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Register a new userA Hilbert space approach to fractional difference equations
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We formulate fractional difference equations of Riemann-Liouville and Caputo type in a functional analytical framework. Main results are existence of solutions on Hilbert space-valued weighted sequence spaces and a condition for stability of linear fractional difference equations. Using a functional calculus, we relate the fractional sum to fractional powers of the operator $1 - tau^{-1}$ with the right shift $tau^{-1}$ on weighted sequence spaces. Causality of the solution operator plays a crucial role for the description of initial value problems.
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We consider general difference equations $u_{n+1} = F(u)_n$ for $n in mathbb{Z}$ on exponentially weighted $ell_2$ spaces of two-sided Hilbert space valued sequences $u$ and discuss initial value problems. As an application of the Hilbert space approach, we characterize exponential stability of linear equations and prove a stable manifold theorem for causal nonlinear difference equations.
We study fractional differential equations of Riemann-Liouville and Caputo type in Hilbert spaces. Using exponentially weighted spaces of functions defined on $mathbb{R}$, we define fractional operators by means of a functional calculus using the Fourier transform. Main tools are extrapolation- and interpolation spaces. Main results are the existence and uniqueness of solutions and the causality of solution operators for non-linear fractional differential equations.
We seek solutions $uinR^n$ to the semilinear elliptic partial difference equation $-Lu + f_s(u) = 0$, where $L$ is the matrix corresponding to the Laplacian operator on a graph $G$ and $f_s$ is a one-parameter family of nonlinear functions. This article combines the ideas introduced by the authors in two papers: a) {it Nonlinear Elliptic Partial Difference Equations on Graphs} (J. Experimental Mathematics, 2006), which introduces analytical and numerical techniques for solving such equations, and b) {it Symmetry and Automated Branch Following for a Semilinear Elliptic PDE on a Fractal Region} wherein we present some of our recent advances concerning symmetry, bifurcation, and automation fo We apply the symmetry analysis found in the SIAM paper to arbitrary graphs in order to obtain better initial guesses for Newtons method, create informative graphics, and be in the underlying variational structure. We use two modified implementations of the gradient Newton-Galerkin algorithm (GNGA, Neuberger and Swift) to follow bifurcation branches in a robust way. By handling difficulties that arise when encountering accidental degeneracies and higher-dimension we can find many solutions of many symmetry types to the discrete nonlinear system. We present a selection of experimental results which demonstrate our algorithms capability to automatically generate bifurcation diagrams and solution graphics starting with only an edgelis of a graph. We highlight interesting symmetry and variational phenomena.
In this paper, we study slow manifolds for infinite-dimensional evolution equations. We compare two approaches: an abstract evolution equation framework and a finite-dimensional spectral Galerkin approximation. We prove that the slow manifolds constructed within each approach are asymptotically close under suitable conditions. The proof is based upon Lyapunov-Perron methods and a comparison of the local graphs for the slow manifolds in scales of Banach spaces. In summary, our main result allows us to change between different characterizations of slow invariant manifolds, depending upon the technical challenges posed by particular fast-slow systems.
We introduce mesoscopic and macroscopic model equations of chemotaxis with anomalous subdiffusion for modelling chemically directed transport of biological organisms in changing chemical environments with diffusion hindered by traps or macro-molecular crowding. The mesoscopic models are formulated using Continuous Time Random Walk master equations and the macroscopic models are formulated with fractional order differential equations. Different models are proposed depending on the timing of the chemotactic forcing. Generalizations of the models to include linear reaction dynamics are also derived. Finally a Monte Carlo method for simulating anomalous subdiffusion with chemotaxis is introduced and simulation results are compared with numerical solutions of the model equations. The model equations developed here could be used to replace Keller-Segel type equations in biological systems with transport hindered by traps, macro-molecular crowding or other obstacles.
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