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The numerical solution of stefan problems with front-tracking and smoothing methods
A variational Bayesian method for inverse problems with impulsive noise
We propose a novel numerical method for solving inverse problems subject to
impulsive noises which possibly contain a large number of outliers. The
approach is of Bayesian type, and it exploits a heavy-tailed t distribution for
data noise to achieve robustness with respect to outliers. A hierarchical model
with all hyper-parameters automatically determined from the given data is
described. An algorithm of variational type by minimizing the Kullback-Leibler
divergence between the true posteriori distribution and a separable
approximation is developed. The numerical method is illustrated on several one-
and two-dimensional linear and nonlinear inverse problems arising from heat
conduction, including estimating boundary temperature, heat flux and heat
transfer coefficient. The results show its robustness to outliers and the fast
and steady convergence of the algorithm.Comment: 20 pages, to appear in J. Comput. Phy
Optimal Perturbation Iteration Method for Bratu-Type Problems
In this paper, we introduce the new optimal perturbation iteration method
based on the perturbation iteration algorithms for the approximate solutions of
nonlinear differential equations of many types. The proposed method is
illustrated by studying Bratu-type equations. Our results show that only a few
terms are required to obtain an approximate solution which is more accurate and
efficient than many other methods in the literature.Comment: 11 pages, 3 Figure
Lie and conditional symmetries of a class of nonlinear (1+2)-dimensional boundary value problems
A new definition of conditional invariance for boundary value problems
involving a wide range of boundary conditions (including initial value problems
as a special case) is proposed. It is shown that other definitions worked out
in order to find Lie symmetries of boundary value problems with standard
boundary conditions, follow as particular cases from our definition. Simple
examples of direct applicability to the nonlinear problems arising in
applications are demonstrated. Moreover, the successful application of the
definition for the Lie and conditional symmetry classification of a class of
(1+2)-dimensional nonlinear boundary value problems governed by the nonlinear
diffusion equation in a semi-infinite domain is realised. In particular, it is
proved that there is a special exponent, , for the power diffusivity
when the problem in question with non-vanishing flux on the boundary
admits additional Lie symmetry operators compared to the case . In
order to demonstrate the applicability of the symmetries derived, they are used
for reducing the nonlinear problems with power diffusivity and a constant
non-zero flux on the boundary (such problems are common in applications and
describing a wide range of phenomena) to (1+1)-dimensional problems. The
structure and properties of the problems obtained are briefly analysed.
Finally, some results demonstrating how Lie invariance of the boundary value
problem in question depends on geometry of the domain are presented.Comment: 25 pages; the main results were presented at the Conference Symmetry,
Methods, Applications and Related Fields, Vancouver, Canada, May 13-16, 201
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
Arc-Length Continuation and Multigrid Techniques for Nonlinear Elliptic Eigenvalue Problems
We investigate multi-grid methods for solving linear systems arising from arc-length continuation techniques applied to nonlinear elliptic eigenvalue problems. We find that the usual multi-grid methods diverge in the neighborhood of singular points of the solution branches. As a result, the continuation method is unable to continue past a limit point in the Bratu problem. This divergence is analyzed and a modified multi-grid algorithm has been devised based on this analysis. In principle, this new multi-grid algorithm converges for elliptic systems, arbitrarily close to singularity and has been used successfully in conjunction with arc-length continuation procedures on the model problem. In the worst situation, both the storage and the computational work are only about a factor of two more than the unmodified multi-grid methods
A new approach for solving nonlinear Thomas-Fermi equation based on fractional order of rational Bessel functions
In this paper, the fractional order of rational Bessel functions collocation
method (FRBC) to solve Thomas-Fermi equation which is defined in the
semi-infinite domain and has singularity at and its boundary condition
occurs at infinity, have been introduced. We solve the problem on semi-infinite
domain without any domain truncation or transformation of the domain of the
problem to a finite domain. This approach at first, obtains a sequence of
linear differential equations by using the quasilinearization method (QLM),
then at each iteration solves it by FRBC method. To illustrate the reliability
of this work, we compare the numerical results of the present method with some
well-known results in other to show that the new method is accurate, efficient
and applicable
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