Quintic Spline Functions Research Articles

A meshless method based on the Laplace transform for multi-term time-space fractional diffusion equation

<abstract><p>Multi-term fractional diffusion equations can be regarded as a generalisation of fractional diffusion equations. In this paper, we develop an efficient meshless method for solving the multi-term time-space fractional diffusion equation. First, we use the Laplace transform method to deal with the multi-term time fractional operator, we transform the time into complex frequency domain by Laplace transform. The properties of the Laplace transform with respect to fractional-order operators are exploited to deal with multi-term time fractional-order operators, overcoming the dependence of fractional-order operators with respect to time and giving better results. Second, we proposed a meshless method to deal with space fractional operators on convex region based on quintic Hermite spline functions based on the theory of polynomial functions dense theorem. Meanwhile, the approximate solution of the equation is obtained through theory of the minimum residual approximate solution, and the error analysis are provided. Third, we obtain the numerical solution of the diffusion equation by inverse Laplace transform. Finally, we first experimented with a single space-time fractional-order diffusion equation to verify the validity of our method, and then experimented with a multi-term time equation with different parameters and regions and compared it with the previous method to illustrate the accuracy of our method.</p></abstract>

Quintic Spline functions and Fredholm integral equation

A new six order method developed for the approximation Fredholm integral equation of the second kind. This method is based on the quintic spline functions (QSF). In our approach, we first formulate the Quintic polynomial spline then the solution of integral equation approximated by this spline. But we need to develop the end conditions which can be associated with the quntic spline. To avoid the reduction accuracy, we formulate the end condition in such a way to obtain the band matrix and also to obtain the same order of accuracy. The convergence of the method is discussed by using matrix algebra. Finally, four test problems have been used for numerical illustration to demonstrate the practical ability of the new method.

Numerical simulation of entropy generation due to natural convection heat transfer using Kernel Derivative-Free (KDF) Incompressible Smoothed Particle Hydrodynamics (ISPH) model

This paper develops and applies a Kernel Derivative-Free (KDF) Incompressible Smoothed Particle Hydrodynamics (ISPH) model for analysis of entropy generation and heat transfer in fluid-structure coupling problems. A modified high order Laplacian operator is applied for the treatment of pressure-velocity coupling (Poisson's equation), while an explicit third-order TVD Runge-Kutta scheme is used for time integration of the momentum, energy and displacement equations. To improve the consistency and stability of the model, a new particle regularization technique based on the particle shifting is also introduced for simulating free-surface flows. The developed KDF-ISPH model is validated and evaluated for a series of challenging benchmark cases, including, dam break, stretching water drop, rotating square patch of fluid, and natural convection in square cavity. Accuracy and applicability of the method are further validated by analyzing entropy generation due to the natural convection heat transfer in three well-known geometries including: square cavity with hot obstacle inside, C-shaped enclosure, and square enclosure containing a pair of hot and cold horizontal pipes (heat exchanger). The results are found to be in good agreement with available numerical and experimental data. The accuracy of the developed KDF-ISPH with new Laplacian operator, for use in prediction of fluid flow and heat transfer characteristics is also proven. Finally, by combining the cosine and signal functions, a new high order smoothing kernel is constructed. The evaluation of this new kernel for the propagation of shock wave in 1D tube demonstrates better global stability and consistency properties compared to two frequently used SPH kernels (i.e. cubic and quintic spline functions).

Composite non-polynomial spline solution of boundary value problems in plate deflection theory

This article presents the numerical solution of a system of linear fourth-order boundary value problems using a different amalgamation of non-polynomial splines. A novel approach was developed using quintic spline function together with exponential and trigonometric functions. Our method is convergent and second-order accurate. Numerical examples show that the method congregates with sufficient accuracy to the exact solutions. Our methodology has the advantages over some existing quintic spline method, direct method, and finite difference method.

A spline-based computational technique applicable for solution of boundary value problem arising in human physiology

Non-polynomial quintic spline functions based algorithms are used for computing an approximation to the nonlinear two point second order singular boundary value problems arising in human physiology. After removing the singularity by L' hospital rule, the resulting boundary value problem is then efficiently treated by employing non-polynomial quintic spline for finding the numerical solution. Two examples have been included and comparison of the numerical results made with cubic extended B-spline method and finite difference method.

A spline-based computational technique applicable for solution of boundary value problem arising in human physiology

Non-polynomial quintic spline functions based algorithms are used for computing an approximation to the nonlinear two point second order singular boundary value problems arising in human physiology. After removing the singularity by L' hospital rule, the resulting boundary value problem is then efficiently treated by employing non-polynomial quintic spline for finding the numerical solution. Two examples have been included and comparison of the numerical results made with cubic extended B-spline method and finite difference method.

The improved element-free Galerkin method for elastoplasticity large deformation problems

Based on the improved moving least-squares (IMLS) approximation, the improved element-free Galerkin (IEFG) method for elastoplasticity large deformation problems is presented. Compared with the moving least-squares (MLS) approximation, by orthogonalizing the basis function, the IMLS approximation can overcome the disadvantage of ill-conditional or singular equations in the MLS approximation. Compared with the meshless methods based on the MLS approximation, under the similar computational accuracy, the ones using the IMLS approximation have higher computational efficiency. The IMLS approximation is used to form the approximation function, the Galerkin weak form of elastoplasticity large deformation problem is used to form the discretized equation system, and the penalty method is used to apply the displacement boundary conditions, then the formulae of the improved element-free Galerkin (IEFG) method for elastoplasticity large deformation problems are obtained, and Newton-Raphson method is used to obtain the solution of the final equation system. Some numerical examples are given to discuss the influences of the weight function, the scale parameter, the penalty factor, the node distribution and the load step number on the computational precision of the numerical solutions. Considering the relative errors when the cubic, quartic and quintic spline functions are used as weight functions, it is shown that the numerical solution using the cubic spline function has higher computational precision. About the scale parameter of the influence domain and the penalty factor, the numerical results show that d max=3.6 and α =1010× E will obtain the solutions with higher precision. The convergence of the method in this paper is analyzed by considering the influence of the node distribution on the computational precision of the solutions, and it is shown that the method in this paper is convergent. And the influence of the total load step number on the computational precision of the solutions is also discussed. Numerical examples show that the method in this paper is effective, and it has the advantage of improving the computational efficiency of the EFG method.

A parametric spline scheme for the nonlinear Schrödinger equation

We develop a numerical method based on parametric adaptive quintic spline functions for solving the nonlinear Schrödinger (NLS) equation. The truncation error is theoretically analyzed. Based on the von Neumann method and the linearization technique, stability analysis of the method is studied and the method is shown to be unconditionally stable. Two invariants of motion related to mass and momentum are calculated to determine the conservation properties of the problem. Finally, some numerical tests are presented to illustrate the method’s efficiency.

Quintic spline smooth semi-supervised support vector classification machine

A semi-supervised vector machine is a relatively new learning method using both labeled and unlabeled data in classification. Since the objective function of the model for an unstrained semi-supervised vector machine is not smooth, many fast optimization algorithms cannot be applied to solve the model. In order to overcome the difficulty of dealing with non-smooth objective functions, new methods that can solve the semi-supervised vector machine with desired classification accuracy are in great demand. A quintic spline function with three-times differentiability at the origin is constructed by a general three-moment method, which can be used to approximate the symmetric hinge loss function. The approximate accuracy of the quintic spline function is estimated. Moreover, a quintic spline smooth semi-support vector machine is obtained and the convergence accuracy of the smooth model to the non-smooth one is analyzed. Three experiments are performed to test the efficiency of the model. The experimental results show that the new model outperforms other smooth models, in terms of classification performance. Furthermore, the new model is not sensitive to the increasing number of the labeled samples, which means that the new model is more efficient.

Research on Spiral Bevel Gear NC Machining Control Method

Based on spiral bevel gear NC manufacture principle and generating motion of the generating gear and the work gear, the NC generation of spiral bevel gear cutting is realized by means of the vector transformation. As sophisticated calculation of the transformation is hard to implement in each interpolation cycle of NC machining, the coordinated motion of NC coordinate axes is expressed as quintic parametric spline functions, which use the rotation angle of the work gear as the parameter. The constrained optimization method of three design variables is established to obtain contacting points of gear tooth surfaces, which are acquired by solving nonlinear equations sets traditionally. The condition of normal coincidence is simplified and the object function of the equal position vectors of mating gears is established. The contacting points of gear tooth surfaces can be used for tooth contact analysis. The parametric curve interpolator is combined with spiral bevel gear NC machining, and is applied to the self-developed spiral bevel gear NC milling machine to control the gear cutting. Experimental results show that the proposed method is feasible and effective in the control of NC machining for the spiral bevel gear.

Numerical algorithm based on quintic nonpolynomial spline for solving third-order boundary value problems associated with draining and coating flows

A numerical algorithm is developed for the approximation of the solution to certain boundary value problems involving the third-order ordinary differential equation associated with draining and coating flows. The authors show that the approximate solutions obtained by the numerical algorithm developed by using nonpolynomial quintic spline functions are better than those produced by other spline and domain decomposition methods. The algorithm is tested on two problems associated with draining and coating flows to demonstrate the practical usefulness of the approach.

Parametric quintic spline solution of third-order boundary value problems

In this paper, we develop parametric quintic spline function to approximate the solution of third-order boundary value problems of the form u″′=f(x, u), a≤x≤b, subject to the boundary conditions u(a)=k 1, u′(a)=k 2 and u(b)=k 3. The class of methods are second-, fourth- and sixth-order accurate. End equations of the splines are derived and truncation error is given. Three numerical examples are presented to illustrate the practical use of our methods as well as their accuracies when compared with some existing spline function methods. It is shown that the new methods give approximations, which are better than those produced by other methods.

Non-polynomial quintic spline solution for the system of third order boundary-value problems

In this paper we develop a non-polynomial quintic spline function to approximate the solution of third order linear and non-linear boundary value problems associated with odd-order obstacle problems. Such problems arise in physical oceanography and can be studied in the framework of variational inequality theory. The class of methods are second and fourth order convergent. End equations of the splines are derived and truncation error is obtained. Two numerical examples are given to illustrate the applicability and efficiency of proposed method. It is shown that the new method gives approximations, which are better than those produced by other methods.

Quintic nonpolynomial spline method for the solution of a second-order boundary-value problem with engineering applications

Nonpolynomial quintic spline functions are used to develop a numerical algorithm for computing an approximation to the solution of a system of second order boundary value problems associated with heat transfer. We show that the approximate solutions obtained by our algorithm are better than those produced by other spline and domain decomposition methods. A comparison of our algorithm with nonpolynomial quadratic spline method is discussed with the help of two numerical examples.

Numerical Treatment of Nonlinear Third Order Boundary Value Problem

In this paper, the boundary value problems for nonlinear third order differential equations are treated. A generic approach based on nonpolynomial quintic spline is developed to solve such boundary value problem. We show that the approximate solutions of such problems obtained by the numerical algorithm developed using nonpolynomial quintic spline functions are better than those produced by other numerical methods. The algorithm is tested on a problem to demonstrate the practical usefulness of the approach.

High order accuracy nonpolynomial spline solutions for 2 μth order two point boundary value problems

Sixth order accurate method based on quintic nonpolynomial spline function for the numerical solution of 2 μth order two point BVPs (where μ is a positive integer) is presented by transforming the problem into a system of μ second order two point BVPs. The nonpolynomial spline function is used to drive some consistency relations for computing approximation to the solution of this problem. The proposed approach gives better approximations than existing polynomial spline and finite difference methods and has a lower computational cost. Convergence analysis of the proposed method is discussed. Numerical examples are included to illustrate the practical usefulness of our method.

Non-polynomial spline for solution of boundary-value problems in plate deflection theory

In this paper a non-polynomial quintic spline function is applied to the numerical solution of a certain fourth-order, two-point boundary-value problem occurring in plate deflection theory. Direct methods of orders two, four, and six have been obtained which lead to five-diagonal linear systems. Boundary formulae of various orders have been developed to retain the bandwidth of the coefficient matrix as five. Convergence analysis of the sixth-order method is given. Numerical results are provided to demonstrate the superiority of our methods.

Refinable bivariate quartic and quintic [formula omitted]-splines for quadrilateral subdivisions

Refinable compactly supported bivariate C 2 quartic and quintic spline function vectors on the four-directional mesh are introduced in this paper to generate matrix-valued templates for approximation and Hermite interpolatory surface subdivision schemes, respectively, for both the 2 and 1-to-4 split quadrilateral topological rules. These splines have their full local polynomial preservation orders. In addition, we extend our study to parametric approach and use the symmetric properties of our refinable quintic spline components as a guideline to reduce the number of free parameters in constructing second order C 2 Hermite interpolatory quadrilateral subdivision schemes with precisely six components.

FIFTH-ORDER CONSERVATIVE SCHEME WITH FLUX CONTROL FOR CONVECTION EQUATIONS ON 3D CURVILINEAR COORDINATES

In the three-dimensional curvilinear coordinates, a fifth-order numerical scheme (FVM-QSI scheme) is proposed for convection equations which are discretized with a finite volume method. In the FVM-QSI scheme, fluxes on the cell boundaries are evaluated with quintic spline functions. In order to prevent numerical oscillations, the calculated fluxes are corrected with the flux control techniques (DC and FP methods), which are newly proposed in the 3D space in this paper. As a result of the computations of cubic cavity flows and its scalar convections, it was shown that the FVM-QSI scheme with the DC method enables us to obtain conservative and accurate results without numerical oscillations.

The quadratic finite element/strip with generalized degrees of freedom and their application

This paper presents a quadratic finite element with generalized degrees of freedom (GDOF) and a quadratic finite strip with GDOF based on the principle that the local displacement fields of elements should be compatible with the global displacement field of the corresponding systems. Firstly, a global displacement field is developed using quadratic B-spline functions. Secondly, local displacement fields of elements and strips are constructed, employing multi-term interpolation polynomials of second degree. Making the local displacement fields of elements and strips be compatible with their corresponding global displacement fields, respectively, models of the finite element with GDOF and the finite strip with GDOF are accordingly generated. On the other hand, the quadratic B-spline function is convenient to derive the explicit form of characteristic equations for computing model because of its lower order, and accordingly simplifies the computation, compared with cubic and quintic spline functions. Several numerical examples demonstrate the accuracy, simplicity and versatility of the present element and strip in the analysis of thin-walled structures.