75,996 research outputs found
Aspects of interval analysis applied to initial-value problems for ordinary differential equations and hyperbolic partial differential equations
Interval analysis is an essential tool in the construction of validated numerical solutions
of Initial Value Problems (IVP) for Ordinary (ODE) and Partial (PDE) Differential
Equations. A validated solution typically consists of guaranteed lower and upper bounds
for the exact solution or set of exact solutions in the case of uncertain data, i.e. it is an
interval function (enclosure) containing all solutions of the problem.
IVP for ODE: The central point of discussion is the wrapping effect. A new concept of
wrapping function is introduced and applied in studying this effect. It is proved that the
wrapping function is the limit of the enclosures produced by any method of certain type
(propagate and wrap type). Then, the wrapping effect can be quantified as the difference
between the wrapping function and the optimal interval enclosure of the solution set (or
some norm of it). The problems with no wrapping effect are characterized as problems for
which the wrapping function equals the optimal interval enclosure. A sufficient condition
for no wrapping effect is that there exist a linear transformation, preserving the intervals,
which reduces the right-hand side of the system of ODE to a quasi-isotone function. This
condition is also necessary for linear problems and "near" necessary in the general case.
Hyperbolic PDE: The Initial Value Problem with periodic boundary conditions for
the wave equation is considered. It is proved that under certain conditions the problem
is an operator equation with an operator of monotone type. Using the established monotone
properties, an interval (validated) method for numerical solution of the problem is
proposed. The solution is obtained step by step in the time dimension as a Fourier series
of the space variable and a polynomial of the time variable. The numerical implementation
involves computations in Fourier and Taylor functoids. Propagation of discontinuo~swaves
is a serious problem when a Fourier series is used (Gibbs phenomenon, etc.). We
propose the combined use of periodic splines and Fourier series for representing discontinuous
functions and a method for propagating discontinuous waves. The numerical implementation involves computations in a Fourier hyper functoid.Mathematical SciencesD. Phil. (Mathematics
Status of the differential transformation method
Further to a recent controversy on whether the differential transformation
method (DTM) for solving a differential equation is purely and solely the
traditional Taylor series method, it is emphasized that the DTM is currently
used, often only, as a technique for (analytically) calculating the power
series of the solution (in terms of the initial value parameters). Sometimes, a
piecewise analytic continuation process is implemented either in a numerical
routine (e.g., within a shooting method) or in a semi-analytical procedure
(e.g., to solve a boundary value problem). Emphasized also is the fact that, at
the time of its invention, the currently-used basic ingredients of the DTM
(that transform a differential equation into a difference equation of same
order that is iteratively solvable) were already known for a long time by the
"traditional"-Taylor-method users (notably in the elaboration of software
packages --numerical routines-- for automatically solving ordinary differential
equations). At now, the defenders of the DTM still ignore the, though much
better developed, studies of the "traditional"-Taylor-method users who, in
turn, seem to ignore similarly the existence of the DTM. The DTM has been given
an apparent strong formalization (set on the same footing as the Fourier,
Laplace or Mellin transformations). Though often used trivially, it is easily
attainable and easily adaptable to different kinds of differentiation
procedures. That has made it very attractive. Hence applications to various
problems of the Taylor method, and more generally of the power series method
(including noninteger powers) has been sketched. It seems that its potential
has not been exploited as it could be. After a discussion on the reasons of the
"misunderstandings" which have caused the controversy, the preceding topics are
concretely illustrated.Comment: To appear in Applied Mathematics and Computation, 29 pages,
references and further considerations adde
Steady and Stable: Numerical Investigations of Nonlinear Partial Differential Equations
Excerpt: Mathematics is a language which can describe patterns in everyday life as well as abstract concepts existing only in our minds. Patterns exist in data, functions, and sets constructed around a common theme, but the most tangible patterns are visual. Visual demonstrations can help undergraduate students connect to abstract concepts in advanced mathematical courses. The study of partial differential equations, in particular, benefits from numerical analysis and simulation
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
Spatial Manifestations of Order Reduction in Runge-Kutta Methods for Initial Boundary Value Problems
This paper studies the spatial manifestations of order reduction that occur
when time-stepping initial-boundary-value problems (IBVPs) with high-order
Runge-Kutta methods. For such IBVPs, geometric structures arise that do not
have an analog in ODE IVPs: boundary layers appear, induced by a mismatch
between the approximation error in the interior and at the boundaries. To
understand those boundary layers, an analysis of the modes of the numerical
scheme is conducted, which explains under which circumstances boundary layers
persist over many time steps. Based on this, two remedies to order reduction
are studied: first, a new condition on the Butcher tableau, called weak stage
order, that is compatible with diagonally implicit Runge-Kutta schemes; and
second, the impact of modified boundary conditions on the boundary layer theory
is analyzed.Comment: 41 pages, 9 figure
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