B-spline Quasi-interpolation Research Articles

Retraction: A cubic B-spline quasi-interpolation method for solving hyperbolic partial differential equations

Retraction: A cubic B-spline quasi-interpolation method for solving hyperbolic partial differential equations

RETRACTED ARTICLE: A cubic B-spline quasi-interpolation method for solving hyperbolic partial differential equations

We, the Editors and Publisher of International Journal of Computer Mathematics have retracted the following article: Sudhir Kumar, R.C. Mittal & Ram Jiwari (2023) A cubic B-spline quasi-interpolation method for solving hyperbolic partial differential equations, International Journal of Computer Mathematics, DOI: 10.1080/00207160.2023.2205963 Since publication, significant concerns have been raised about the fact that this article has substantial overlap with the following article which was not acknowledged, particularly in Sections 1-4 where there were verbatim sentences and sentence fragments: Mittal, R. C., Kumar, S., & Jiwari, R. (2021). A cubic B-spline quasi-interpolation algorithm to capture the pattern formation of coupled reaction-diffusion models. Engineering with Computers, 1-17 https://doi.org/10.1007/s00366-020-01278-3. Upon query, the authors were not able to address the concerns with their paper or provide a satisfactory explanation for this significant level of overlap. As this is a serious breach of our Editorial Policies, we are retracting the article from the journal. The authors do not agree with the retraction. We have been informed in our decision-making by our policy on publishing ethics and integrity and the COPE guidelines on retractions. The retracted article will remain online to maintain the scholarly record, but it will be digitally watermarked on each page as ‘Retracted’.

A meshless quasi-interpolation method for solving hyperbolic conservation laws based on the essenstially non-oscillatory reconstruction

In this paper, a quasi-interpolation method is proposed for solving hyperbolic conservation laws based on the essentially non-oscillatory (ENO) scheme. The hyperbolic equation is discretized in space with the finite difference ENO method and then the semi-discrete system is integrated by the strong stability preserving Runge-Kutta scheme. However, in the step of ENO reconstruction, the finite difference method is replaced by several quasi-interpolation schemes, including the multiquadric quasi-interpolation, the integral-type multiquadric quasi-interpolation and the cubic B-spline quasi-interpolation. Our quasi-interpolation method is simple and easy to implement since it doesn't need to solve any linear system. Moreover, it is also suitable for nonuniform grids and noisy sampling data. Nonlinear hyperbolic problems we target include one-dimensional and two-dimensional Burgers' equations and also the Euler equation. Numerical results demonstrate that the proposed quasi-interpolation ENO method is stable and has good accuracy.

A difference scheme based on cubic B-spline quasi-interpolation for the solution of a fourth-order time-fractional partial integro-differential equation with a weakly singular kernel

A difference scheme based on cubic B-spline quasi-interpolation for the solution of a fourth-order time-fractional partial integro-differential equation with a weakly singular kernel

A θ-finite difference scheme based on cubic B-spline quasi-interpolation for the time fractional Cattaneo equation with Caputo–Fabrizio operator

In this paper, a θ-finite difference scheme based on cubic B-spline quasi-interpolation has been derived for the solution of time fractional Cattaneo equation. The fractional derivative of the mentioned equation has been described in the Caputo–Fabrizio sense. Time fractional derivative is approximated by a scheme of order . The spatial second derivative in two consecutive time steps has been approximated using the second derivative of the cubic B-spline quasi-interpolation. For this scheme, the Fourier analysis method is used to discuss the stability and convergence. It has also been shown that the method is the convergence of order . Finally, five numerical examples have been illustrated to verify the efficiency and high accuracy of the proposed scheme.

A cubic B-spline quasi-interpolation algorithm to capture the pattern formation of coupled reaction-diffusion models

Diffusion plays a significant role in complex pattern formulations occurred in biological and chemical reactions. In this work, the authors study the effect of diffusion in coupled reaction-diffusion systems named the Gray-Scott model for complex pattern formation with the help of cubic B-spline quasi-interpolation (CBSQI) method and capture various formates of these patterns. The idea of Kronecker product is used first time with CBSQI method for 2D problems. Linear stability analysis of the reaction-diffusion system as well as stability of the proposed method is studied. Four test problems are considered to check the accuracy and efficiency of the method and found the stable patterns.

A B-Spline Quasi Interpolation Crank–Nicolson Scheme for Solving the Coupled Burgers Equations with the Caputo–Fabrizio Derivative

In this paper, a Crank–Nicolson finite difference scheme based on cubic B-spline quasi-interpolation has been derived for the solution of the coupled Burgers equations with the Caputo–Fabrizio derivative. The first- and second-order spatial derivatives have been approximated by first and second derivatives of the cubic B-spline quasi-interpolation. The discrete scheme obtained in this way constitutes a system of algebraic equations associated with a bi-pentadiagonal matrix. We show that the proposed scheme is unconditionally stable. Numerical examples are provided to verify the efficiency of the method.

Application of cubic B-spline quasi-interpolation for solving timefractional partial differential equation

The purpose of this paper is to present a numerical scheme for solving time-fractional partial differential equation based on cubic B-spline quasi-interpolation. For this purpose, first we will approximate the time-fractional derivative by Laplace transform method and then by using of cubic B-spline quasi-interpolation, the spatial derivatives are approximated. Moreover, the stability of this method is studied. Finally, European call and put options are priced and we will show that the results are good agreement with the other methods. The main advantage of the resulting scheme is that the algorithm is very simple, so it is very easy to implement.

Estimation of effluent parameters of slaughterhouse wastewater treatment with artificial neural network and B-spline quasi interpolation

Effluent of slaughterhouse wastewater treatment by combined up-flow anaerobic sludge blanket (UASB) reactor and extended aeration reactor was estimated through artificial neural networks (ANN), ANN-genetic algorithm (GA) and B-spline quasi interpolation. The overall system operated at two runs with average total chemical oxygen demand (TCOD) of 1514.65 and 3160.2 mg/L for the first and second run, respectively; with two overall hydraulic retention times of 73 and 104 h for each run. The overall system could remove TCOD, ammonia, phosphate and turbidity to a high extent. The multilayer perceptron artificial neural network (MLPANN) trained by Levenberge–Marquardt algorithm was employed to predict the TCOD, ammonia, phosphate and turbidity of the effluent which resulted in R of 0.8257, 0.6274, 0.7961 and 0.6884, respectively. The optimization of MLPANN by GA performed better than MLPANN with R of 0.8390, 0.7650, 0.8107 and 0.7365 for TCOD, ammonia, phosphate and turbidity, respectively. The B-spline quasi interpolation indicates a more accurate prediction due to its R of 0.9619, 0.8806, 0.8307 and 0.7856 for TCOD, ammonia, phosphate and turbidity, respectively. The B-spline quasi interpolation operation time is notably lower than ANN and ANN-GA. In addition, it has a simple algorithm and is implemented easier than the artificial neural network model.

A Comparative Study of Cubic B-spline-Based Quasi-interpolation and Differential Quadrature Methods for Solving Fourth-Order Parabolic PDEs

In this work, we present two approaches for simulation of fourth-order parabolic partial differential equations. In the first method, cubic B-spline quasi-interpolation is used to approximate the spatial derivative of the dependent variable and forward difference to approximate the time derivative. In the second method, we have used modified cubic B-spline functions-based differential quadrature method (DQM) for space discretization to get a system of ODEs and then this system is solved by SSP-RK43 method to get the results at knots. The numerical results demonstrate the accuracy of the proposed method. The stability analysis of the methods has also been discussed. It is observed that quasi-interpolation-based method is unconditionally stable, whereas for DQM, the stability has to be checked for a large number of space points. Moreover, for the small number of grid points, DQM gives better results, while for a large number of grid points, quasi-interpolation-based method is better.

An iterated quasi-interpolation approach for derivative approximation

Given discrete function values sampled at uniform centers, the iterated quasi-interpolation approach for approximating the m th derivative consists of two steps. The first step adopts m successive applications of the operator DQ (the quasi-interpolation operator Q first, and then the differentiation operator D) to get approximated values of the m th derivative at uniform centers. Then, by one further application of the quasi-interpolation operator Q to corresponding approximated derivative values gives the final approximation of the m th derivative. The most salient feature of the approach is that it approximates all derivatives with the same convergence rate. In addition, it is valid for a general multivariate function, compared with the existing iterated interpolation approaches that are only valid for periodic functions, so far. Numerical examples of approximating high-order derivatives using both the iterated and direct approach based on B-spline quasi-interpolation and multiquadric quasi-interpolation are presented at the end of the paper, which demonstrate that the iterated quasi-interpolation approach provides higher approximation orders than the corresponding direct approach.

A cubic B-spline quasi-interpolation method for solving two-dimensional unsteady advection diffusion equations

PurposeThe purpose of this study is to extend the cubic B-spline quasi-interpolation (CBSQI) method via Kronecker product for solving 2D unsteady advection-diffusion equation. The CBSQI method has been used for solving 1D problems in literature so far. This study seeks to use the idea of a Kronecker product to extend the method for 2D problems.Design/methodology/approachIn this work, a CBSQI is used to approximate the spatial partial derivatives of the dependent variable. The idea of the Kronecker product is used to extend the method for 2D problems. This produces the system of ordinary differential equations (ODE) with initial conditions. The obtained system of ODE is solved by strong stability preserving the Runge–Kutta method (SSP-RK-43).FindingsIt is found that solutions obtained by the proposed method are in good agreement with the analytical solution. Further, the results are also compared with available numerical results in the literature, and a reasonable degree of compliance is observed.Originality/valueTo the best of the authors’ knowledge, the CBSQI method is used for the first time for solving 2D problems and can be extended for higher-dimensional problems.

Dimension-dependent error estimates for sampling recovery on Smolyak grids based on B-spline quasi-interpolation

We investigate dimension-dependent estimates of the approximation error for linear algorithms of sampling recovery on Smolyak grids parametrized by m∈N, of periodic d-variate functions from the space with Hölder–Zygmund mixed smoothness α>0. We consider two subsets of the unit ball in this space. The first one is the subset of functions with homogeneous condition, and the second one is the subset of functions depending on ν active variables (1≤ν≤d). For these classes, we prove some upper bounds and lower bounds (for α≤2) of the error of the optimal sampling recovery on Smolyak grids, explicit in d, ν, m when d and m may be large.

Biological treatment of slaughterhouse wastewater: kinetic modeling and prediction of effluent.

In this study three modeling approaches consisting Modified Stover-Kincannon, multilayer perceptron neural network (MLPANN) and B-Spline quasi interpolation were applied in order to predict effluent of up-flow anaerobic sludge blanket (UASB) reactor and also to find the reaction kinetics. At first run, the average total chemical oxygen demand (TCOD) removal efficiency was 48.3% with hydraulic retention time (HRT) of 26h and 63.8% with HRT of 37h, at OLR of 0.77-1.66kg TCOD/m3 d. At the second run, UASB reactor operated with OLR of 1.94-3.1kg TCOD/m3 d and achieved the average TCOD removal efficiency of 64.74 and 72.48% with HRT of 26 and 37h, respectively. The Modified Stover-Kincannon performed well in terms of kinetic determination with a high value of regression coefficient over 0.98. The B-Spline quasi interpolation and MLPANN indicated a great fit for effluent prediction with average R of 0.9984 and 0.9986, and MSE of 157.6050 and 129.7796, respectively; however, they gave no information about reactions occurred in the system.

Solving Linear Optimal Control Problems Using Cubic B-spline Quasi-interpolation

In this article, we apply an impressive method for solving linear optimal control problem based on cubic B-spline quasi-interpolation. Hamilton-Jacobi equation are applied to linear optimal control problem convert to systems of first-order equations. The main idea of our scheme is approximation derivative with cubic B-spline quasi-interpolation. This method is straightforward, without restrictive assumptions.The results of scheme are made in pleasant agreement with analytic solutions. The accuracy of the proposed method is demonstrated by absolute error. Our scheme is simple to implement because its algorithm is easy and it's one of the advantages of the proposed method.

Cubic B-Spline Quasi-Interpolation Method For Regularized Long Wave Equation

In this study, we present a numerical method to solve the Regularized Long Wave (RLW) equation, based on cubic B-spline quasi-interpolation for the space integration and Crank-Nicolson method for the time integration. The method is tested on the problems of propagation of a solitary wave and interaction of two solitary waves. The three conservation quantities of the motion are calculated to determine the conservation properties of the proposed algorithm.

Modified cubic B-spline quasi-interpolation numerical scheme for hyperbolic conservation laws

ABSTRACTIn general, B-spline quasi-interpolation (BSQI)-based numerical schemes for hyperbolic conservation laws are unstable in nature. In the present work, we have developed the stable modified version of the cubic B-spline quasi-interpolation (CBSQI) numerical scheme for the hyperbolic conservation laws in one space dimension. In order to stabilize the CBSQI scheme, we have added an adaptive artificial viscosity term and derived the conditions under which the modified scheme is monotone. Although the CBSQI scheme can be of at most fourth order accurate in space, the monotone condition restricts the resulting scheme to be of at most order of one, as expected. In order to improve the order of accuracy, we adopt a procedure, which we call the weighted adaptive viscosity approach, through which we achieved the fourth order accuracy both in space and time in the modified CBSQI scheme. Further, linear -stability of the modified CBSQI scheme is analyzed using von-Neumann analysis. Numerical experiments are performed to validate the efficiency and stability of the modified CBSQI scheme.

Numerical solution of the Degasperis–Procesi equation by the cubic B-spline quasi-interpolation method

In this paper, a numerical scheme is presented to solve the non-dissipative Degasperis–Procesi equation based on the u-p formulation. The cubic B-spline quasi-interpolation coupled with the finite difference method is applied to approximate the spatial derivatives and an optimal third order TVD Runge–Kutta method to estimate the time derivative of the dependent variable. The accuracy and effectiveness of the proposed method are validated by six classical problems. Numerical results indicate that the proposed scheme is simple, easy to implement with high accuracy.

B-Spline Quasi-Interpolation Sampling Representation and Sampling Recovery in Sobolev Spaces of Mixed Smoothness

We proved direct and inverse theorems on B-spline quasi-interpolation sampling representation with a Littlewood-Paley-type norm equivalence in Sobolev spaces ${W^{r}_{p}}$ of mixed smoothness r. Based on this representation, we established estimates of the approximation error of recovery in L q -norm of functions from the unit ball ${U^{r}_{p}}$ in the spaces ${W^{r}_{p}}$ by linear sampling algorithms and the asymptotic optimality of these sampling algorithms in terms of Smolyak sampling width ${r^{s}_{n}}({U^{r}_{p}}, L_{q})$ and sampling width $r_{n}({U^{r}_{p}}, L_{q})$ .

Adaptive semi-discrete formulation of BSQI–WENO scheme for the modified Burgers’ equation

In this article, we have proposed a septic B-spline quasi-interpolation (SeBSQI) based numerical scheme for the modified Burgers’ equation. The SeBSQI scheme maintains eighth order accuracy for the smooth solution, but fails to maintain a non-oscillatory profile when the solution has discontinuities or sharp variations. To ensure the non-oscillatory profile of the solution, we have proposed an adaptive SeBSQI (ASeBSQI) scheme for the modified Burgers’ equation. The ASeBSQI scheme maintains higher order accuracy in the smooth regions using SeBSQI approximation and in regions with discontinuities or sharp variations, 5th order weighted essentially non-oscillatory (WENO) reconstruction is used to preserve a non-oscillatory profile. To identify discontinuous or sharp variation regions, a weak local truncation error based smooth indicator is proposed for the modified Burgers’ equation. For the temporal derivative, we have considered the Runge–Kutta method of order four. We have shown numerically that the ASeBSQI scheme preserves the convergence rate of the SeBSQI and it converges to the exact solution with convergence rate eight. We have performed numerical experiments to validate the proposed scheme. The numerical experiments demonstrate an improvement in accuracy and efficiency of the proposed schemes over the WENO5 and septic B-spline collocation schemes. The ASeBSQI scheme is also tested for one-dimensional Euler equations.