Abstract: In this paper, a modified harmonic balance method based an analytical technique has been developed to determine higher-order approximate periodic solutions of a conservative nonlinear oscillator for which the elastic force term is proportional to x1/3. Usually, a set of nonlinear algebraic equations is solved in this method. However, analytical solutions of these algebraic equations are not always possible, especially in the case of a large oscillation. In this article, different parameters of the same nonlinear problems are found, for which the power series produces desired results even for the large oscillation. We find a modified harmonic balance method works very well for the whole range of initial amplitudes, and the excellent agreement of the approximate frequencies and periodic solutions with the exact ones has been demonstrated and discussed. Besides these, a suitable truncation formula is found in which the solution measures better results than existing solutions. The method is mainly illustrated by the x1/3 force nonlinear oscillator but it is also useful for many other nonlinear problems.
Abstract: The objective of this paper is to present a
comparative study of Homotopy Perturbation Method (HPM),
Variational Iteration Method (VIM) and Homotopy Analysis
Method (HAM) for the semi analytical solution of Kortweg-de
Vries (KdV) type equation called KdV. The study have been
highlighted the efficiency and capability of aforementioned methods
in solving these nonlinear problems which has been arisen from a
number of important physical phenomenon.
Abstract: The Helmholtz equation often arises in the study of physical problems involving partial differential equation. Many researchers have proposed numerous methods to find the analytic or approximate solutions for the proposed problems. In this work, the exact analytical solutions of the Helmholtz equation in spherical polar coordinates are presented using the Nikiforov-Uvarov (NU) method. It is found that the solution of the angular eigenfunction can be expressed by the associated-Legendre polynomial and radial eigenfunctions are obtained in terms of the Laguerre polynomials. The special case for k=0, which corresponds to the Laplace equation is also presented.
Abstract: A fully implicit finite-difference method has been proposed for the numerical solutions of one dimensional coupled nonlinear Burgers’ equations on the uniform mesh points. The method forms a system of nonlinear difference equations which is to be solved at each iteration. Newton’s iterative method has been implemented to solve this nonlinear assembled system of equations. The linear system has been solved by Gauss elimination method with partial pivoting algorithm at each iteration of Newton’s method. Three test examples have been carried out to illustrate the accuracy of the method. Computed solutions obtained by proposed scheme have been compared with analytical solutions and those already available in the literature by finding L2 and L∞ errors.
Abstract: The Marangoni convective instability in a horizontal
fluid layer with the insoluble surfactant and nondeformable free
surface is investigated. The surface tension at the free surface is
linearly dependent on the temperature and concentration gradients.
At the bottom surface, the temperature conditions of uniform
temperature and uniform heat flux are considered. By linear stability
theory, the exact analytical solutions for the steady Marangoni
convection are derived and the marginal curves are plotted. The
effects of surfactant or elasticity number, Lewis number and Biot
number on the marginal Marangoni instability are assessed. The
surfactant concentration gradients and the heat transfer mechanism at
the free surface have stabilizing effects while the Lewis number
destabilizes fluid system. The fluid system with uniform temperature
condition at the bottom boundary is more stable than the fluid layer
that is subjected to uniform heat flux at the bottom boundary.
Abstract: In order to obtain an accurate result of the heat transfer
of the rib in the internal cooling Rectangular channel, using separation
of variables, analytical solutions of three dimensional steady-state heat
conduction in rectangular ribs are given by solving three dimensional
steady-state function of the rectangular ribs. Therefore, we can get
solution of three dimensional temperature field in the rib. Based on the
solution, we can get how the Bi number affected on heat transfer.
Furthermore, comparisons of the analytical and numerical results
indicate agreement on temperature field in the rib.
Abstract: The analytical solutions for geodesic acoustic
eigenmodes in tokamak plasmas with circular concentric magnetic
surfaces are found. In the frame of ideal magnetohydrodynamics the
dispersion relation taking into account the toroidal coupling between
electrostatic perturbations and electromagnetic perturbations with
poloidal mode number |m| = 2 is derived. In the absence of such
a coupling the dispersion relation gives the standard continuous
spectrum of geodesic acoustic modes. The analysis of the existence
of global eigenmodes for plasma equilibria with both off-axis
and on-axis maximum of the local geodesic acoustic frequency is
performed.
Abstract: This paper analyses the unsteady, two-dimensional
stagnation point flow of an incompressible viscous fluid over a flat
sheet when the flow is started impulsively from rest and at the same
time, the sheet is suddenly stretched in its own plane with a velocity
proportional to the distance from the stagnation point. The partial
differential equations governing the laminar boundary layer forced
convection flow are non-dimensionalised using semi-similar
transformations and then solved numerically using an implicit finitedifference
scheme known as the Keller-box method. Results
pertaining to the flow and heat transfer characteristics are computed
for all dimensionless time, uniformly valid in the whole spatial region
without any numerical difficulties. Analytical solutions are also
obtained for both small and large times, respectively representing the
initial unsteady and final steady state flow and heat transfer.
Numerical results indicate that the velocity ratio parameter is found
to have a significant effect on skin friction and heat transfer rate at
the surface. Furthermore, it is exposed that there is a smooth
transition from the initial unsteady state flow (small time solution) to
the final steady state (large time solution).
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: 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: In this paper, collocation based cubic B-spline and
extended cubic uniform B-spline method are considered for
solving one-dimensional heat equation with a nonlocal initial
condition. Finite difference and θ-weighted scheme is used for
time and space discretization respectively. The stability of the
method is analyzed by the Von Neumann method. Accuracy of
the methods is illustrated with an example. The numerical results
are obtained and compared with the analytical solutions.
Abstract: In this paper, the telegraph equation is solved numerically by cubic B-spline quasi-interpolation .We obtain the numerical scheme, by using the derivative of the quasi-interpolation to approximate the spatial derivative of the dependent variable and a low order forward difference to approximate the temporal derivative of the dependent variable. The advantage of the resulting scheme is that the algorithm is very simple so it is very easy to implement. The results of numerical experiments are presented, and are compared with analytical solutions by calculating errors L2 and L∞ norms to confirm the good accuracy of the presented scheme.
Abstract: The main goal of the present work is to decrease the
computational burden for optimum design of steel frames with
frequency constraints using a new type of neural networks called
Wavelet Neural Network. It is contested to train a suitable neural
network for frequency approximation work as the analysis program.
The combination of wavelet theory and Neural Networks (NN)
has lead to the development of wavelet neural networks.
Wavelet neural networks are feed-forward networks using
wavelet as activation function. Wavelets are mathematical
functions within suitable inner parameters, which help them to
approximate arbitrary functions. WNN was used to predict the
frequency of the structures. In WNN a RAtional function with
Second order Poles (RASP) wavelet was used as a transfer
function. It is shown that the convergence speed was faster
than other neural networks. Also comparisons of WNN with
the embedded Artificial Neural Network (ANN) and with
approximate techniques and also with analytical solutions are
available in the literature.
Abstract: Based on the standard finite element method, a new
finite element method which is known as nonlocal finite element
method (NL-FEM) is numerically implemented in this article to
study the nonlocal effects for solving 1D nonlocal elastic problem.
An Eringen-type nonlocal elastic model is considered. In this model,
the constitutive stress-strain law is expressed interms of integral
equation which governs the nonlocal material behavior. The new
NL-FEM is adopted in such a way that the postulated nonlocal elastic
behavior of material is captured by a finite element endowed with a
set of (cross-stiffness) element itself by the other elements in mesh.
An example with their analytical solutions and the relevant numerical
findings for various load and boundary conditions are presented and
discussed in details. It is observed from the numerical solutions that
the torsional deformation angle decreases with increasing nonlocal
nanoscale parameter. It is also noted that the analytical solution fails
to capture the nonlocal effect in some cases where numerical
solutions handle those situation effectively which prove the
reliability and effectiveness of numerical techniques.
Abstract: The paper deals with the estimation of amplitude and phase of an analogue multi-harmonic band-limited signal from irregularly spaced sampling values. To this end, assuming the signal fundamental frequency is known in advance (i.e., estimated at an independent stage), a complexity-reduced algorithm for signal reconstruction in time domain is proposed. The reduction in complexity is achieved owing to completely new analytical and summarized expressions that enable a quick estimation at a low numerical error. The proposed algorithm for the calculation of the unknown parameters requires O((2M+1)2) flops, while the straightforward solution of the obtained equations takes O((2M+1)3) flops (M is the number of the harmonic components). It is applied in signal reconstruction, spectral estimation, system identification, as well as in other important signal processing problems. The proposed method of processing can be used for precise RMS measurements (for power and energy) of a periodic signal based on the presented signal reconstruction. The paper investigates the errors related to the signal parameter estimation, and there is a computer simulation that demonstrates the accuracy of these algorithms.
Abstract: In this paper, we investigate the study of techniques
for scheduling users for resource allocation in the case of multiple
input and multiple output (MIMO) packet transmission systems. In
these systems, transmit antennas are assigned to one user or
dynamically to different users using spatial multiplexing. The
allocation of all transmit antennas to one user cannot take full
advantages of multi-user diversity. Therefore, we developed the case
when resources are allocated dynamically. At each time slot users
have to feed back their channel information on an uplink feedback
channel. Channel information considered available in the schedulers
is the zero forcing (ZF) post detection signal to interference plus
noise ratio. Our analysis study concerns the round robin and the
opportunistic schemes.
In this paper, we present an overview and a complete capacity
analysis of these schemes. The main results in our study are to give
an analytical form of system capacity using the ZF receiver at the
user terminal. Simulations have been carried out to validate all
proposed analytical solutions and to compare the performance of
these schemes.
Abstract: In this paper, using a model transformation approach a system of linear delay differential equations (DDEs) with multiple delays is converted to a non-delayed initial value problem. The variational iteration method (VIM) is then applied to obtain the approximate analytical solutions. Numerical results are given for several examples involving scalar and second order systems. Comparisons with the classical fourth-order Runge-Kutta method (RK4) verify that this method is very effective and convenient.
Abstract: In this work the numerical simulation of transient heat
transfer in a cylindrical probe is done. An experiment was conducted
introducing a steel cylinder in a heating chamber and registering its
surface temperature along the time during one hour. In parallel, a
mathematical model was solved for one dimension transient heat
transfer in cylindrical coordinates, considering the boundary
conditions of the test. The model was solved using finite difference
method, because the thermal conductivity in the cylindrical steel bar
and the convection heat transfer coefficient used in the model are
considered temperature dependant functions, and both conditions
prevent the use of the analytical solution. The comparison between
theoretical and experimental results showed the average deviation is
below 2%. It was concluded that numerical methods are useful in
order to solve engineering complex problems. For constant k and h,
the experimental methodology used here can be used as a tool for
teaching heat transfer in mechanical engineering, using mathematical
simplified models with analytical solutions.
Abstract: This paper at first presents approximate analytical
solutions for systems of fractional differential equations using the
differential transform method. The application of differential
transform method, developed for differential equations of integer
order, is extended to derive approximate analytical solutions of
systems of fractional differential equations. The solutions of our
model equations are calculated in the form of convergent series with
easily computable components. After that a drive-response
synchronization method with linear output error feedback is
presented for “generalized projective synchronization" for a class of
fractional-order chaotic systems via a scalar transmitted signal.
Genesio_Tesi and Duffing systems are used to illustrate the
effectiveness of the proposed synchronization method.
Abstract: In this article two algorithms, one based on variation iteration method and the other on Adomian's decomposition method, are developed to find the numerical solution of an initial value problem involving the non linear integro differantial equation where R is a nonlinear operator that contains partial derivatives with respect to x. Special cases of the integro-differential equation are solved using the algorithms. The numerical solutions are compared with analytical solutions. The results show that these two methods are efficient and accurate with only two or three iterations