2,529 research outputs found
Efficient multistep methods for tempered fractional calculus: Algorithms and Simulations
In this work, we extend the fractional linear multistep methods in [C.
Lubich, SIAM J. Math. Anal., 17 (1986), pp.704--719] to the tempered fractional
integral and derivative operators in the sense that the tempered fractional
derivative operator is interpreted in terms of the Hadamard finite-part
integral. We develop two fast methods, Fast Method I and Fast Method II, with
linear complexity to calculate the discrete convolution for the approximation
of the (tempered) fractional operator. Fast Method I is based on a local
approximation for the contour integral that represents the convolution weight.
Fast Method II is based on a globally uniform approximation of the trapezoidal
rule for the integral on the real line. Both methods are efficient, but
numerical experimentation reveals that Fast Method II outperforms Fast Method I
in terms of accuracy, efficiency, and coding simplicity. The memory requirement
and computational cost of Fast Method II are and ,
respectively, where is the number of the final time steps and is the
number of quadrature points used in the trapezoidal rule. The effectiveness of
the fast methods is verified through a series of numerical examples for
long-time integration, including a numerical study of a fractional
reaction-diffusion model
Fast and oblivious convolution quadrature
We give an algorithm to compute steps of a convolution quadrature
approximation to a continuous temporal convolution using only
multiplications and active memory. The method does not require
evaluations of the convolution kernel, but instead evaluations of
its Laplace transform, which is assumed sectorial.
The algorithm can be used for the stable numerical solution with
quasi-optimal complexity of linear and nonlinear integral and
integro-differential equations of convolution type. In a numerical example we
apply it to solve a subdiffusion equation with transparent boundary conditions
A Gauss-Jacobi Kernel Compression Scheme for Fractional Differential Equations
A scheme for approximating the kernel of the fractional -integral
by a linear combination of exponentials is proposed and studied. The scheme is
based on the application of a composite Gauss-Jacobi quadrature rule to an
integral representation of . This results in an approximation of in an
interval , with , which converges rapidly in the number
of quadrature nodes associated with each interval of the composite rule.
Using error analysis for Gauss-Jacobi quadratures for analytic functions, an
estimate of the relative pointwise error is obtained. The estimate shows that
the number of terms required for the approximation to satisfy a prescribed
error tolerance is bounded for all , and that is bounded
for , , and
Error Estimates for Approximations of Distributed Order Time Fractional Diffusion with Nonsmooth Data
In this work, we consider the numerical solution of an initial boundary value
problem for the distributed order time fractional diffusion equation. The model
arises in the mathematical modeling of ultra-slow diffusion processes observed
in some physical problems, whose solution decays only logarithmically as the
time tends to infinity. We develop a space semidiscrete scheme based on the
standard Galerkin finite element method, and establish error estimates optimal
with respect to data regularity in and norms for both smooth
and nonsmooth initial data. Further, we propose two fully discrete schemes,
based on the Laplace transform and convolution quadrature generated by the
backward Euler method, respectively, and provide optimal convergence rates in
the norm, which exhibits exponential convergence and first-order
convergence in time, respectively. Extensive numerical experiments are provided
to verify the error estimates for both smooth and nonsmooth initial data, and
to examine the asymptotic behavior of the solution.Comment: 25 pages, 2 figure
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