1,458 research outputs found
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
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Localized boundary-domain integral formulations for problems with variable coefficients
Specially constructed localized parametrixes are used in this paper instead of a fundamental solution to reduce a boundary value problem with variable coefficients to a localized boundary-domain integral or integro-differential equation (LBDIE or LBDIDE). After discretization, this results in a sparsely populated system of linear algebraic equations, which can be solved by well-known efficient methods. This make the method competitive with the finite element method for such problems. Some methods of the parametrix localization are discussed and the corresponding LBDIEs and LBDIDEs are introduced. Both mesh-based and meshless algorithms for the localized equations discretization are described
Application of Operator Splitting Methods in Finance
Financial derivatives pricing aims to find the fair value of a financial
contract on an underlying asset. Here we consider option pricing in the partial
differential equations framework. The contemporary models lead to
one-dimensional or multidimensional parabolic problems of the
convection-diffusion type and generalizations thereof. An overview of various
operator splitting methods is presented for the efficient numerical solution of
these problems.
Splitting schemes of the Alternating Direction Implicit (ADI) type are
discussed for multidimensional problems, e.g. given by stochastic volatility
(SV) models. For jump models Implicit-Explicit (IMEX) methods are considered
which efficiently treat the nonlocal jump operator. For American options an
easy-to-implement operator splitting method is described for the resulting
linear complementarity problems.
Numerical experiments are presented to illustrate the actual stability and
convergence of the splitting schemes. Here European and American put options
are considered under four asset price models: the classical Black-Scholes
model, the Merton jump-diffusion model, the Heston SV model, and the Bates SV
model with jumps
Numerical simulations of time resolved quantum electronics
This paper discusses the technical aspects - mathematical and numerical -
associated with the numerical simulations of a mesoscopic system in the time
domain (i.e. beyond the single frequency AC limit). After a short review of the
state of the art, we develop a theoretical framework for the calculation of
time resolved observables in a general multiterminal system subject to an
arbitrary time dependent perturbation (oscillating electrostatic gates, voltage
pulses, time-vaying magnetic fields) The approach is mathematically equivalent
to (i) the time dependent scattering formalism, (ii) the time resolved Non
Equilibrium Green Function (NEGF) formalism and (iii) the partition-free
approach. The central object of our theory is a wave function that obeys a
simple Schrodinger equation with an additional source term that accounts for
the electrons injected from the electrodes. The time resolved observables
(current, density. . .) and the (inelastic) scattering matrix are simply
expressed in term of this wave function. We use our approach to develop a
numerical technique for simulating time resolved quantum transport. We find
that the use of this wave function is advantageous for numerical simulations
resulting in a speed up of many orders of magnitude with respect to the direct
integration of NEGF equations. Our technique allows one to simulate realistic
situations beyond simple models, a subject that was until now beyond the
simulation capabilities of available approaches.Comment: Typographic mistakes in appendix C were correcte
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Proceedings of the Eleventh UK Conference on Boundary Integral Methods (UKBIM 11), 10-11 July 2017, Nottingham Conference Centre, Nottingham Trent University
This book contains the abstracts and papers presented at the Eleventh UK Conference on Boundary Integral Methods (UKBIM 11), held at Nottingham Trent University in July 2017. The work presented at the conference, and published in this volume, demonstrates the wide range of work that is being carried out in the UK, as well as from further afield
Analytical solutions forfuzzysystem using power series approach
The aim of the present paper is present a relatively new analytical method, called residual power series (RPS) method, for solving system of fuzzy initial value problems under strongly generalized differentiability. The technique methodology provides the solution in the form of a rapidly convergent series with easily computable components using symbolic computation software. Several computational experiments are given to show the good performance and potentiality of the proposed procedure. The results reveal that the present simulated method is very effective, straightforward and powerful methodology to solve such fuzzy equations
One approach to solve a nonlinear boundary value problem for the Fredholm integro-differential equation
A quasilinear boundary value problem for a Fredholm integro-differential equation is considered. The
interval is divided into N parts and the values of the solution to the equation at the left end points of the subintervals are introduced as additional parameters. New unknown functions are introduced on the subintervals and a special Cauchy problem with parameters is solved with respect to a system of such functions. By means of the solution to this problem, a new general solution to the quasilinear Fredholm integro-differential equation is constructed. The conditions of the existence of a unique new general solution to the equation under consideration are obtained. A new general solution is used to create a system of nonlinear algebraic equations in parameters introduced. The conditions for the existence of a unique solution to this system are established. This ensures the existence of a unique solution to original problem
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