Numerical studies on a novel split-step quadratic B-spline finite element method for the coupled Schrödinger–KdV equations

We propose a novel split-step quadratic B-spline finite element method for solving the initial-boundary value problem of the coupled Schrödinger–KdV equations. A full-discrete finite element scheme is constructed. The conserved properties of the full-discrete scheme are proved. Detailed numerical results show the efficiency of our scheme.     

A Petrov-Galerkin method for solving the generalized equal width (GEW) equation

The generalized equal width (GEW) equation is solved numerically by the Petrov-Galerkin method using a linear hat function as the test function and a quadratic B-spline function as the trial function. Product approximation has been used in this method. A linear stability analysis of the scheme shows it to be conditionally stable. Test problems including the single soliton and the interaction of solitons are used to validate the suggested method, which is found to be accurate and efficient. Finally, the Maxwellian initial condition pulse is studied.     

A parameter-uniform numerical method for time-dependent singularly perturbed differential-difference equations

A numerical study is made for solving a class of time-dependent singularly perturbed convection–diffusion problems with retarded terms which often arise in computational neuroscience. To approximate the retarded terms, a Taylor’s series expansion has been used and the resulting time-dependent singularly perturbed differential equation is approximated using parameter-uniform numerical methods comprised of a standard implicit finite difference scheme to discretize in the temporal direction on a uniform mesh by means of Rothe’s method and a B-spline collocation method in the spatial direction on a piecewise-uniform mesh of Shishkin type. The method is shown to be accurate of order O(M⁻¹ + N⁻² ln³N), where M and N are the number of mesh points used in the temporal direction and in the spatial direction respectively. An extensive amount of analysis has been carried out to prove the uniform convergence with respect to the singular perturbation parameter. Numerical results are given to illustrate the parameter-uniform convergence of the numerical approximations. Comparisons of the numerical solutions are performed with an upwind and midpoint upwind finite difference scheme on a piecewise-uniform mesh to demonstrate the efficiency of the method.     

Obtaining a banded system of equations in complete spline interpolation problem via B-spline basis

We study the problem of interpolation by a complete spline of 2n − 1 degree given in B-spline representation. Explicit formulas for the first nand the last ncoefficients of B-spline decomposition are found. It is shown that other B-spline coefficients can be computed as a solution of a banded system of an equitype linear equations.     

A class of algebraic-trigonometric blended splines

This paper presents a new kind of algebraic-trigonometric blended spline curve, called xyB curves, generated over the space {1,t,sint,cost,sin²t,sin³t,cos³t}. The new curves not only inherit most properties of usual cubic B-spline curves in polynomial space, but also enjoy some other advantageous properties for modeling. For given control points, the shape of the new curves can be adjusted by using the parameters x and y. When the control points and the parameters are chosen appropriately, the new curves can represent some conics and transcendental curves. In addition, we present methods of constructing an interpolation xyB-spline curve and an xyB-spline curve which is tangent to the given control polygon. The generation of tensor product surfaces by these new spline curves is straightforward. Many properties of the curves can be easily extended to the surfaces. The new surfaces can exactly represent the rotation surfaces as well as the surfaces with elliptical or circular sections.     

Local modification of Pythagorean-hodograph quintic spline curves using the B-spline form

The problems of determining the B–spline form of a C ² Pythagorean–hodograph (PH) quintic spline curve interpolating given points, and of using this form to make local modifications, are addressed. To achieve the correct order of continuity, a quintic B–spline basis constructed on a knot sequence in which each (interior) knot is of multiplicity 3 is required. C ² quintic bases on uniform triple knots are constructed for both open and closed C ² curves, and are used to derive simple explicit formulae for the B–spline control points of C ² PH quintic spline curves. These B-spline control points are verified, and generalized to the case of non–uniform knots, by applying a knot removal scheme to the Bézier control points of the individual PH quintic spline segments, associated with a set of six–fold knots. Based on the B–spline form, a scheme for the local modification of planar PH quintic splines, in response to a control point displacement, is proposed. Only two contiguous spline segments are modified, but to preserve the PH nature of the modified segments, the continuity between modified and unmodified segments must be relaxed from C ² to C ¹. A number of computed examples are presented, to compare the shape quality of PH quintic and “ordinary” cubic splines subject to control point modifications.     

Selection of smoothing parameters in<Emphasis Type="Italic">B</Emphasis>-spline nonparametric regression models using information criteria

We consider the use ofB-spline nonparametric regression models estimated by the maximum penalized likelihood method for extracting information from data with complex nonlinear structure. Crucial points inB-spline smoothing are the choices of a smoothing parameter and the number of basis functions, for which several selectors have been proposed based on cross-validation and Akaike information criterion known as AIC. It might be however noticed that AIC is a criterion for evaluating models estimated by the maximum likelihood method, and it was derived under the assumption that the ture distribution belongs to the specified parametric model. In this paper we derive information criteria for evaluatingB-spline nonparametric regression models estimated by the maximum penalized likelihood method in the context of generalized linear models under model misspecification. We use Monte Carlo experiments and real data examples to examine the properties of our criteria including various selectors proposed previously.     

A parameter-uniform scheme for singularly perturbed partial differential equations with a time lag

A numerical scheme for a class of singularly perturbed delay parabolic partial differential equations which has wide applications in the various branches of science and engineering is suggested. The solution of these problems exhibits a parabolic boundary layer on the lateral side of the rectangular domain which continuously depends on the perturbation parameter. For the small perturbation parameter, the standard numerical schemes for the solution of these problems fail to resolve the boundary layer(s) and the oscillations occur near the boundary layer. Thus, in this paper to resolve the boundary layer the extended cubic B-spline basis functions consisting of a free parameter λ are used on a fitted-mesh. The extended B-splines are the extension of classical B-splines. To find the best value of λ the optimization technique is adopted. The extended cubic B-splines are an advantage over the classical B-splines as for some optimized value of λ the solution obtained by the extended B-splines is better than the solution obtained by classical B-splines. The method is shown to be first-order accurate in t and almost the second-order accurate in x. It is also shown that this method is better than some existing methods. Several test problems are encountered to validate the theoretical results.     

Splines and efficiency in dynamic programming

It is shown how one can use splines, represented in the B-spline basis, to reduce the difficulties of large storage requirements in dynamic programming via approximations to the minimum-return function without the inefficiency associated with using polynomials to the same end.     

Iterative Approximation of Statistical Distributions and Relation to Information Geometry

The optimal control of stochastic processes through sensor estimation of probability density functions is given a geometric setting via information theory and the information metric. Information theory identifies the exponential distribution as the maximum entropy distribution if only the mean is known and the Γ distribution if also the mean logarithm is known. The surface representing Γ models has a natural Riemannian information metric. The exponential distributions form a one-dimensional subspace of the two-dimensional space of all Γ distributions, so we have an isometric embedding of the random model as a subspace of the Γ models. This geometry provides an appropriate structure on which to represent the dynamics of a process and algorithms to control it. This short paper presents a comparative study on the parameter estimation performance between the geodesic equation and the B-spline function approximations when they are used to optimize the parameters of the Γ family distributions. In this case, the B-spline functions are first used to approximate the Γ probability density function on a fixed length interval; then the coefficients of the approximation are related, through mean and variance calculations, to the two parameters (i.e. μ and β) in Γ distributions. A gradient based parameter tuning method has been used to produce the trajectories for (μ, β) when B-spline functions are used, and desired results have been obtained which are comparable to the trajectories obtained from the geodesic equation. This revised version was published online in August 2006 with corrections to the Cover Date.     

Boundary elements and β-spline surface modeling for medical applications

The aim of this paper is to present and discuss an approach based on the integration of the boundary element method (BEM) with β-spline geometric modeling of the different surfaces involved in the external bone remodeling phenomena. The purpose of combining these two techniques is to avoid the jagged edges shapes and thus, to increase the convergence speed of the bone remodeling function. In this study, the external bone remodeling model proposed by Fridez et al. [P. Fridez, L. Rakotomanana, A. Terrier, P.F. Leyvraz, Three dimensional model of bone external adaptation, Comput. Methods Biomech. Biomed. Eng. 2 (1998) 189–196] is used. This model shows the change of the external bone surface remodeling at a boundary point, as a function of the stimulus variable Ψ. This variable is related to the stress tensor and the normal vector to that point. The β-spline surfaces were used because they are simple and reliable to smooth the contour by using the less possible number of geometric constraints. Some numerical examples are presented and discussed in order to show the versatility of the proposed approach.     

Sufficient conditions for the comonotone interpolation of cubic C 2-splines

We consider the problem of interpolation of a function under the condition of the preservation of the nature of its piecewise monotonicity. We give sufficient conditions for the comonotone interpolation by a classical cubic C ²-spline in the representation based on the expansion of its first derivative in a basis consisting of B-splines. These conditions allow to determine whether the soobtained spline is comonotone without solving the interpolation problem.     

Quintic B-spline collocation method for numerical solution of the Kuramoto–Sivashinsky equation

In this paper, the quintic B-spline collocation scheme is implemented to find numerical solution of the Kuramoto–Sivashinsky equation. The scheme is based on the Crank–Nicolson formulation for time integration and quintic B-spline functions for space integration. The accuracy of the proposed method is demonstrated by four test problems. The numerical results are found to be in good agreement with the exact solutions. Results are also shown graphically and are compared with results given in the literature.     

Numerical comparison of methods for solving second-order ordinary initial value problems

In this paper, we apply Adomian decomposition method (shortly, ADM) to develop a fast and accurate algorithm of a special second-order ordinary initial value problems. The ADM does not require discretization and consequently of massive computations. This paper is particularly concerned with the ADM and the results obtained are compared with previously known results using the Quintic C²-spline integration methods. The numerical results demonstrate that the ADM is relatively accurate and easily implemented.     

Mixed interpolating-smoothing splines and the ν-spline

In their monograph, Bezhaev and Vasilenko have characterized the “mixed interpolating-smoothing spline” in the abstract setting of a Hilbert space. In this paper, we derive a similar characterization under slightly more general conditions. This is specialized to the finite-dimensional case, and applied to a few well-known problems, including the ν-spline (a piecewise polynomial spline in tension) and near-interpolation, as well as interpolation and smoothing. In particular, one of the main objectives in this paper is to show that the ν-spline is actually a mixed spline, an observation that we believe was not known prior to this work. We also show that the ν-spline is a limiting case of smoothing splines as certain weights increase to infinity, and a limiting case of near-interpolants as certain tolerances decrease to zero. We conclude with an iteration used to construct curvature-bounded ν-spline curves.     

Solution of governing differential equations of vibrating cylindrical shells using B-spline functions

The cubic B₃-spline functions and eigenfunctions are used to obtain the approximate solution for the vibration of cylindrical shells in this paper. Unified computational schemes suited for various types of boundary conditions are formulated here. In comparison with the conventional finite elements method and finite strip method, the main features of the present method are higher accuracy, fewer unknowns, ease in programming, and economy in computer solution. The numerical results are given and compared with other numerical solutions.     

Determining tension parameters in rational gamma-spline interpolation

Inspired by the classic γ-spline, we propose a method for constructing a G² rational γ-spline curve that interpolates a given set of distinct ordered data-points (planar or spatial). The only input of our method is just these data-points. We also present a procedure to solve the key problem of determining the tension parameters γ_i which are computed in terms of exponential functions that determine the eccentricities of the common conic osculants at the junction points while keeping in geometrical agreement with data-points. This allows the resulting curve to be modified in the close vicinity of each data-point.     

A note on the evaluation of bounded L2-functionals at B-splines and its application to singular integral equations

An algorithm computing recursively the values of ∫g(t)v(t) dt, whereg is anL ₂-function andv is aB-spline, is presented. For the functionsg _s(t)=log∥s?t∥ the starting values of the recursion formula can be computed analytically. The problem is related to the numerical solution of integral equations with a logarithmic singularity.     

A High-Order <Emphasis Type="Italic">B</Emphasis>-Spline Collocation Method for Solving Nonlinear Singular Boundary Value Problems Arising in Engineering and Applied Science

In this paper, a high-order B-spline collocation method on a uniform mesh is presented for solving nonlinear singular two-point boundary value problems with Neumann and Robin boundary conditions:

$$\begin{aligned} (p(x)y')'= & {} p(x)f(x,y), \quad 0<x\le 1, \\ y'(0)= & {} 0,\quad ay(1)+by'(1)=c, \end{aligned}$$

where \(p(x)=x^{\alpha }g(x),\alpha \ge 0\) is a general class of non-negative function. The error analysis for the quartic B-spline interpolation is discussed. To demonstrate the applicability and efficiency of our method we consider eight numerical examples, seven of which arise in various branches of applied science and engineering: (1) equilibrium of isothermal gas sphere; (2) thermal explosion; (3) thermal distribution in the human head; (4) oxygen diffusion in a spherical cell; (5) stress distribution on shallow membrane cap; (6) reaction diffusion process in a spherical permeable catalyst; (7) heat and mass transfer in a spherical catalyst. It is shown that our method has fourth-order convergence and is more accurate than finite difference methods (Chawla et al., in BIT 28:88–97, 1988; Pandey et al. in J Comput Appl Math 224:734–742, 2009) and B-spline collocation methods (Abukhaled et al. in Int J Numer Anal Model 8:353–363, 2011; Khuri and Sayfy in Int J Comput Methods 11(1):1350052, 2014).