephson Junction. Computations are done using the software package AUTO.