On integro quartic spline interpolation

In this paper, we use quartic B-spline to construct an approximating function to agree with the given integral values of a univariate real-valued function over the same intervals. It is called integro quartic spline interpolation. Our interpolation method is new and easy to implement. Moreover, it can work successfully even without any boundary conditions. The interpolation errors are studied. The super convergence (sixth order and fourth order, respectively) in approximating function values and second-order derivative values at the knots is proved. Numerical examples illustrate that our method is very effective and our integro-interpolating quartic spline has higher approximation ability than others.     

Implicitly enriched Galerkin methods for numerical solutions of fourth‐order partial differential equations containing singularities

Highlights are the following:

• For any integer , we construct ‐continuous partition of unity (PU) functions with flat‐top from B‐spline functions to have numerical solutions of fourth‐order equations with singularities. B‐spline functions are modified to satisfy clamped boundary conditions.
• To handle singularity arising in fourth‐order elliptic differential equations, these modified B‐spline functions are enriched either by introducing enrichment basis functions implicitly through particular geometric mappings or by adding singular basis functions explicitly.
• To show the effectiveness of the proposed implicit enrichment methods (mapping method), the accuracy, the number of degrees of freedom (DOF), and matrix condition numbers are computed and compared in the h‐refinement, the p‐refinement, and the k‐refinement of the approximation space of B‐spline basis functions.

Using Partition of unity (PU) functions with flat‐top, B‐spline functions are modified to satisfy boundary conditions of the fourth‐order equations. Since the standard isogeometric analysis (IGA) as well as the conventional FEM have limitations in handling fourth‐order differential equations containing singularities, we consider two enrichment methods (explicit and implicit) in the framework of the p‐, the k, and the h‐refinements of IGA. We demonstrate that both enrichment methods yield good approximate solutions, but explicit enrichment methods give large (almost singular) matrix condition numbers and face integrating singular functions. Because of these limitations of external enrichment methods, we extensively investigate implicit enrichment methods (mapping methods) that virtually convert fourth‐order elliptic problems with singularities to problems with no influence of the singularities. Effectiveness of the proposed mapping method extensively tested to one‐dimensional fourth‐order equation with singularities. The implicit enrichment (mapping) method is extended to the two‐dimensional cases and test it to fourth‐order partial differential equations on cracked domains.     

Numerical solution of a system of fourth order boundary value problems using variational iteration method

In this paper, the variational iteration method is used to solve a system of fourth order boundary value problems associated with obstacle, unilateral and contact problems. Numerical solution obtained by the method is of high accuracy. Moreover, the higher-order derivatives of numerical solution can also approximate the higher-order derivatives of exact solution well. Five examples compared with those considered by Siddiqi and Akram [S.S. Siddiqi, G. Akram, Numerical solution of a system of fourth order boundary value problems using cubic non-polynomial spline method, Applied Mathematics and Computation 190 (2007) 652–661] show that the method is more efficient.     

Galerkin's finite element formulation of the system of fourth‐order boundary‐value problems

In this article, a Galerkin's finite element approach based on weighted‐residual is presented to find approximate solutions of a system of fourth‐order boundary‐value problems associated with obstacle, unilateral and contact problems. The approach utilizes a piece‐wise cubic approximations utilizing cubic Hermite interpolation polynomials. Numerical studies have shown the superior accuracy and lesser computational cost of the scheme in comparison to cubic spline, non‐polynomial spline and cubic non‐polynomial spline methods. Numerical examples are presented to illustrate the applicability of the method. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27: 1551–1560, 2011     

A new cubic B-spline method for linear fifth order boundary value problems

In this paper, we mainly study the numerical solution of linear fifth order boundary value problems by using cubic B-splines. Our algorithm develops not only the cubic spline approximation solution but also the approximation derivatives of first order to fourth order of the analytic solution at the same time. This new method has lower computational cost than many other methods and is second order convergent. Numerical examples are given to demonstrate the effectiveness of our method.     

Quintic nonpolynomial spline solutions for fourth order two-point boundary value problem

In this paper, we develop quintic nonpolynomial spline methods for the numerical solution of fourth order two-point boundary value problems. Using this spline function a few consistency relations are derived for computing approximations to the solution of the problem. The present approach gives better approximations and generalizes all the existing polynomial spline methods up to order four. This approach has less computational cost. Convergence analysis of these methods is discussed. Two numerical examples are included to illustrate the practical usefulness of our methods.     

Quintic spline technique for time fractional fourth‐order partial differential equation

Higher order non‐Fickian diffusion theories involve fourth‐order linear partial differential equations and their solutions. A quintic polynomial spline technique is used for the numerical solutions of fourth‐order partial differential equations with Caputo time fractional derivative on a finite domain. These equations occur in many applications in real life problems such as modeling of plates and thin beams, strain gradient elasticity, and phase separation in binary mixtures, which are basic elements in engineering structures and are of great practical significance to civil, mechanical, and aerospace engineering. The quintic polynomial spline technique is used for space discretization and the time‐stepping is done using a backward Euler method based on the L1 approximation to the Caputo derivative. The stability and convergence analysis are also discussed. The numerical results are given, which demonstrate the effectiveness and accuracy of the numerical method. The numerical results obtained in this article are also compared favorably well with the results of (S. S. Siddiqi and S. Arshed, Int. J. Comput. Math. 92 (2015), 1496–1518). © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 445–466, 2017     

Local approximation on surfaces with discontinuities,given limited order Fourier coefficients

We present a spline approximation method for a piece of a surface where jump discontinuities occur along curves. The data for the surface is assumed to be Fourier coefficients which are limited in order and possibly contaminated with noise. The support of the approximation is bounded by three sides of a rectangle with a fourth boundary possibly curved. Discontinuities of the surface may occur across the curved side and linear sides adjacent to it. The approximation uses a small number of lines through the support and parallel to the straight boundary lines that are adjacent to the curve. Along each line a one-dimensional spline approximation is done for a section of the surface over the line. This approximation uses two-dimensional Fourier coefficient data, localizing spline functions, and a technique which we developed earlier for one-dimensional analogues of the problem. We use a spline quasi-interpolation scheme to create a surface approximation from the section approximations. The result is accurate even when the surface is discontinuous across the curved boundary and adjacent side boundaries.     

A high‐order B‐spline collocation scheme for solving a nonhomogeneous time‐fractional diffusion equation

A high‐accuracy numerical approach for a nonhomogeneous time‐fractional diffusion equation with Neumann and Dirichlet boundary conditions is described in this paper. The time‐fractional derivative is described in the sense of Riemann‐Liouville and discretized by the backward Euler scheme. A fourth‐order optimal cubic B‐spline collocation (OCBSC) method is used to discretize the space variable. The stability analysis with respect to time discretization is carried out, and it is shown that the method is unconditionally stable. Convergence analysis of the method is performed. Two numerical examples are considered to demonstrate the performance of the method and validate the theoretical results. It is shown that the proposed method is of order O(Δx⁴ + Δt^{2 ? α}) convergence, where α ∈ (0,1) . Moreover, the impact of fractional‐order derivative on the solution profile is investigated. Numerical results obtained by the present method are compared with those obtained by the method based on standard cubic B‐spline collocation method. The CPU time for present numerical method and the method based on cubic B‐spline collocation method are provided.     

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.     

TAGE iterative algorithm and nonpolynomial spline basis for the solution of nonlinear singular second order ordinary differential equations

In the present paper, we discuss three point difference method based on nonpolynomial spline basis for the second order ordinary differential equation. Difference schemes are derived for linear and nonlinear case and are used to solve via two parameter alternating group explicit iterative algorithm. The schemes have a fourth and second order of uniform convergence for the choice of the parameters involved in the method. Computational results are presented comparing the two methods in terms of accuracy and execution times. The results indicate the advantage of using parallel implementation of the new method.     

An efficient numerical treatment of fourth‐order fractional diffusion‐wave problems

In this paper, we consider the numerical treatment of a fourth‐order fractional diffusion‐wave problem. Our proposed method includes the use of parametric quintic spline in the spatial dimension and the weighted shifted Grünwald‐Letnikov approximation of fractional integral. The solvability, stability, and convergence of the numerical scheme are rigorously proved. It is shown that the theoretical convergence order improves those of earlier work. Simulation is further carried out to demonstrate the numerical efficiency of the proposed scheme and to compare with other methods.     

Explicit finite element schemes for first order symmetric hyperbolic systems

Summary Finite element methods of up to fourth order accuracy admitting explicit discrete equations are constructed for linear symmetric first order hyperbolic equations having sufficiently smooth solutions. Lumping of the mass matrix at the forward time level is achieved by the addition of a differential operator, which for smooth spline spaces is dissipative and strongly enhances the stability properties of the resulting scheme.     

Smooth spline approximations for the solution of a boundary value problem with engineering applications

Methods of order 2, and 4 are developed for the continuous approximation of the solution of a two-point boundary value problem involving a fourth order linear differential equation via quintic and sextic spline functions. In three typical numerical examples, the results are briefly summarized to demonstrate the practical usefulness of the methods.     

Bivariate spline method for numerical solution of time evolution Navier‐Stokes equations over polygons in stream function formulation

We use a bivariate spline method to solve the time evolution Navier‐Stokes equations numerically. The bivariate splines we use in this article are in the spline space of smoothness r and degree 3r over triangulated quadrangulations. The stream function formulation for the Navier‐Stokes equations is employed. Galerkin's method is applied to discretize the space variables of the nonlinear fourth‐order equation, Crank‐Nicholson's method is applied to discretize the time variable, and Newton's iterative method is then used to solve the resulting nonlinear system. We show the existence and uniqueness of the weak solution in L₂(0, T; H²(Ω)) ∩ L_∞(0, T; H¹(Ω)) of the 2D nonlinear fourth‐order problem and give an estimate of how fast the numerical solution converges to the weak solution. The C¹ cubic splines are implemented in MATLAB for solving the Navier‐Stokes equations numerically. Our numerical experiments show that the method is effective and efficient. © 2003 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 19: 776–827, 2003.     

A smooth approximation based on exponential spline solutions for nonlinear fourth order two point boundary value problems

Methods of order two, four and six based on Exponential spline functions consisting of a polynomial part of degree three and an exponential part is developed to find approximation of linear and nonlinear fourth order two point boundary value problems. It is shown that the free parameter k of the exponential part can be used to raise the order of accuracy of the new scheme. Convergence analysis of these methods is shown along with numerical examples each for linear and nonlinear are included to illustrate the practical usefulness of our methods.     

High order approximations using spline-based differential quadrature method: Implementation to the multi-dimensional PDEs

A new differential quadrature method based on cubic B-spline is developed for the numerical solution of differential equations. In order to develop the new approach, the B-spline basis functions are used on the grid and midpoints of a uniform partition. Some error bounds are obtained by help of cubic spline collocation, which show that the method in its classic form is second order convergent. In order to derive higher accuracy, high order perturbations of the problem are generated and applied to construct the numerical algorithm. A new fourth order method is developed for the numerical solution of systems of second order ordinary differential equations. By solving some test problems, the performance of the proposed methods is examined. Also the implementation of the method for multi-dimensional time dependent partial differential equations is presented. The stability of the proposed methods is examined via matrix analysis. To demonstrate the applicability of the algorithms, we solve the 2D and 3D coupled Burgers’ equations and 2D sine-Gordon equation as test problems. Also the coefficient matrix of the methods for multi-dimensional problems is described to analyze the stability.     

Quintic hyperbolic nonpolynomial spline and finite difference method for nonlinear second order differential equations and its application

An efficient numerical method based on quintic nonpolynomial spline basis and high order finite difference approximations has been presented. The scheme deals with the space containing hyperbolic and polynomial functions as spline basis. With the help of spline functions we derive consistency conditions and high order discretizations of the differential equation with the significant first order derivative. The error analysis of the new method is discussed briefly. The new method is analyzed for its efficiency using the physical problems. The order and accuracy of the proposed method have been analyzed in terms of maximum errors and root mean square errors.     

Modified nodal cubic spline collocation for biharmonic equations

We formulate a modified nodal cubic spline collocation scheme for the solution of the biharmonic Dirichlet problem on the unit square. We prove existence and uniqueness of a solution of the scheme and show how the scheme can be solved on an N × N uniform partition of the square at a cost O(N ² log₂ N + mN ²) using fast Fourier transforms and m iterations of the preconditioned conjugate gradient method. We demonstrate numerically that m proportional to log₂ N guarantees the desired convergence rates. Numerical results indicate the fourth order accuracy of the approximations in the global maximum norm and the fourth order accuracy of the approximations to the first order partial derivatives at the partition nodes.      

Least-squares Spline Regression with Block-diagonal Variance Matrices

A numerical method of solution is presented for the least squaresfitting of experimental data by spline functions in the casewhere the data errors are correlated and for which the variancematrix is specified. The method is general in that it permits(a) splines of any order, (b) the knots of the spline to bearbitrary in number and position, and (c) variance matricesthat are block diagonal in form. Since limiting forms of (c)are diagonal and full variance matrices, the method can handle,as special cases, both conventional spline regression problemsand spline regression problems with general, unstructured variancematrices. An application to gamma spectrometry, in which theblocks of the variance matrix have special structure, is fullytreated.