Quartic B-spline Research Articles

Quartic B-Spline Method for Non-Linear Second Order Singularly Perturbed Delay Differential Equations

Quartic B-Spline Method for Non-Linear Second Order Singularly Perturbed Delay Differential Equations

Numerical investigation of electroosmotic driven transport of MHD Casson fluid over an exponential stretching sheet using quartic B-spline method

The stretching surface flow phenomenon plays novel significance in the differential industrial sectors like manufacturing processes, food processing, polymers, petroleum engineering, glass fiber production, metal spinning, and many others. This research focuses on investigating the flow of electroosmotic-driven transport of magnetic Casson fluid over an exponential stretching sheet. The study employs mathematical modeling based on partial differential equations which are further truncated into ordinary system under the implementation of dimensionless variables. The quartic B-spline method is applied for obtaining solutions. Key findings indicate that electroosmotic and Helmholtz–Smoluchowski effects enhance fluid velocity, while magnetic forces reduce velocity. Additionally, an increase in the Hartmann parameter leads to a decrease in the skin friction coefficient.

Numerical solution of the Whitham-Broer-Kaup shallow water equation by quartic B-spline collocation method

This paper introduces a novel quartic B-spline collocation method to address the coupled Whitham–Broer–Kaup (WBK) problem. The WBK problem is a topic of interest in the study of nonlinear wave phenomena and has applications in various fields, including fluid dynamics, plasma physics, and nonlinear optics. The method combines spatial quartic B-spline scheme discretization, and Crank–Nicolson temporal discretization. It is unconditionally stable as proven by the Von-Neumann technique. Numerical examples demonstrate the method’s superior accuracy compared to existing solutions. Error analysis employs and norms, while the method exhibits high computational efficiency. The nonlinearity is managed through Rubin-Graves linearization. Comparisons with prior approaches highlight its efficiency, stability, adaptability to complex problems. The quartic B-spline method is well-suited for simulating fluid flow phenomena in shallow water scenarios, offering high accuracy and low computational cost.

Improved explicit quartic B-spline time integration scheme for dynamic response analysis of viscoelastic systems

A quartic B-spline based explicit scheme is extended for dynamic analysis of viscoelastic systems. In the calculation formulation, different integration approximation schemes for exponential damping models and other damping models are presented. The mathematical derivation of the convolution solving strategy is formulated for viscoelastic systems. For the damping models composed of exponential function, this scheme only requires once convolution calculation in each time step, even though the dynamic equations are used twice. This significantly reduces the calculation time for the damping models composed of exponential function. Numerical examples show that the proposed quartic B-spline based explicit scheme achieves higher accuracy and efficiency compared to other competitive explicit schemes, not only for damping models composed of exponential functions, but also for other viscoelastic damping models.

A Shape Preserving Class of Two-Frequency Trigonometric B-Spline Curves

This paper proposes a new approach to define two frequency trigonometric spline curves with interesting shape preserving properties. This construction requires the normalized B-basis of the space U4(Iα)=span{1,cost,sint,cos2t,sin2t} defined on compact intervals Iα=[0,α], where α is a global shape parameter. It will be shown that the normalized B-basis can be regarded as the equivalent in the trigonometric space U4(Iα) to the Bernstein polynomial basis and shares its well-known symmetry properties. In fact, the normalized B-basis functions converge to the Bernstein polynomials as α→0. As a consequence, the convergence of the obtained piecewise trigonometric curves to uniform quartic B-Spline curves will be also shown. The proposed trigonometric spline curves can be used for CAM design, trajectory-generation, data fitting on the sphere and even to define new algebraic-trigonometric Pythagorean-Hodograph curves and their piecewise counterparts allowing the resolution of C(3 Hermite interpolation problems.

Viscous Effect on Solitary Kelvin Wave in Open Cylindrical Channel under Precession

Viscous effect is introduced into the system of Navier–Stokes equations, that were derived to study the solitary Kelvin mode in an open cylindrical channel that precesses. Accordingly, three new weakly nonlinear models were derived: Korteweg–de Vries-Burgers, and two new Benjamin–Bona–Mahony-Burgers. The first was solved analytically by discussing the phase solution and numerically using an implicite finite difference method to track the solution with time under diffusion effect. The second two models were solved numerically only using the Quartic B-Spline collocation method. By manipulating the scaling the first model included only the gravity force effect, and the second included both gravity and Coriolis forces. The numerical method is tested experimentally by comparing the velocity solutions with ADV signal extracted from the ADV measurements under bore conditions, and the amplitude solution with the solitary kelvin mode.

An efficient computational technique for solving a fractional-order model describing dynamics of neutron flux in a nuclear reactor

This paper focuses on the development of a computational technique for numerical solution of the time-fractional neutron diffusion (FND) equation, which describes dynamics of neutron flux in a nuclear reactor. The time-fractional derivatives are discretized by means of L1 scheme, while the space derivative is discretized by employing a collocation method based on quartic B-spline (QUBS) basis functions. Numerical experiments are presented to illustrate the performance of proposed technique. Results reveal that the proposed technique yields a reliable approximation to the solution of the underlying model problem and has O(k+h4)−order convergence, with k and h representing the step sizes in time and space directions, respectively. It is important to point out that the numerical method for the underlying problem has not yet been discussed in the literature. The effect of fractional order derivative on the behavior of neutron flux is investigated.

Numerical approximation to the MEW equation for the single solitary wave and different types of interactions of the solitary waves

The main motivation of the current study is to find out better approximate solutions of the modified equal width wave (MEW) equation. In order to achieve this aim, the power of two numerical methods are combined and an extended literature survey has been carried out. Quartic B-spline base functions have been utilized since the first-order and second-order weighting coefficients that are needed for space discretization are obtained directly. As test problems, twelve different applications of single solitary wave and four different applications of the interaction between the two solitary waves are solved successfully. All of the approximate solutions have been compared to nearly fifty various earlier applications existing in the literature. Also, the rate of the convergence is given with error norms. Comparisons show the fact that the current method obtains improved results than most of the common earlier methods.

Versatile Tool for Parametric Smooth Turbomachinery Blades

Designing blades for efficient energy transfer by turning the flow and angular momentum change is both an art and iterative multidisciplinary engineering process. A robust parametric design tool with few inputs to create 3D blades for turbomachinery and rotating or non-rotating energy converters is described in this paper. The parameters include axial–radial coordinates of the leading/trailing edges, construction lines (streamlines), metal angles, thickness-to-chord ratio, standard, and user-defined airfoil type among others. Using these, 2D airfoils are created, conformally mapped to 3D stream surfaces, stacked radially with multiple options, and they are transformed to a 3D Cartesian coordinate system. Smooth changes in blade curvature are essential to ensure a smooth pressure distribution and attached flow. B-splines are used to control meanline curvature, thickness, leading edge shape, sweep-lean, and other parameters chordwise and spanwise, making the design iteration quick and easy. C2 curve continuity is achieved through parametric segments of cubic and quartic B-splines and is better than G2. New geometries using an efficient parametric scheme and minimal CAD interaction create watertight solid bodies and optional fluid domains. Several examples of ducted axial and radial turbomachinery with special airfoil shapes or otherwise, unducted rotors including propellers and wind and hydrokinetic turbines are presented to demonstrate versatility and robustness of the tool and can be easily tied to any automation chain and optimizer.

An effective approach based on Smooth Composite Chebyshev Finite Difference Method and its applications to Bratu-type and higher order Lane–Emden problems

The Smooth Composite Chebyshev Finite Difference method is generalized for higher order initial and boundary value problems. Round-off and truncation error analyses and convergence analysis of the method are also extended to higher order. The proposed method is applied to obtain the highly precise numerical solutions of boundary or initial value problems of the Bratu and higher order Lane Emden types. To visualize the competency of the presented method, the obtained results are compared with nine different methods, namely, Bezier curve method, Adomian decomposition method, Operational matrix collocation method, Direct collocation method, Haar Wavelet Collocation, Bernstein Collocation Method, Improved decomposition method, Quartic B-Spline method and New Cubic B-spline method. The comparisons show that the presented method is highly accurate than the other numerical methods and also gets rid of the singularity of the given problems.

N-Dimensional quartic B-spline collocation method to solve different types of n-dimensional partial differential equations

In this article, we introduce a new structure to the quartic B-spline functions in n-dimensions to solve various mathematical problems of partial differential equations in n-dimensions. We use quartic B-spline functions in one dimension to generate multi-dimensional quartic B-spline functions as two and three-dimensional. We measure the efficiency and accuracy of the proposed collocation method as shown in five problems in two and three dimensions. We compare our new method with other different methods.

A computational technique for solving the time‐fractional Fokker‐Planck equation

In this paper, a high‐order computational scheme is constructed for the time‐fractional Fokker‐Planck (TFFP) equation. The formula is used to approximate the fractional derivative in the model problem. The space derivatives in the resulting semi‐discrete equation are approximated by a collocation technique based on quartic B‐spline (QBS) basis functions. The method is rigorously analyzed for its convergence. Two examples are provided to show the applicability and robustness of the present method. Our results are compared with the results obtained using finite difference method (FDM) and high‐order compact finite difference method (HCFDM).

A fourth-order numerical method for solving a class of derivative-dependent nonlinear singular boundary value problems

In this paper, a high order numerical scheme based on quartic B-spline functions is proposed to solve derivative-dependent nonlinear singular boundary value problems. Convergence of the method is analyzed. Five test problems are considered to illustrate the accuracy and efficiency of the method. The results obtained by present method are compared with those obtained by the uniform mesh cubic B-spline collocation (UCS) method [1] and non-uniform mesh cubic B-spline collocation (NCS) method [1]. The CPU time of proposed method is compared with that of the NCS method.

Solutions of Fluid Flow Problem over a Generalized Stretching or Shrinking Sheet with Heat Transfer Using Cubic and Quartic B-Spline Collocation Methods

Solutions of Fluid Flow Problem over a Generalized Stretching or Shrinking Sheet with Heat Transfer Using Cubic and Quartic B-Spline Collocation Methods

A Galerkin approach to solitary wave propagation for the second-order nonlinear evolution equation based on quartic B-spline functions

This article aims to produce efficient numerical results using the Galerkin approach and quartic B-spline functions for the one-dimensional second-order nonlinear evolution equation. The quartic B-spline functions are used as the shape and weight functions for the Galerkin method. The time derivative and space parameters are discretized by the finite difference and Crank–Nicolson schemes, respectively. A Galerkin approach is investigated by propagating waves at various times and comparisons are made with published papers. The obtained results show that the scheme presented good results and conservations laws are all adequately obeyed.

Solitary waves of the RLW equation via least squares method

Abstract Integration using least squares method in space and Crank–Nicolson approach in time is managed to set up an algorithm to solve the RLW equation numerically. Trial functions in the least square method consist of a combination of the quartic B-spline functions. Integration of the RLW equation gives a system of algebraic equations. The solutions consisting of a combination of the quartic B-splines are given for some initial and boundary value problems of RLW equation.

An inverse problem for the damped generalized regularized long wave equation

In this article, we consider an inverse problem related to the damped generalized regularized long wave equation with noisy data. We study numerically this problem by applying the quartic B-spline and Haar wavelet methods combined with the Tikhonov regularization method. We investigate the stability and the convergence analysis of this problem and show that and are the convergence rates of the quartic B-spline method and the Haar wavelet method, respectively. The numerical results, by a comparative study, show that an excellent estimation of the unknown functions of the mentioned inverse problem.

A fourth order one step method for numerical solution of good Boussinesq equation

In this paper, we investigate the numerical solution of "good" Boussinesq equation by using the quartic B-spline Galerkin method for space discretization and the fourth order one-step method for time discretization.The proposed numerical scheme is analyzed for truncation error. Four test problems are studied. The accuracy and efficiency are measured by computing error norm $L_{\infty }$ and the order of convergence for the proposed method. The results of numerical experiments confirm that the proposed method has a higher accuracy.

An improved quartic B-spline based explicit time integration algorithm for structural dynamics

In this study, an improved explicit time integration method based on quartic B-spline interpolation is generalized for linear and nonlinear dynamics. The accuracy order of the proposed method is analytically obtained as well as its spectral radius, period elongation, and algorithmic damping. The analysis shows the proposed method achieves third-order and at least second-order accuracy for displacement and velocity, respectively. With one algorithmic parameter, the proposed method can adjust numerical dissipation and accuracy. Linear dynamic examples demonstrate that the effectiveness of the proposed method as well as its high-order accuracy. Nonlinear dynamic problems show the proposed method can provide desirable solutions. Numerical results demonstrate the proposed method can provide more stable and accurate solutions than other classical explicit methods.

A Comparative Study of Approximate Solution of Fifth-order Boundary-value Problems by Applying HaarWavelets Technique and Galerkin Method with Quartic B-splines

A Comparative Study of Approximate Solution of Fifth-order Boundary-value Problems by Applying HaarWavelets Technique and Galerkin Method with Quartic B-splines