40 research outputs found
Numerical methods for time-fractional evolution equations with nonsmooth data: a concise overview
Over the past few decades, there has been substantial interest in evolution
equations that involving a fractional-order derivative of order
in time, due to their many successful applications in
engineering, physics, biology and finance. Thus, it is of paramount importance
to develop and to analyze efficient and accurate numerical methods for reliably
simulating such models, and the literature on the topic is vast and fast
growing. The present paper gives a concise overview on numerical schemes for
the subdiffusion model with nonsmooth problem data, which are important for the
numerical analysis of many problems arising in optimal control, inverse
problems and stochastic analysis. We focus on the following aspects of the
subdiffusion model: regularity theory, Galerkin finite element discretization
in space, time-stepping schemes (including convolution quadrature and L1 type
schemes), and space-time variational formulations, and compare the results with
that for standard parabolic problems. Further, these aspects are showcased with
illustrative numerical experiments and complemented with perspectives and
pointers to relevant literature.Comment: 24 pages, 3 figure
Decay estimates for time-fractional and other non-local in time subdiffusion equations in
We prove optimal estimates for the decay in time of solutions to a rather
general class of non-local in time subdiffusion equations in . An
important special case is the time-fractional diffusion equation, which has
seen much interest during the last years, mostly due to its applications in the
modeling of anomalous diffusion processes. We follow three different approaches
and techniques to study this particular case: (A) estimates based on the
fundamental solution and Young's inequality, (B) Fourier multiplier methods,
and (C) the energy method. It turns out that the decay behaviour is markedly
different from the heat equation case, in particular there occurs a {\em
critical dimension phenomenon}. The general subdiffusion case is treated by
method (B) and relies on a careful estimation of the underlying relaxation
function. Several examples of kernels, including the ultraslow diffusion case,
illustrate our results.Comment: 34 page
Fundamental Solutions and Decay of Fully Non-local Problems
In this paper, we study a fully non-local reaction-diffusion equation which
is non-local both in time and space. We apply subordination principles to
construct the fundamental solutions of this problem, which we use to find a
representation of the mild solutions. Moreover, using techniques of Harmonic
Analysis and Fourier Multipliers, we obtain the temporal decay rates for the
mild solutions
Representation of solutions and large-time behavior for fully nonlocal diffusion equations
We study the Cauchy problem for a nonlocal heat equation, which is of
fractional order both in space and time. We prove four main theorems:
(i) a representation formula for classical solutions,
(ii) a quantitative decay rate at which the solution tends to the fundamental
solution,
(iii) optimal -decay of mild solutions in all dimensions,
(iv) -decay of weak solutions via energy methods.
The first result relies on a delicate analysis of the definition of classical
solutions. After proving the representation formula we carefully analyze the
integral representation to obtain the quantitative decay rates of (ii).
Next we use Fourier analysis techniques to obtain the optimal decay rate for
mild solutions. Here we encounter the critical dimension phenomenon where the
decay rate attains the decay rate of that in a bounded domain for large enough
dimensions. Consequently, the decay rate does not anymore improve when the
dimension increases. The theory is markedly different from that of the standard
caloric functions and this substantially complicates the analysis.
Finally, we use energy estimates and a comparison principle to prove a
quantitative decay rate for weak solutions defined via a variational
formulation. Our main idea is to show that the -norm is actually a
subsolution to a purely time-fractional problem which allows us to use the
known theory to obtain the result
Long-time behaviour of non-local in time Fokker-Planck equations via the entropy method
We consider a rather general class of non-local in time Fokker-Planck
equations and show by means of the entropy method that as the
solution converges in to the unique steady state. Important special cases
are the time-fractional and ultraslow diffusion case. We also prove estimates
for the rate of decay. In contrast to the classical (local) case, where the
usual time derivative appears in the Fokker-Planck equation, the obtained decay
rate depends on the entropy, which is related to the integrability of the
initial datum. It seems that higher integrability of the initial datum leads to
better decay rates and that the optimal decay rate is reached, as we show, when
the initial datum belongs to a certain weighted space. We also show how
our estimates can be adapted to the discrete-time case thereby improving known
decay rates from the literature.Comment: 25 page
Bounded weak solutions of time-fractional porous medium type and more general nonlinear and degenerate evolutionary integro-differential equations
We prove existence of a bounded weak solution to a degenerate quasilinear
subdiffusion problem with bounded measurable coefficients that may explicitly
depend on time. The kernel in the involved integro-differential operator w.r.t.
time belongs to the large class of kernels. In particular, the case
of a fractional time derivative of order less than 1 is included. A key
ingredient in the proof is a new compactness criterion of Aubin-Lions type
which involves function spaces defined in terms of the integro-differential
operator in time. Boundedness of the solution is obtained by the De Giorgi
iteration technique. Sufficiently regular solutions are shown to be unique by
means of an -contraction estimate.Comment: 21 page
A PDE Approach to Numerical Fractional Diffusion
Fractional diffusion has become a fundamental tool for the modeling of
multiscale and heterogeneous phenomena. However, due to its nonlocal nature,
its accurate numerical approximation is delicate. We survey our research
program on the design and analysis of efficient solution techniques for
problems involving fractional powers of elliptic operators. Starting from a
localization PDE result for these operators, we develop local techniques for
their solution: a priori and a posteriori error analyses, adaptivity and
multilevel methods. We show the flexibility of our approach by proposing and
analyzing local solution techniques for a space-time fractional parabolic
equation.Comment: The final version of this overview appeared in the Proceedings of the
ICIAM, 201
Asymptotic behaviour for the Fractional Heat Equation in the Euclidean space
We consider weak solutions of the fractional heat equation posed in the whole
-dimensional space, and establish their asymptotic convergence to the
fundamental solution as under the assumption that the initial
datum is an integrable function, or a finite Radon measure. Convergence with
suitable rates is obtained for solutions with a finite first initial moment,
while for solutions with compactly supported initial data convergence in
relative error holds. The results are applied to the fractional Fokker-Planck
equation. Brief mention of other techniques and related equations is made.Comment: 16 pages. Improved version accepted by Complex Variables and Elliptic
Equation
Decay estimates for evolution equations with classical and fractional time-derivatives
Using energy methods, we prove some power-law and exponential decay estimates
for classical and nonlocal evolutionary equations. The results obtained are
framed into a general setting, which comprise, among the others, equations
involving both standard and Caputo time-derivative, complex valued magnetic
operators, fractional porous media equations and nonlocal Kirchhoff operators.
Both local and fractional space diffusion are taken into account, possibly in a
nonlinear setting. The different quantitative behaviors, which distinguish
polynomial decays from exponential ones, depend heavily on the structure of the
time-derivative involved in the equation
Fractional Calculus and the Future of Science
Newton foresaw the limitations of geometry’s description of planetary behavior and developed fluxions (differentials) as the new language for celestial mechanics and as the way to implement his laws of mechanics. Two hundred years later Mandelbrot introduced the notion of fractals into the scientific lexicon of geometry, dynamics, and statistics and in so doing suggested ways to see beyond the limitations of Newton’s laws. Mandelbrot’s mathematical essays suggest how fractals may lead to the understanding of turbulence, viscoelasticity, and ultimately to end of dominance of the Newton’s macroscopic world view.Fractional Calculus and the Future of Science examines the nexus of these two game-changing contributions to our scientific understanding of the world. It addresses how non-integer differential equations replace Newton’s laws to describe the many guises of complexity, most of which lay beyond Newton’s experience, and many had even eluded Mandelbrot’s powerful intuition. The book’s authors look behind the mathematics and examine what must be true about a phenomenon’s behavior to justify the replacement of an integer-order with a noninteger-order (fractional) derivative. This window into the future of specific science disciplines using the fractional calculus lens suggests how what is seen entails a difference in scientific thinking and understanding