Abstract: In this paper, numerical solution of system of
Fredholm and Volterra integral equations by means of the Spline
collocation method is considered. This approximation reduces the
system of integral equations to an explicit system of algebraic
equations. The solution is collocated by cubic B-spline and the
integrand is approximated by the Newton-Cotes formula. The error
analysis of proposed numerical method is studied theoretically. The
results are compared with the results obtained by other methods to
illustrate the accuracy and the implementation of our method.
Abstract: The generalized wave equation models various
problems in sciences and engineering. In this paper, a new three-time
level implicit approach based on cubic trigonometric B-spline for the
approximate solution of wave equation is developed. The usual finite
difference approach is used to discretize the time derivative while
cubic trigonometric B-spline is applied as an interpolating function in
the space dimension. Von Neumann stability analysis is used to
analyze the proposed method. Two problems are discussed to exhibit
the feasibility and capability of the method. The absolute errors and
maximum error are computed to assess the performance of the
proposed method. The results were found to be in good agreement
with known solutions and with existing schemes in literature.
Abstract: In this paper, a backward semi-Lagrangian scheme
combined with the second-order backward difference formula
is designed to calculate the numerical solutions of nonlinear
advection-diffusion equations. The primary aims of this paper are
to remove any iteration process and to get an efficient algorithm
with the convergence order of accuracy 2 in time. In order to achieve
these objects, we use the second-order central finite difference and the
B-spline approximations of degree 2 and 3 in order to approximate
the diffusion term and the spatial discretization, respectively. For the
temporal discretization, the second order backward difference formula
is applied. To calculate the numerical solution of the starting point
of the characteristic curves, we use the error correction methodology
developed by the authors recently. The proposed algorithm turns out
to be completely iteration free, which resolves the main weakness
of the conventional backward semi-Lagrangian method. Also, the
adaptability of the proposed method is indicated by numerical
simulations for Burgers’ equations. Throughout these numerical
simulations, it is shown that the numerical results is in good
agreement with the analytic solution and the present scheme offer
better accuracy in comparison with other existing numerical schemes.
Abstract: In this paper, Semi-orthogonal B-spline scaling
functions and wavelets and their dual functions are presented
to approximate the solutions of integro-differential equations.The
B-spline scaling functions and wavelets, their properties and the
operational matrices of derivative for this function are presented to
reduce the solution of integro-differential equations to the solution of
algebraic equations. Here we compute B-spline scaling functions of
degree 4 and their dual, then we will show that by using them we have
better approximation results for the solution of integro-differential
equations in comparison with less degrees of scaling functions
Abstract: In this paper the Kuramoto-Sivashinsky equation is solved numerically by collocation method. The solution is approximated as a linear combination of septic B-spline functions. Applying the Von-Neumann stability analysis technique, we show that the method is unconditionally stable. The method is applied on some test examples, and the numerical results have been compared with the exact solutions. The global relative error and L∞ in the solutions show the efficiency of the method computationally.
Abstract: In this paper, numerical solutions of the nonlinear Benjamin-Bona-Mahony-Burgers (BBMB) equation are obtained by a method based on collocation of cubic B-splines. Applying the Von-Neumann stability analysis, the proposed method is shown to be unconditionally stable. The method is applied on some test examples, and the numerical results have been compared with the exact solutions. The L∞ and L2 in the solutions show the efficiency of the method computationally.
Abstract: In the present work, the performance of the particle
swarm optimization and the genetic algorithm compared as a typical
geometry design problem. The design maximizes the heat transfer
rate from a given fin volume. The analysis presumes that a linear
temperature distribution along the fin. The fin profile generated using
the B-spline curves and controlled by the change of control point
coordinates. An inverse method applied to find the appropriate fin
geometry yield the linear temperature distribution along the fin
corresponds to optimum design. The numbers of the populations, the
count of iterations and time to convergence measure efficiency.
Results show that the particle swarm optimization is most efficient
for geometry optimization.
Abstract: Linear two-point boundary value problems of order
two are solved using cubic trigonometric B-spline interpolation
method (CTBIM). Cubic trigonometric B-spline is a piecewise
function consisting of trigonometric equations. This method is tested
on some problems and the results are compared with cubic B-spline
interpolation method (CBIM) from the literature. CTBIM is found to
approximate the solution slightly more accurately than CBIM if the
problems are trigonometric.
Abstract: Linear two-point boundary value problem of order two is solved using extended cubic B-spline interpolation method. There is one free parameters, λ, that control the tension of the solution curve. For some λ, this method produced better results than cubic B-spline interpolation method.
Abstract: This paper presents the application of a signal intensity independent registration criterion for non-rigid body registration of medical images. The criterion is defined as the weighted ratio image of two images. The ratio is computed on a voxel per voxel basis and weighting is performed by setting the ratios between signal and background voxels to a standard high value. The mean squared value of the weighted ratio is computed over the union of the signal areas of the two images and it is minimized using the Chebyshev polynomial approximation. The geometric transformation model adopted is a local cubic B-splines based model.
Abstract: The aim of the current study is to develop a numerical
tool that is capable of achieving an optimum shape and design of
hyperbolic cooling towers based on coupling a non-linear finite
element model developed in-house and a genetic algorithm
optimization technique. The objective function is set to be the
minimum weight of the tower. The geometric modeling of the tower
is represented by means of B-spline curves. The finite element
method is applied to model the elastic buckling behaviour of a tower
subjected to wind pressure and dead load. The study is divided into
two main parts. The first part investigates the optimum shape of the
tower corresponding to minimum weight assuming constant
thickness. The study is extended in the second part by introducing the
shell thickness as one of the design variables in order to achieve an
optimum shape and design. Design, functionality and practicality
constraints are applied.
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: This paper presents a method to detect multiple cracks
based on frequency information. When a structure is subjected to
dynamic or static loads, cracks may develop and the modal
frequencies of the cracked structure may change. To detect cracks in a
structure, we construct a high precision wavelet finite element (EF)
model of a certain structure using the B-spline wavelet on the interval
(BSWI). Cracks can be modeled by rotational springs and added to the
FE model. The crack detection database will be obtained by solving
that model. Then the crack locations and depths can be determined
based on the frequency information from the database. The
performance of the proposed method has been numerically verified by
a rotor example.
Abstract: In this paper, a simple active contour based visual
tracking algorithm is presented for outdoor AGV application which is
currently under development at the USM robotic research group
(URRG) lab. The presented algorithm is computationally low cost
and able to track road boundaries in an image sequence and can
easily be implemented on available low cost hardware. The proposed
algorithm used an active shape modeling using the B-spline
deformable template and recursive curve fitting method to track the
current orientation of the road.
Abstract: Recently, it is found that telegraph equation is more suitable than ordinary diffusion equation in modelling reaction diffusion for such branches of sciences. In this paper, a numerical solution for the one-dimensional hyperbolic telegraph equation by using the collocation method using the septic splines is proposed. The scheme works in a similar fashion as finite difference methods. Test problems are used to validate our scheme by calculate L2-norm and L∞-norm. The accuracy of the presented method is demonstrated by two test problems. The numerical results are found to be in good agreement with the exact 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: In the present paper, we use generalized B-Spline curve in trigonometric form on circular domain, to capture the transcendental nature of circle involute curve and uncertainty characteristic of design. The required involute curve get generated within the given tolerance limit and is useful in gear design.