Abstract: The aim of this paper is to investigate the
performance of the developed two point block method designed for
two processors for solving directly non stiff large systems of higher
order ordinary differential equations (ODEs). The method calculates
the numerical solution at two points simultaneously and produces
two new equally spaced solution values within a block and it is
possible to assign the computational tasks at each time step to a
single processor. The algorithm of the method was developed in C
language and the parallel computation was done on a parallel shared
memory environment. Numerical results are given to compare the
efficiency of the developed method to the sequential timing. For
large problems, the parallel implementation produced 1.95 speed-up
and 98% efficiency for the two processors.
Abstract: This paper presents investigation effects of a sharp edged gust on aeroelastic behavior and time-domain response of a typical section model using Jones approximate aerodynamics for pure plunging motion. Flutter analysis has been done by using p and p-k methods developed for presented finite-state aerodynamic model for a typical section model (airfoil). Introduction of gust analysis as a linear set of ordinary differential equations in a simplified procedure has been carried out by using transformation into an eigenvalue problem.
Abstract: The aim of this paper is to investigate twodimensional unsteady flow of a viscous incompressible fluid about stagnation point on permeable stretching sheet in presence of time dependent free stream velocity. Fluid is considered in the influence of transverse magnetic field in the presence of radiation effect. Rosseland approximation is use to model the radiative heat transfer. Using time-dependent stream function, partial differential equations corresponding to the momentum and energy equations are converted into non-linear ordinary differential equations. Numerical solutions of these equations are obtained by using Runge-Kutta Fehlberg method with the help of Newton-Raphson shooting technique. In the present work the effect of unsteadiness parameter, magnetic field parameter, radiation parameter, stretching parameter and the Prandtl number on flow and heat transfer characteristics have been discussed. Skin-friction coefficient and Nusselt number at the sheet are computed and discussed. The results reported in the paper are in good agreement with published work in literature by other researchers.
Abstract: An evolutionary computing technique for solving initial value problems in Ordinary Differential Equations is proposed in this paper. Neural network is used as a universal approximator while the adaptive parameters of neural networks are optimized by genetic algorithm. The solution is achieved on the continuous grid of time instead of discrete as in other numerical techniques. The comparison is carried out with classical numerical techniques and the solution is found with a uniform accuracy of MSE ≈ 10-9 .
Abstract: Solving Ordinary Differential Equations (ODEs) by
using Partitioning Block Intervalwise (PBI) technique is our aim in
this paper. The PBI technique is based on Block Adams Method and
Backward Differentiation Formula (BDF). Block Adams Method
only use the simple iteration for solving while BDF requires Newtonlike
iteration involving Jacobian matrix of ODEs which consumes a
considerable amount of computational effort. Therefore, PBI is
developed in order to reduce the cost of iteration within acceptable
maximum error
Abstract: An important task in solving second order linear ordinary differential equations by the finite difference is to choose a suitable stepsize h. In this paper, by using the stochastic arithmetic, the CESTAC method and the CADNA library we present a procedure to estimate the optimal stepsize hopt, the stepsize which minimizes the global error consisting of truncation and round-off error.
Abstract: The RK5GL3 method is a numerical method for solving
initial value problems in ordinary differential equations, and is based
on a combination of a fifth-order Runge-Kutta method and 3-point
Gauss-Legendre quadrature. In this paper we describe the propagation
of local errors in this method, and show that the global order of
RK5GL3 is expected to be six, one better than the underlying Runge-
Kutta method.
Abstract: The main aim of this study is to describe and introduce a method of numerical analysis in obtaining approximate solutions for the SIR-SI differential equations (susceptible-infectiverecovered for human populations; susceptible-infective for vector populations) that represent a model for dengue disease transmission. Firstly, we describe the ordinary differential equations for the SIR-SI disease transmission models. Then, we introduce the numerical analysis of solutions of this continuous time, discrete space SIR-SI model by simplifying the continuous time scale to a densely populated, discrete time scale. This is followed by the application of this numerical analysis of solutions of the SIR-SI differential equations to the estimation of relative risk using continuous time, discrete space dengue data of Kuala Lumpur, Malaysia. Finally, we present the results of the analysis, comparing and displaying the results in graphs, table and maps. Results of the numerical analysis of solutions that we implemented offers a useful and potentially superior model for estimating relative risks based on continuous time, discrete space data for vector borne infectious diseases specifically for dengue disease.
Abstract: The effect of variable chemical reaction on heat and mass transfer characteristics over unsteady stretching surface embedded in a porus medium is studied. The governing time dependent boundary layer equations are transformed into ordinary differential equations containing chemical reaction parameter, unsteadiness parameter, Prandtl number and Schmidt number. These equations have been transformed into a system of first order differential equations. MATHEMATICA has been used to solve this system after obtaining the missed initial conditions. The velocity gradient, temperature, and concentration profiles are computed and discussed in details for various values of the different parameters.
Abstract: In this paper we propose a class of second derivative multistep methods for solving some well-known classes of Lane- Emden type equations which are nonlinear ordinary differential equations on the semi-infinite domain. These methods, which have good stability and accuracy properties, are useful in deal with stiff ODEs. We show superiority of these methods by applying them on the some famous Lane-Emden type equations.
Abstract: In this paper, linear multistep technique using power
series as the basis function is used to develop the block methods
which are suitable for generating direct solution of the special second
order ordinary differential equations with associated initial or
boundary conditions. The continuous hybrid formulations enable us
to differentiate and evaluate at some grids and off – grid points to
obtain two different four discrete schemes, each of order (5,5,5,5)T,
which were used in block form for parallel or sequential solutions of
the problems. The computational burden and computer time wastage
involved in the usual reduction of second order problem into system
of first order equations are avoided by this approach. Furthermore, a
stability analysis and efficiency of the block methods are tested on
linear and non-linear ordinary differential equations and the results
obtained compared favorably with the exact solution.
Abstract: As we know, most differential equations concerning
physical phenomenon could not be solved by analytical method. Even if we use Series Method, some times we need an appropriate change of variable, and even when we can, their closed form solution may be
so complicated that using it to obtain an image or to examine the structure of the system is impossible. For example, if we consider Schrodinger equation, i.e.,
We come to a three-term recursion relations, which work with it takes, at least, a little bit time to get a series solution[6]. For this
reason we use a change of variable such as or when we consider the orbital angular momentum[1], it will be
necessary to solve. As we can observe, working with this equation is tedious. In this paper, after introducing Clenshaw method, which is a kind of Spectral method, we try to solve some of such equations.
Abstract: This paper examines the forced convection flow of
incompressible, electrically conducting viscous fluid past a sharp
wedge in the presence of heat generation or absorption with an
applied magnetic field. The system of partial differential equations
governing Falkner - Skan wedge flow and heat transfer is first
transformed into a system of ordinary differential equations using
similarity transformations which is later solved using an implicit
finite - difference scheme, along with quasilinearization technique.
Numerical computations are performed for air (Pr = 0.7) and
displayed graphically to illustrate the influence of pertinent physical
parameters on local skin friction and heat transfer coefficients and,
also on, velocity and temperature fields. It is observed that the
magnetic field increases both the coefficients of skin friction and heat
transfer. The effect of heat generation or absorption is found to be
very significant on heat transfer, but its effect on the skin friction is
negligible. Indeed, the occurrence of overshoot is noticed in the
temperature profiles during heat generation process, causing the
reversal in the direction of heat transfer.
Abstract: This paper presents the vibrations suppression of a thermoelastic beam subject to sudden heat input by a distributed piezoelectric actuators. An optimization problem is formulated as the minimization of a quadratic functional in terms of displacement and velocity at a given time and with the least control effort. The solution method is based on a combination of modal expansion and variational approaches. The modal expansion approach is used to convert the optimal control of distributed parameter system into the optimal control of lumped parameter system. By utilizing the variational approach, an explicit optimal control law is derived and the determination of the corresponding displacement and velocity is reduced to solving a set of ordinary differential equations.
Abstract: The RK1GL2X3 method is a numerical method for solving initial value problems in ordinary differential equations, and is based on the RK1GL2 method which, in turn, is a particular case of the general RKrGLm method. The RK1GL2X3 method is a fourth-order method, even though its underlying Runge-Kutta method RK1 is the first-order Euler method, and hence, RK1GL2X3 is considerably more efficient than RK1. This enhancement is achieved through an implementation involving triple-nested two-point Gauss- Legendre quadrature.
Abstract: In this paper, a self starting two step continuous block
hybrid formulae (CBHF) with four Off-step points is developed using
collocation and interpolation procedures. The CBHF is then used to
produce multiple numerical integrators which are of uniform order
and are assembled into a single block matrix equation. These
equations are simultaneously applied to provide the approximate
solution for the stiff ordinary differential equations. The order of
accuracy and stability of the block method is discussed and its
accuracy is established numerically.
Abstract: The RK5GL3 method is a numerical method for solving
initial value problems in ordinary differential equations, and is
based on a combination of a fifth-order Runge-Kutta method and
3-point Gauss-Legendre quadrature. In this paper we describe an
effective local error control algorithm for RK5GL3, which uses local
extrapolation with an eighth-order Runge-Kutta method in tandem
with RK5GL3, and a Hermite interpolating polynomial for solution
estimation at the Gauss-Legendre quadrature nodes.
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: In this paper, a direct method based on variable step
size Block Backward Differentiation Formula which is referred as
BBDF2 for solving second order Ordinary Differential Equations
(ODEs) is developed. The advantages of the BBDF2 method over the
corresponding sequential variable step variable order Backward
Differentiation Formula (BDFVS) when used to solve the same
problem as a first order system are pointed out. Numerical results are
given to validate the method.
Abstract: We seek exact solutions of the coupled Klein-Gordon-Schrödinger equation with variable coefficients with the aid of Lie classical approach. By using the Lie classical method, we are able to derive symmetries that are used for reducing the coupled system of partial differential equations into ordinary differential equations. From reduced differential equations we have derived some new exact solutions of coupled Klein-Gordon-Schrödinger equations involving some special functions such as Airy wave functions, Bessel functions, Mathieu functions etc.