B-spline Scheme Research Articles

On the numerical investigation of the interaction in plasma between (high & low) frequency of (Langmuir & ion-acoustic) waves

In this paper, the Zakharov (Z.) equation in the dimensionless form is numerically investigated via (Cubic & Quantic & Septic) B-spline schemes to demonstrate the fidelity of the calculated computational solutions. The Z equation depicts the interaction in plasma between (high & low) frequency of (Langmuir & ion-acoustic) waves. This interaction is expounded in the prompts of the coastal engineering, electromagnetic field, signal handling in the optical fibres, plasma physics, and fluid dynamics. Three different computational schemes were applied to the Z equation for constructing many novel analytical solutions. In our paper, we try to check the accuracy of these solutions via the above-mentioned numerical schemes. Moreover, some separate sketches are given to indicate more physical features of this interaction. The originality of the obtained solutions is investigated by showing the similarities and differences between our obtained solutions and that was purchased in previously published papers.

Generation of a local lung respiratory motion model using a weighted sparse algorithm and motion prior-based registration

Respiration-introduced tumor location uncertainty is a challenge in lung percutaneous interventions, especially for the respiratory motion estimation of the tumor and surrounding vessel structures. In this work, a local motion modeling method is proposed based on whole-chest computed tomography (CT) and CT-fluoroscopy (CTF) scans. A weighted sparse statistical modeling (WSSM) method that can accurately capture location errors for each landmark point is proposed for lung motion prediction. By varying the sparse weight coefficients of the WSSM method, newly input motion information is approximately represented by a sparse linear combination of the respiratory motion repository and employed to serve as prior knowledge for the following registration process. We have also proposed an adaptive motion prior-based registration method to improve the motion prediction accuracy of the motion model in the region of interest (ROI). This registration method adopts a B-spline scheme to interactively weight the relative influence of the prior knowledge, model surface and image intensity information by locally controlling the deformation in the CTF image region. The proposed method has been evaluated on 15 image pairs between the end-expiratory (EE) and end-inspiratory (EI) phases and 31 four-dimensional CT (4DCT) datasets. The results reveal that the proposed WSSM method achieved a better motion prediction performance than other existing lung statistical motion modeling methods, and the motion prior-based registration method can generate more accurate local motion information in the ROI.

ON THE NEW EXPLICIT SOLUTIONS OF THE FRACTIONAL NONLINEAR SPACE-TIME NUCLEAR MODEL

In this research, the analytical and numerical solutions of the fractional nonlinear space-time Phi-four model are investigated by employing two systematic schemes and the B-spline schemes. A new fractional operator definition is applied to this model to convert the model from its fractional formula to an integer-order nonlinear ordinary differential equation. The considered model is of major interest for studying the nuclear interaction, elementary particles in a condensed medium, and propagation of dislocations in crystals. Explicit wave solutions are obtained.

Abundant analytical and numerical solutions of the fractional microbiological densities model in bacteria cell as a result of diffusion mechanisms

In this paper, an analytical scheme [the generalized Sinh–Gordon equation method) with a new fractional operator (ABR fractional operator] is employed to find novel computational solutions of the nonlinear fractional Kolmogorov–Petrovskii–Piskunov (FKPP) equation. These solutions are used to evaluate the initial and boundary conditions under the numerical scheme (B-spline schemes) to get the numerical solutions of the same model. This equation describes the behavior of genetic models in the increase of microorganisms. Usually, it is used as a biological model to investigate the microbiological densities in bacteria cells as a result of diffusion mechanisms in terms of space–time. Some novel computational solutions are obtained, and the accuracy of them is investigated by calculating the absolute value of error between the obtained computational and numerical solutions. The comparison between the distinct types of obtained solutions is shown by calculating the absolute value of error.

Analytical, semi-analytical, and numerical solutions for the Cahn\u2013Allen equation

This paper studies the analytical, semi-analytical, and numerical solutions of the Cahn–Allen equation, which plays a vital role in describing the structure of the dynamics for phase separation in Fe–Cr–X (X=Mo,Cu) ternary alloys. The modified Khater method, the Adomian decomposition method, and the quintic B-spline scheme are implemented on our suggested model to get distinct kinds of solutions. These solutions describe the dynamics of the phase separation in iron alloys and are also used in solidification and nucleation problems. The applications of this model arise in many various fields such as plasma physics, quantum mechanics, mathematical biology, and fluid dynamics. The comparison between the obtained solutions is represented by using figures and tables to explain the value of the error between exact and numerical solutions. All solutions are verified by using Mathematica software.

A Class of Refinement Schemes With Two Shape Control Parameters

A subdivision scheme defines a smooth curve or surface as the limit of a sequence of successive refinements of given polygon or mesh. These schemes take polygons or meshes as inputs and produce smooth curves or surfaces as outputs. In this paper, a class of combine refinement schemes with two shape control parameters is presented. These even and odd rules of these schemes have complexity three and four respectively. The even rule is designed to modify the vertices of the given polygon, whereas the odd rule is designed to insert a new point between every edge of the given polygon. These schemes can produce high order of continuous shapes than existing combine binary and ternary family of schemes. It has been observed that the schemes have interpolating and approximating behaviors depending on the values of parameters. These schemes have an interproximate behavior in the case of non-uniform setting of the parameters. These schemes can be considered as the generalized version of some of the interpolating and B-spline schemes. The theoretical as well as the numerical and graphical analysis of the shapes produced by these schemes are also presented.

Abundant numerical and analytical solutions of the generalized formula of Hirota-Satsuma coupled KdV system

Abstract This research paper investigates the analytical and numerical solutions of the generalized formula of Hirota-Satsuma coupled KdV system which is also known as the generalized KdV equation that is derived by R. Hirota and J. Satsuma. The modified Khater method and B-spline scheme are used to earn abundant of computational and approximate solutions on this model. This equation characterizes an interaction of two long undulations with diverse dispersion kinsmen. The comparison between our obtained computational and numerical solutions to clarify the convergence of solutions is explained and discussed. For more explanation of the model’s physical properties, some of the obtained solutions are sketched in different types, and the comparison between the computational and numerical solutions are explained by showing the values of absolute error between them. The performance of both used method is effective, powerful, and shows its ability to apply to many nonlinear evolution equations.

A New Scheme Using Cubic B-Spline to Solve Non-Linear Differential Equations Arising in Visco-Elastic Flows and Hydrodynamic Stability Problems

This study deals with the numerical solution of the non-linear differential equations (DEs) arising in the study of hydrodynamics and hydro-magnetic stability problems using a new cubic B-spline scheme (CBS). The main idea is that we have modified the boundary value problems (BVPs) to produce a new system of linear equations. The algorithm developed here is not only for the approximation solutions of the 10th order BVPs but also estimate from 1st derivative to 10th derivative of the exact solution as well. Some examples are illustrated to show the feasibility and competence of the proposed scheme.

A Family of Binary Univariate Nonstationary Quasi-Interpolatory Subdivision Reproducing Exponential Polynomials

In this paper, by suitably using the so-called push-back operation, a connection between the approximating and interpolatory subdivision, a new family of nonstationary subdivision schemes is presented. Each scheme of this family is a quasi-interpolatory scheme and reproduces a certain space of exponential polynomials. This new family of schemes unifies and extends quite a number of the existing interpolatory schemes reproducing exponential polynomials and noninterpolatory schemes like the cubic exponential B-spline scheme. For these new schemes, we investigate their convergence, smoothness, and accuracy and show that they can reach higher smoothness orders than the interpolatory schemes with the same reproduction property and better accuracy than the exponential B-spline schemes. Several examples are given to illustrate the performance of these new schemes.

Analytical and Approximate Solutions for Complex Nonlinear Schrödinger Equation via Generalized Auxiliary Equation and Numerical Schemes

This article studies the performance of analytical, semi-analytical and numerical scheme on the complex nonlinear Schrödinger (NLS) equation. The generalized auxiliary equation method is surveyed to get the explicit wave solutions that are used to examine the semi-analytical and numerical solutions that are obtained by the Adomian decomposition method, and B-spline schemes (cubic, quantic, and septic). The complex NLS equation relates to many physical phenomena in different branches of science like a quantum state, fiber optics, and water waves. It describes the evolution of slowly varying packets of quasi-monochromatic waves, wave propagation, and the envelope of modulated wave groups, respectively. Moreover, it relates to Bose-Einstein condensates which is a state of matter of a dilute gas of bosons cooled to temperatures very close to absolute zero. Some of the obtained solutions are studied under specific conditions on the parameters to constitute and study the dynamical behavior of this model in two and three-dimensional.

A non-stationary Catmull–Clark subdivision scheme with shape control

In this paper, a non-stationary Catmull–Clark subdivision scheme is presented which enables the user to change the shape of the surface obtained from a given initial mesh. Such a non-stationary subdivision scheme is constructed by taking tensor product of a properly modified cubic exponential B-spline scheme in the regular mesh and giving suitable rules in the neighborhoods of extraordinary points. For this new scheme, we show that it can generate tangent plane continuous surfaces. Besides, we propose a generalization, which could locally control the limit surface for the purpose of generating more flexible surfaces and providing some features control. Some examples are given to show the performance of the new schemes.

Analytical and numerical simulations for the kinetics of phase separation in iron (Fe–Cr–X (X=Mo, Cu)) based on ternary alloys

In this paper, we investigate the physical behavior of the basic elements that related to phase decomposition in ternary alloys of (Fe–Cr–Mo) and (Fe–Cr–Cu) according to analytical and approximate simulation. We study the dynamic of the separation phase for the ternary alloys of iron. The dynamical process of this separation has been described in a mathematical model called the Cahn–Hilliard equation. The minor element behavior in the process has been described by the Cahn–Hilliard equation. It describes the process of phase separation for two components of a binary fluid in ternary alloys of (Fe–Cr–Mo) and (Fe–Cr–Cu). We implement a modified auxiliary equation method and the cubic B-spline scheme on this mathematical model to show the dynamical process of phase separation and the concentration of one of two components in a system. We try obtaining the solitary and approximate solutions of this model to show the relation between the components in this phase. We discuss our solutions in view of a Stefan, Thomas-Windle, and Navier–Stokes models. Whereas, these models describe the motion of viscous fluid substance.

A Generalized Cubic Exponential B-Spline Scheme with Shape Control

In this paper, a generalized cubic exponential B-spline scheme is presented, which can generate different kinds of curves, including the conics. Such a scheme is obtained by generalizing the cubic exponential B-spline scheme based on an iteration from the generation of exponential polynomials and a suitable function with two parameters μ and ν. By changing the values of μ and ν, the sensitivity of the shape of the subdivision curve to the initial control value v0 can be changed and different kinds of curves can then be obtained by adjusting the value of v0. For this new scheme, we show that, with any admissible choice of μ and ν, it owns the same smoothness order and support as the cubic exponential B-spline scheme. Besides, based on a different iteration and another suitable function, we construct a similar nonstationary scheme to generate more curves with different shapes and show the role of iterations and suitably chosen functions in the construction and analysis of such schemes. Several examples are given to illustrate the performance of our new schemes.

Improved Hydrodynamic Analysis of 3-D Hydrofoil and Marine Propeller Using the Potential Panel Method Based on B-Spline Scheme

In this paper, the hydrodynamic performance of lift-body marine propellers and hydrofoils is analyzed using a B-spline potential-based panel method. The potential panel method, based on a combination of two singularity elements, is proposed, and a B-spline curve interpolation method is integrated with the interpolation of the corner points and collocation points to ensure accuracy and continuity of the interpolation points. The B-spline interpolation is used for the distribution of the singularity elements on a complex surface to ensure continuity of the results for the intensity of the singular points and to reduce the possibility of abrupt changes in the surface velocity potential to a certain extent. A conventional cubic spline method is also implemented as a comparison of the proposed method. The surface pressure coefficient and lift the performance of 2-D and 3-D hydrofoils of sweepback and dihedral type with different aspect ratios are analyzed to verify the rationality and feasibility of the present method. The surface pressure distribution and coefficients of thrust and torque are calculated for different marine propellers and compared with the experimental data. A parametric study on the propeller wake model was carried out. The validated results show that it is practical to improve the accuracy of hydrodynamic performance prediction using the improved potential panel method proposed.

On exact and approximate solutions of (2+1)-dimensional Konopelchenko-Dubrovsky equation via modified simplest equation and cubic B-spline schemes

This paper studies the analytical and numerical solutions for (2+1)-D Konopelchenko-Dubrovsky equation. It also examines the performance of the modified simplest equation method and the cubic B-spline scheme on this model. Many explicit wave solutions are found by using the analytical technique. These solutions allow studying the physical properties of this model. The comparison between the analytical and numerical solutions are discussed to show which one of cubic B-spline scheme families is more accurate in finding the numerical solutions of this model.

Interpolatory subdivision schemes with the optimal approximation order

In this paper, we show that the univariate interpolatory subdivision schemes with the optimal approximation order, which are the D-D interpolatory ones, can be obtained from the cubic B-spline scheme. Such schemes are derived in this paper by suitably using the push-back operation, a connection between the approximating and interpolatory subdivision. Then, the process to obtain the D-D schemes is generalized to the bivariate case and bivariate interpolatory schemes with the optimal approximation order are generated based on the triangular meshes. Compared with the existing bivariate interpolatory schemes with the optimal approximation order, the newly constructed ones own some advantages, such as a better performance in terms of the number of nonzero coefficients in the mask. Several examples are given and explicitly computed.

Symbols and exact regularity of symmetric pseudo-splines of any arity

Pseudo-splines form a family of subdivision schemes that provide a natural blend between interpolating schemes and approximating schemes, including the Dubuc-Deslauriers schemes and B-spline schemes. Using a generating function approach, we derive expressions for the symbols of the symmetric $m$-ary pseudo-spline subdivision schemes. We show that their masks have positive Fourier transform, making it possible to compute the exact H\"older regularity algebraically as a logarithm of the spectral radius of a matrix. We apply this method to compute the regularity explicitly in some special cases, including the symmetric binary, ternary, and quarternary pseudo-spline schemes.

Myocardial Motion Estimation From Medical Images Using the Monogenic Signal

We present a method for the analysis of heart motion from medical images. The algorithm exploits monogenic signal theory, recently introduced as an N-dimensional generalization of the analytic signal. The displacement is computed locally by assuming the conservation of the monogenic phase over time. A local affine displacement model is considered to account for typical heart motions as contraction/expansion and shear. A coarse-to-fine B-spline scheme allows a robust and effective computation of the model's parameters, and a pyramidal refinement scheme helps to handle large motions. Robustness against noise is increased by replacing the standard point-wise computation of the monogenic orientation with a robust least-squares orientation estimate. Given its general formulation, the algorithm is well suited for images from different modalities, in particular for those cases where time variant changes of local intensity invalidate the standard brightness constancy assumption. This paper evaluates the method's feasibility on two emblematic cases: cardiac tagged magnetic resonance and cardiac ultrasound. In order to quantify the performance of the proposed method, we made use of realistic synthetic sequences from both modalities for which the benchmark motion is known. A comparison is presented with state-of-the-art methods for cardiac motion analysis. On the data considered, these conventional approaches are outperformed by the proposed algorithm. A recent global optical-flow estimation algorithm based on the monogenic curvature tensor is also considered in the comparison. With respect to the latter, the proposed framework provides, along with higher accuracy, superior robustness to noise and a considerably shorter computation time.

Multiscale optical flow computation from the monogenic signal

We have developed an algorithm for the estimation of cardiac motion from medical images. The algorithm exploits monogenic signal theory, recently introduced as an N-dimensional generalization of the analytic signal. The displacement is computed locally by assuming the conservation of the monogenic phase over time. A local affine displacement model replaces the standard translation model to account for more complex motions as contraction/expansion and shear. A coarse-to-fine B-spline scheme allows a robust and effective computation of the models parameters and a pyramidal refinement scheme helps handle large motions. Robustness against noise is increased by replacing the standard pointwise computation of the monogenic orientation with a more robust least-squares orientation estimate. This paper reviews the results obtained on simulated cardiac images from different modalities, namely 2D and 3D cardiac ultrasound and tagged magnetic resonance. We also show how the proposed algorithm represents a valuable alternative to state-of-the-art algorithms in the respective fields.

Efficient numerical solution of Fisher's equation by using B-spline method

This paper seeks to develop an efficient B-spline scheme for solving Fisher's equation, which is a nonlinear reaction–diffusion equation describing the relation between the diffusion and nonlinear multiplication of a species. To find the solution, domain is partitioned into a uniform mesh and then cubic B-spline function is applied to Fisher's equation. The method yields stable and accurate solutions. The results obtained are acceptable and in good agreement with some earlier studies. An important advantage is that the method is capable of greatly reducing the size of computational work.