A septic B-spline collocation method is implemented to find the numerical solution of the modified regularized long wave (MRLW) equation. Three test problems including the single soliton and interaction of two and three solitons are studied to validate the proposed method by calculating the error norms $L_2$ and $L_{\mathrm{\infty }}$ and the invariants $I_1$, $I_2$, and $I_3$. Also, we have studied the Maxwellian initial condition pulse. The numerical results obtained by the method show that the present method is accurate and efficient. Results are compared with some earlier results given in the literature. A linear stability analysis of the method is also investigated.

We discuss the feasible interval of the parameter $k$ and a general expression of matrix $X$ which satisfies the rank equation $r(A-BXC)=k$. With these results, we study two problems under the rank constraint $r(A-BXC)=k$. The first one is to determine the maximal and minimal ranks under the rank constraint $r(A-BXC)=k$. The second one is to derive the least squares solutions of ${\parallel BXC-A\parallel }_F=\min$ under the rank constraint $r(A-BXC)=k$.

Using fixed-point techniques and Faber-Schauder systems in adequate Banach spaces, we approximate the solution of a system of nonlinear Fredholm integrodifferential equations of the second kind.

A new interpolation spline with two parameters, called EH interpolation spline, is presented in this paper, which is the extension of the standard cubic Hermite interpolation spline, and inherits the same properties of the standard cubic Hermite interpolation spline. Given the fixed interpolation conditions, the shape of the proposed splines can be adjusted by changing the values of the parameters. Also, the introduced spline could approximate to the interpolated function better than the standard cubic Hermite interpolation spline and the quartic Hermite interpolation splines with single parameter by a new algorithm.

General interpolation formulae for bivariate interpolation are established by introducing multiple parameters, which are extensions and improvements of those studied by Tan and Fang. The general interpolation formulae include general interpolation formulae of symmetric branched continued fraction, general interpolation formulae of univariate and bivariate interpolation, univariate block based blending rational interpolation, bivariate block based blending rational interpolation and their dual schemes, and some interpolation form studied by many scholars in recent years. We discuss the interpolation theorem, algorithms, dual interpolation, and special cases and give many kinds of interpolation scheme. Numerical examples are given to show the effectiveness of the method.

The aim of this paper is to solve an elliptic interface problem with a discontinuous coefficient and a singular source term by the spectral collocation method. First, we develop an algorithm for the elliptic interface problem defined in a rectangular domain with a line interface. By using the Gordon-Hall transformation, we generalize it to a domain with a curve boundary and a curve interface. The spectral element collocation method is then employed to complex geometries; that is, we decompose the domain into some nonoverlaping subdomains and the spectral collocation solution is sought in each subdomain. We give some numerical experiments to show efficiency of our algorithm and its spectral convergence.

we present a first supercloseness analysis for higher order FEM/LDG coupled method for solving singularly perturbed convection-diffusion problem. Based on piecewise polynomial approximations of degree $k$ $(k\ge 1)$, a supercloseness property of $k+1/2$ in DG norm is established on S-type mesh. Numerical experiments complement the theoretical results.

We first give the block style spectral decomposition of arbitrary block skew circulant matrix with skew circulant blocks. Secondly, we obtain the singular value of block skew circulant matrix with skew circulant blocks as well. Finally, based on the block style spectral decomposition, we deal with the optimal backward perturbation analysis for the block skew circulant linear system with skew circulant blocks.

Based on the modified Hermitian and skew-Hermitian splitting (MHSS) and preconditioned MHSS (PMHSS) methods, a generalized preconditioned MHSS (GPMHSS) method for a class of complex symmetric linear systems is presented. Theoretical analysis gives an upper bound for the spectral radius of the iteration matrix. From a practical point of view, we have analyzed and implemented inexact GPMHSS (IGPMHSS) iteration, which employs Krylov subspace methods as its inner processes. Numerical experiments are reported to confirm the efficiency of the proposed methods.

The Ostrowski inequality expresses bounds on the deviation of a function from its integral mean. The aim of this paper is to establish some new inequalities similar to the Ostrowski's inequality. The current paper obtains bounds for the deviation of a function from a combination of integral means over the end intervals covering the entire interval in terms of the norms of the second derivative of the function. Some new perturbed results are obtained. Application for cumulative distribution function is also discussed.

We employ Legendre-Galerkin spectral methods to solve state-constrained optimal control problems. The constraint on the state variable is an integration form. We choose one-dimensional case to illustrate the techniques. Meanwhile, we investigate the explicit formulae of constants within a posteriori error indicator.

We present a new stabilized finite element method for incompressible flows based on Brezzi-Pitkäranta stabilized method. The stability and error estimates of finite element solutions are derived for classical one-level method. Combining the techniques of two-level discretizations, we propose two-level Stokes/Oseen/Newton iteration methods corresponding to three different linearization methods and show the stability and error estimates of these three methods. We also propose a new Newton correction scheme based on the above two-level iteration methods. Finally, some numerical experiments are given to support the theoretical results and to check the efficiency of these two-level iteration methods.

We show that Schröder’s processes of the first kind and of the second kind to obtain a simple root of a nonlinear equation are related by polynomial and rational approximations.

We introduce a $q$-generalization of Szász-Mirakjan operators $S_{n,q}$ and discuss their properties for fixed $q\in (\mathrm{0,1})$. We show that the $q$-Szász-Mirakjan operators $S_{n,q}$ have good shape-preserving properties. For example, $S_{n,q}$ are variation-diminishing, and preserve monotonicity, convexity, and concave modulus of continuity. For fixed $q\in (\mathrm{0,1})$, we prove that the sequence $\{S_{n,q}\left(f\right)\}$ converges to $B_{\infty ,q}(f)$ uniformly on $[\mathrm{0,1}]$ for each $f\in C[0, 1/(1-q)]$, where $B_{\infty ,q}$ is the limit $q$-Bernstein operator. We obtain the estimates for the rate of convergence for $\{S_{n,q}\left(f\right)\}$ by the modulus of continuity of $f$, and the estimates are sharp in the sense of order for Lipschitz continuous functions.

The structural properties of LM-g splines are investigated by optimization and optimal control theory. The continuity and structure of LM-g splines are derived by using a class of necessary conditions with state constraints of optimal control and the relationship between LM-g interpolating splines and the corresponding L-g interpolating splines. This work provides a new method for further exploration of LM-g interpolating splines and its applications in the optimal control.

The main purpose of this paper is to introduce and investigate a class of $q$-Bernoulli, $q$-Euler, and $q$-Genocchi polynomials. The $q$-analogues of well-known formulas are derived. In addition, the $q$-analogue of the Srivastava-Pintér theorem is obtained. Some new identities, involving $q$-polynomials, are proved.

The aim of this paper is twofold. First, a matrix iteration for finding approximate inverses of nonsingular square matrices is constructed. Second, how the new method could be applied for computing the Drazin inverse is discussed. It is theoretically proven that the contributed method possesses the convergence rate nine. Numerical studies are brought forward to support the analytical parts.

We discuss the rate of convergence of the Lupas $q$-analogues of the Bernstein operators $R_{n,q}(f;x)$ which were given by Lupas in 1987. We obtain the estimates for the rate of convergence of $R_{n,q}(f)$ by the modulus of continuity of $f$, and show that the estimates are sharp in the sense of order for Lipschitz continuous functions.

This paper deals with approximating properties of the newly defined $q$-generalization of the genuine Bernstein-Durrmeyer polynomials in the case $q>1$, which are no longer positive linear operators on $C\left[\mathrm{0,1}\right]$. Quantitative estimates of the convergence, the Voronovskaja-type theorem, and saturation of convergence for complex genuine $q$-Bernstein-Durrmeyer polynomials attached to analytic functions in compact disks are given. In particular, it is proved that, for functions analytic in , $R>q$, the rate of approximation by the genuine $q$-Bernstein-Durrmeyer polynomials $\left(q>1\right)$ is of order $q^{-n}$ versus $1/n$ for the classical genuine Bernstein-Durrmeyer polynomials. We give explicit formulas of Voronovskaja type for the genuine $q$-Bernstein-Durrmeyer for $q>1$. This paper represents an answer to the open problem initiated by Gal in (2013, page 115).

Let $B_{n,q}(f;x)$, $q\in (0,\infty )$ be the $q$-Bernstein polynomials of a function $f\in C[\mathrm{0,1}]$. It has been known that, in general, the sequence $\left(B_{n,q_n}(f)\right)$ with $q_n\to 1+$ is not an approximating sequence for $f\in C[\mathrm{0,1}]$, in contrast to the standard case $q_n\to 1-$. In this paper, we give the sufficient and necessary condition under which the sequence $\left(B_{n,q_n}(f)\right)$ approximates $f$ for any $f\in C[\mathrm{0,1}]$ in the case $q_n>1$. Based on this condition, we get that if for sufficiently large $n$, then $\left(B_{n,q_n}(f)\right)$ approximates $f$ for any $f\in C[\mathrm{0,1}]$. On the other hand, if $\left(B_{n,q_n}(f)\right)$ can approximate $f$ for any $f\in C[\mathrm{0,1}]$ in the case $q_n>1$, then the sequence $(q_n)$ satisfies ${\overline{\text{lim}}}_{n\to \mathrm{\infty }}n(q_n-1)\le \text{ln}2$.

The truncated $q$-Bernstein polynomials $B_{n,m,q}\left(f;x\right)$, $n\in \mathbb{N}$, and $\mathrm{\hspace{0.17em}m}\in {\mathbb{N}}_0$ emerge naturally when the $q$-Bernstein polynomials of functions vanishing in some neighbourhood of 0 are considered. In this paper, the convergence of the truncated $q$-polynomials on $\left[0,1\right]$ is studied. To support the theoretical results, some numerical examples are provided.