Abstract: This paper considers the development of a two-point
predictor-corrector block method for solving delay differential
equations. The formulae are represented in divided difference form
and the algorithm is implemented in variable stepsize variable order
technique. The block method produces two new values at a single
integration step. Numerical results are compared with existing
methods and it is evident that the block method performs very well.
Stability regions of the block method are also investigated.
Abstract: The objective of this research is to examine the shear thinning behaviour of mixing flow of non-Newtonian fluid like toothpaste in the dissolution container with rotating stirrer. The problem under investigation is related to the chemical industry. Mixing of fluid is performed in a cylindrical container with rotating stirrer, where stirrer is eccentrically placed on the lid of the container. For the simulation purpose the associated motion of the fluid is considered as revolving of the container, with stick stirrer. For numerical prediction, a time-stepping finite element algorithm in a cylindrical polar coordinate system is adopted based on semi-implicit Taylor-Galerkin/pressure-correction scheme. Numerical solutions are obtained for non-Newtonian fluids employing power law model. Variations with power law index have been analysed, with respect to the flow structure and pressure drop.
Abstract: In this paper we study some numerical methods to solve a model one-dimensional convection–diffusion equation. The semi-discretisation of the space variable results into a system of ordinary differential equations and the solution of the latter involves the evaluation of a matrix exponent. Since the calculation of this term is computationally expensive, we study some methods based on Krylov subspace and on Restrictive Taylor series approximation respectively. We also consider the Chebyshev Pseudospectral collocation method to do the spatial discretisation and we present the numerical solution obtained by these methods.
Abstract: This paper investigates the nature of the development
of two-dimensional laminar flow of an incompressible fluid at the
reversed stagnation-point. ". In this study, we revisit the problem
of reversed stagnation-point flow over a flat plate. Proudman and
Johnson (1962) first studied the flow and obtained an asymptotic
solution by neglecting the viscous terms. This is no true in neglecting
the viscous terms within the total flow field. In particular it is pointed
out that for a plate impulsively accelerated from rest to a constant
velocity V0 that a similarity solution to the self-similar ODE is
obtained which is noteworthy completely analytical.
Abstract: Both the minimum energy consumption and
smoothness, which is quantified as a function of jerk, are generally
needed in many dynamic systems such as the automobile and the
pick-and-place robot manipulator that handles fragile equipments.
Nevertheless, many researchers come up with either solely
concerning on the minimum energy consumption or minimum jerk
trajectory. This research paper proposes a simple yet very interesting
when combining the minimum energy and jerk of indirect jerks
approaches in designing the time-dependent system yielding an
alternative optimal solution. Extremal solutions for the cost functions
of the minimum energy, the minimum jerk and combining them
together are found using the dynamic optimization methods together
with the numerical approximation. This is to allow us to simulate
and compare visually and statistically the time history of state inputs
employed by combining minimum energy and jerk designs. The
numerical solution of minimum direct jerk and energy problem are
exactly the same solution; however, the solutions from problem of
minimum energy yield the similar solution especially in term of
tendency.
Abstract: Flow movement in unsaturated soil can be expressed
by a partial differential equation, named Richards equation. The
objective of this study is the finding of an appropriate implicit
numerical solution for head based Richards equation. Some of the
well known finite difference schemes (fully implicit, Crank Nicolson
and Runge-Kutta) have been utilized in this study. In addition, the
effects of different approximations of moisture capacity function,
convergence criteria and time stepping methods were evaluated. Two
different infiltration problems were solved to investigate the
performance of different schemes. These problems include of vertical
water flow in a wet and very dry soils. The numerical solutions of
two problems were compared using four evaluation criteria and the
results of comparisons showed that fully implicit scheme is better
than the other schemes. In addition, utilizing of standard chord slope
method for approximation of moisture capacity function, automatic
time stepping method and difference between two successive
iterations as convergence criterion in the fully implicit scheme can
lead to better and more reliable results for simulation of fluid
movement in different unsaturated soils.
Abstract: In this paper we improve the quasilinearization method by barycentric Lagrange interpolation because of its numerical stability and computation speed to achieve a stable semi analytical solution. Then we applied the improved method for solving the Fin problem which is a nonlinear equation that occurs in the heat transferring. In the quasilinearization approach the nonlinear differential equation is treated by approximating the nonlinear terms by a sequence of linear expressions. The modified QLM is iterative but not perturbative and gives stable semi analytical solutions to nonlinear problems without depending on the existence of a smallness parameter. Comparison with some numerical solutions shows that the present solution is applicable.
Abstract: A numerical method for Riccati equation is presented in this work. The method is based on the replacement of unknown functions through a truncated series of hybrid of block-pulse functions and Chebyshev polynomials. The operational matrices of derivative and product of hybrid functions are presented. These matrices together with the tau method are then utilized to transform the differential equation into a system of algebraic equations. Corresponding numerical examples are presented to demonstrate the accuracy of the proposed method.
Abstract: In this work we adopt a combination of Laplace
transform and the decomposition method to find numerical solutions
of a system of multi-pantograph equations. The procedure leads to a
rapid convergence of the series to the exact solution after computing a
few terms. The effectiveness of the method is demonstrated in some
examples by obtaining the exact solution and in others by computing
the absolute error which decreases as the number of terms of the series
increases.
Abstract: In this work, we apply the Modified Laplace
decomposition algorithm in finding a numerical solution of Blasius’
boundary layer equation for the flat plate in a uniform stream. The
series solution is found by first applying the Laplace transform to the
differential equation and then decomposing the nonlinear term by the
use of Adomian polynomials. The resulting series, which is exactly the
same as that obtained by Weyl 1942a, was expressed as a rational
function by the use of diagonal padé approximant.
Abstract: Analysis for the propagation of elastic waves in
arbitrary anisotropic plates is investigated, commencing with a
formal analysis of waves in a layered plate of an arbitrary anisotropic
media, the dispersion relations of elastic waves are obtained by
invoking continuity at the interface and boundary of conditions on
the surfaces of layered plate. The obtained solutions can be used for
material systems of higher symmetry such as monoclinic,
orthotropic, transversely isotropic, cubic, and isotropic as it is
contained implicitly in the analysis. The cases of free layered plate
and layered half space are considered separately. Some special cases
have also been deduced and discussed. Finally numerical solution of
the frequency equations for an aluminum epoxy is carried out, and
the dispersion curves for the few lower modes are presented. The
results obtained theoretically have been verified numerically and
illustrated graphically.
Abstract: The mixed oxide nuclear fuel (MOX) of U and Pu contains several percent of fission products and minor actinides, such as neptunium, americium and curium. It is important to determine accurately the decay heat from Curium isotopes as they contribute significantly in the MOX fuel. This heat generation can cause samples to melt very quickly if excessive quantities of curium are present. In the present paper, we introduce a new approach that can predict the decay heat from curium isotopes. This work is a part of the project funded by King Abdulaziz City of Science and Technology (KASCT), Long-Term Comprehensive National Plan for Science, Technology and Innovations, and take place in King Abdulaziz University (KAU), Saudi Arabia. The approach is based on the numerical solution of coupled linear differential equations that describe decays and buildups of many nuclides to calculate the decay heat produced after shutdown. Results show the consistency and reliability of the approach applied.
Abstract: In this paper, a numerical solution based on nonpolynomial
cubic spline functions is used for finding the solution of
boundary value problems which arise from the problems of calculus
of variations. This approximation reduce the problems to an explicit
system of algebraic equations. Some numerical examples are also
given to illustrate the accuracy and applicability of the presented
method.
Abstract: In the numerical solution of the forward dynamics of a
multibody system, the positions and velocities of the bodies in the
system are obtained first. With the information of the system state
variables at each time step, the internal and external forces acting on
the system are obtained by appropriate contact force models if the
continuous contact method is used instead of a discrete contact
method. The local deformation of the bodies in contact, represented
by penetration, is used to compute the contact force. The ability and
suitability with current cylindrical contact force models to describe
the contact between bodies with cylindrical geometries with
particular focus on internal contacting geometries involving low
clearances and high loads simultaneously is discussed in this paper.
A comparative assessment of the performance of each model under
analysis for different contact conditions, in particular for very
different penetration and clearance values, is presented. It is
demonstrated that some models represent a rough approximation to
describe the conformal contact between cylindrical geometries
because contact forces are underestimated.
Abstract: In this paper, a new dependable algorithm based on an adaptation of the standard variational iteration method (VIM) is used for analyzing the transition from steady convection to chaos for lowto-intermediate Rayleigh numbers convection in porous media. The solution trajectories show the transition from steady convection to chaos that occurs at a slightly subcritical value of Rayleigh number, the critical value being associated with the loss of linear stability of the steady convection solution. The VIM is treated as an algorithm in a sequence of intervals for finding accurate approximate solutions to the considered model and other dynamical systems. We shall call this technique as the piecewise VIM. Numerical comparisons between the piecewise VIM and the classical fourth-order Runge–Kutta (RK4) numerical solutions reveal that the proposed technique is a promising tool for the nonlinear chaotic and nonchaotic systems.
Abstract: In this paper, we have applied the homotopy perturbation
method (HPM) for obtaining the analytical solution of unsteady
flow of gas through a porous medium and we have also compared the
findings of this research with some other analytical results. Results
showed a very good agreement between results of HPM and the
numerical solutions of the problem rather than other analytical solutions
which have previously been applied. The results of homotopy
perturbation method are of high accuracy and the method is very
effective and succinct.
Abstract: Fick's second law equations for unsteady state
diffusion of salt into the potato tissues were solved numerically. The
set of equations resulted from implicit modeling were solved using
Thomas method to find the salt concentration profiles in solid phase.
The needed effective diffusivity and equilibrium distribution
coefficient were determined experimentally. Cylindrical samples of
potato were infused with aqueous NaCl solutions of 1-3%
concentrations, and variations in salt concentrations of brine were
determined over time. Solute concentrations profiles of samples were
determined by measuring salt uptake of potato slices. For the studied
conditions, equilibrium distribution coefficients were found to be
dependent on salt concentrations, whereas the effective diffusivity
was slightly affected by brine concentration.
Abstract: Unsteady magnetohydrodynamics (MHD) boundary
layer flow and heat transfer over a continuously stretching surface in
the presence of radiation is examined. By similarity transformation,
the governing partial differential equations are transformed to a set of
ordinary differential equations. Numerical solutions are obtained by
employing the Runge-Kutta-Fehlberg method scheme with shooting
technique in Maple software environment. The effects of
unsteadiness parameter, radiation parameter, magnetic parameter and
Prandtl number on the heat transfer characteristics are obtained and
discussed. It is found that the heat transfer rate at the surface
increases as the Prandtl number and unsteadiness parameter increase
but decreases with magnetic and radiation parameter.
Abstract: The analytical prediction of the decay heat results
from the fast neutron fission of actinides was initiated under a project, 10-MAT1134-3, funded by king Abdulaziz City of Science
and Technology (KASCT), Long-Term Comprehensive National Plan for Science, Technology and Innovations, managed by a team
from King Abdulaziz University (KAU), Saudi Arabia, and
supervised by Argonne National Laboratory (ANL) has collaborated
with KAU's team to assist in the computational analysis. In this paper, the numerical solution of coupled linear differential equations
that describe the decays and buildups of minor fission product MFA, has been used to predict the total decay heat and its components from the fast neutron fission of 235U and 239Pu. The reliability of the present approach is illustrated via systematic
comparisons with the measurements reported by the University of
Tokyo, in YAYOI reactor.
Abstract: Stochastic models of biological networks are well established in systems biology, where the computational treatment of such models is often focused on the solution of the so-called chemical master equation via stochastic simulation algorithms. In contrast to this, the development of storage-efficient model representations that are directly suitable for computer implementation has received significantly less attention. Instead, a model is usually described in terms of a stochastic process or a "higher-level paradigm" with graphical representation such as e.g. a stochastic Petri net. A serious problem then arises due to the exponential growth of the model-s state space which is in fact a main reason for the popularity of stochastic simulation since simulation suffers less from the state space explosion than non-simulative numerical solution techniques. In this paper we present transition class models for the representation of biological network models, a compact mathematical formalism that circumvents state space explosion. Transition class models can also serve as an interface between different higher level modeling paradigms, stochastic processes and the implementation coded in a programming language. Besides, the compact model representation provides the opportunity to apply non-simulative solution techniques thereby preserving the possible use of stochastic simulation. Illustrative examples of transition class representations are given for an enzyme-catalyzed substrate conversion and a part of the bacteriophage λ lysis/lysogeny pathway.