11,324 research outputs found
A fractional B-spline collocation method for the numerical solution of fractional predator-prey models
We present a collocation method based on fractional B-splines for the solution of fractional differential problems. The key-idea is to use the space generated by the fractional B-splines, i.e., piecewise polynomials of noninteger degree, as approximating space. Then, in the collocation step the fractional derivative of the approximating function is approximated accurately and efficiently by an exact differentiation rule that involves the generalized finite difference operator. To show the effectiveness of the method for the solution of nonlinear dynamical systems of fractional order, we solved the fractional Lotka-Volterra model and a fractional predator-pray model with variable coefficients. The numerical tests show that the method we proposed is accurate while keeping a low computational cost
Numerical Solution of ODEs and the Columbus' Egg: Three Simple Ideas for Three Difficult Problems
On computers, discrete problems are solved instead of continuous ones. One
must be sure that the solutions of the former problems, obtained in real time
(i.e., when the stepsize h is not infinitesimal) are good approximations of the
solutions of the latter ones. However, since the discrete world is much richer
than the continuous one (the latter being a limit case of the former), the
classical definitions and techniques, devised to analyze the behaviors of
continuous problems, are often insufficient to handle the discrete case, and
new specific tools are needed. Often, the insistence in following a path
already traced in the continuous setting, has caused waste of time and efforts,
whereas new specific tools have solved the problems both more easily and
elegantly. In this paper we survey three of the main difficulties encountered
in the numerical solutions of ODEs, along with the novel solutions proposed.Comment: 25 pages, 4 figures (typos fixed
Solving seismic wave propagation in elastic media using the matrix exponential approach
Three numerical algorithms are proposed to solve the time-dependent
elastodynamic equations in elastic solids. All algorithms are based on
approximating the solution of the equations, which can be written as a matrix
exponential. By approximating the matrix exponential with a product formula, an
unconditionally stable algorithm is derived that conserves the total elastic
energy density. By expanding the matrix exponential in Chebyshev polynomials
for a specific time instance, a so-called ``one-step'' algorithm is constructed
that is very accurate with respect to the time integration. By formulating the
conventional velocity-stress finite-difference time-domain algorithm (VS-FDTD)
in matrix exponential form, the staggered-in-time nature can be removed by a
small modification, and higher order in time algorithms can be easily derived.
For two different seismic events the accuracy of the algorithms is studied and
compared with the result obtained by using the conventional VS-FDTD algorithm.Comment: 13 pages revtex, 6 figures, 2 table
Non-finite-difference algorithm for integrating Newton's motion equations
We have presented some practical consequences on the molecular-dynamics
simulations arising from the numerical algorithm published recently in paper
Int. J. Mod. Phys. C 16, 413 (2005). The algorithm is not a finite-difference
method and therefore it could be complementary to the traditional numerical
integrating of the motion equations. It consists of two steps. First, an
analytic form of polynomials in some formal parameter (we put
after all) is derived, which approximate the solution of the system
of differential equations under consideration. Next, the numerical values of
the derived polynomials in the interval, in which the difference between them
and their truncated part of smaller degree does not exceed a given accuracy
, become the numerical solution. The particular examples, which we
have considered, represent the forced linear and nonlinear oscillator and the
2D Lennard-Jones fluid. In the latter case we have restricted to the
polynomials of the first degree in formal parameter .
The computer simulations play very important role in modeling materials with
unusual properties being contradictictory to our intuition. The particular
example could be the auxetic materials. In this case, the accuracy of the
applied numerical algorithms as well as various side-effects, which might
change the physical reality, could become important for the properties of the
simulated material.Comment: 11 page
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