128,475 research outputs found
Numerical Methods for Solving Fractional Differential Equations
Department of Mathematical SciencesIn this thesis, several efficient numerical methods are proposed to solve initial value problems and boundary value problems of fractional di???erential equations.
For fractional initial value problems, we propose a new type of the predictorevaluate-corrector-evaluate method based on the Caputo fractional derivative operator.
Furthermore, we propose a new type of the Caputo fractional derivative operator that does not have a di???erential form of a solution. However, with some fractional orders, there are problems that a solution blows up and the scheme has a low convergence. Thus, we identify new treatments for these values. Then, we can expect a significant improvement for all fractional orders. The advantages and improvements are shown by testing various numerical examples.
For fractional BVPs, we propose an explicit method that dramatically reduces the computational time for solving a dense matrix system. Moreover, by adopting
high-order predictor-corrector methods which have uniform convergence rates O(h2) or O(h3) for all fractional orders [8], we propose a second-order method and a third-order method by using the Newton???s method and the Halley method, respectively. We show its advantage by testing various numerical examples.clos
Rigorous Simulations of 3D Patterns on Extreme Ultraviolet Lithography Masks
Simulations of light scattering off an extreme ultraviolet lithography mask
with a 2D-periodic absorber pattern are presented. In a detailed convergence
study it is shown that accurate results can be attained for relatively large 3D
computational domains and in the presence of sidewall-angles and
corner-roundings.Comment: SPIE Europe Optical Metrology, Conference Proceeding
Multigrid waveform relaxation for the time-fractional heat equation
In this work, we propose an efficient and robust multigrid method for solving
the time-fractional heat equation. Due to the nonlocal property of fractional
differential operators, numerical methods usually generate systems of equations
for which the coefficient matrix is dense. Therefore, the design of efficient
solvers for the numerical simulation of these problems is a difficult task. We
develop a parallel-in-time multigrid algorithm based on the waveform relaxation
approach, whose application to time-fractional problems seems very natural due
to the fact that the fractional derivative at each spatial point depends on the
values of the function at this point at all earlier times. Exploiting the
Toeplitz-like structure of the coefficient matrix, the proposed multigrid
waveform relaxation method has a computational cost of
operations, where is the number of time steps and is the number of
spatial grid points. A semi-algebraic mode analysis is also developed to
theoretically confirm the good results obtained. Several numerical experiments,
including examples with non-smooth solutions and a nonlinear problem with
applications in porous media, are presented
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