Bivariate Lagrange Interpolation Research Articles

A bivariate spectral linear partition method for solving nonlinear evolution equations

This work develops a method for solving nonlinear evolution equations. The method, termed a bivariate spectral linear partition method (BSLPM), combines the Chebyshev spectral collocation method, bivariate Lagrange interpolation, and a linear partition technique as an underlying linearization method. It is developed for an th‐order nonlinear differential equation and then used to solve three known evolution problems. The results are compared with known exact solutions from literature. The method's applicability, reliability, and accuracy are confirmed by the congruence between the numerical and exact solutions. Tables, error graphs, and convergence graphs were generated using MATLAB (R2015a), to confirm the order of accuracy of the method and verify its convergence. The performance of the method is also observed against other methods performing well in these types of differential equations and is found to be comparable in terms of accuracy. The proposed method is also efficient as it uses minimal computation time.

Upper Bound for Lebesgue Constant of Bivariate Lagrange Interpolation Polynomial on the Second Kind Chebyshev Points

In the paper, we study the upper bound estimation of the Lebesgue constant of the bivariate Lagrange interpolation polynomial based on the common zeros of product Chebyshev polynomials of the second kind on the square [−1,1]2. And, we prove that the growth order of the Lebesgue constant is O((n + 2)2). This result is different from the Lebesgue constant of Lagrange interpolation polynomial on the unit disk, the growth order of which is . And, it is different from the Lebesgue constant of the Lagrange interpolation polynomial based on the common zeros of product Chebyshev polynomials of the first kind on the square [−1,1]2, the growth order of which is O((lnn)2).

Numerical Investigation of Natural Convection Viscoelastic Jeffrey’s Nanofluid Flow from a Vertical Permeable Flat Plate with Heat Generation, Thermal Radiation, and Chemical Reaction

The boundary layer flow of an incompressible viscoelastic Jeffrey’s nanofluid from a vertical permeable flat plate is investigated. We consider the effects of heat generation, thermal radiation, and chemical reaction on the fluid flow. The nonlinear transformed coupled differential equations that describe the transport processes are solved numerically using a multidomain bivariate spectral quasilinearization method (MD-BSQLM). This innovative method involves blending the quasilinearization idea with the bivariate Lagrange interpolation. The solutions of the resulting system of equations are then obtained sequentially on multiple intervals using the Chebyshev spectral collocation method. The method is shown to give accurate solutions for boundary layer-type equations. The influence of various physical parameters on velocity, temperature, and nanoparticle concentration fields, as well as on the skin friction and heat and mass transfer coefficients, is shown and discussed in detail. The range of the values of the governing parameters considered in this study is between 0 , 4 . For qualitative validation of the results and the numerical method used, calculations were carried out to graphically obtain the velocity, temperature, and nanoparticle concentration fields for selected physical parameter values. The results obtained were found to correlate with the results from published literature. For quantitative verification of our findings, the MD-BSQLM numerical solutions were again confirmed against published results reported in the literature, and the results were observed to be in perfect agreement. This study’s findings indicate that the Deborah number and suction parameter have related effects on the velocity profile, which is to suppress both the flow velocity and the momentum boundary layer thickness. Increasing the heat generation and thermal radiation parameters enhances both the temperature and thermal boundary layer depths. In contrast, an increase in the chemical reaction parameter causes a decrease in the fluid concentration.

Bivariate Lagrange interpolation based on Chebyshev points of the second kind

We give compact formulae of the Lagrange interpolation polynomials and cubature formulae based on the common zeros of product Chebyshev polynomials of the second kind. Further, for $$0<p\leq2$$ , we study the mean convergence of the Lagrange interpolation polynomials.

On the Multidomain Bivariate Spectral Local Linearisation Method for Solving Systems of Nonsimilar Boundary Layer Partial Differential Equations

In this work, a novel approach for solving systems of nonsimilar boundary layer equations over a large time domain is presented. The method is a multidomain bivariate spectral local linearisation method (MD-BSLLM), Legendre-Gauss-Lobatto grid points, a local linearisation technique, and the spectral collocation method to approximate functions defined as bivariate Lagrange interpolation. The method is developed for a general system of n nonlinear partial differential equations. The use of the MD-BSLLM is demonstrated by solving a system of nonlinear partial differential equations that describe a class of nonsimilar boundary layer equations. Numerical experiments are conducted to show applicability and accuracy of the method. Grid independence tests establish the accuracy, convergence, and validity of the method. The solution for the limiting case is used to validate the accuracy of the MD-BSLLM. The proposed numerical method performs better than some existing numerical methods for solving a class of nonsimilar boundary layer equations over large time domains since it converges faster and uses few grid points to achieve accurate results. The proposed method uses minimal computation time and its accuracy does not deteriorate with an increase in time.

On a Bivariate Spectral Homotopy Analysis Method for Unsteady Mixed Convection Boundary Layer Flow, Heat, and Mass Transfer due to a Stretching Surface in a Rotating Fluid

A bivariate spectral homotopy analysis method (BSHAM) is extended to solutions of systems of nonlinear coupled partial differential equations (PDEs). The method has been used successfully to solve a nonlinear PDE and is now tested with systems. The method is based on a new idea of finding solutions that obey a rule of solution expression that is defined in terms of the bivariate Lagrange interpolation polynomials. The BSHAM is used to solve a system of coupled nonlinear partial differential equations modeling the unsteady mixed convection boundary layer flow, heat, and mass transfer due to a stretching surface in a rotating fluid, taking into consideration the effect of buoyancy forces. Convergence of the numerical solutions was monitored using the residual error of the PDEs. The effects of the flow parameters on the local skin-friction coefficient, the Nusselt number, and the Sherwood number were presented in graphs.

A numerical study of unsteady non-Newtonian Powell-Eyring nanofluid flow over a shrinking sheet with heat generation and thermal radiation

In this paper we investigate the unsteady boundary-layer flow of an incompressible Powell-Eyring nanofluid over a shrinking surface. The effects of heat generation and thermal radiation on the fluid flow are taken into account. Numerical solutions of the nonlinear differential equations that describe the transport processes are obtained using a multi-domain bivariate spectral quasilinearization method. This innovative technique involves coupling bivariate Lagrange interpolation with quasilinearization. The solutions of the resulting system of equations are then obtained in a piecewise manner in a sequence of multiple intervals using the Chebyshev spectral collocation method. A parametric study shows how various parameters influence the flow and heat transfer processes. The validation of the results, and the method used here, has been achieved through a comparison of the current results with previously published results for selected parameter values. In general, an excellent agreement is observed. The results from this study show that the fluid parameters ε and δ reduce the flow velocity and the momentum boundary-layer thickness. The heat generation and thermal radiation parameters are found to enhance both the temperature and thermal boundary-layer thicknesses.

On a bivariate spectral relaxation method for unsteady magneto-hydrodynamic flow in porous media.

The paper presents a significant improvement to the implementation of the spectral relaxation method (SRM) for solving nonlinear partial differential equations that arise in the modelling of fluid flow problems. Previously the SRM utilized the spectral method to discretize derivatives in space and finite differences to discretize in time. In this work we seek to improve the performance of the SRM by applying the spectral method to discretize derivatives in both space and time variables. The new approach combines the relaxation scheme of the SRM, bivariate Lagrange interpolation as well as the Chebyshev spectral collocation method. The technique is tested on a system of four nonlinear partial differential equations that model unsteady three-dimensional magneto-hydrodynamic flow and mass transfer in a porous medium. Computed solutions are compared with previously published results obtained using the SRM, the spectral quasilinearization method and the Keller-box method. There is clear evidence that the new approach produces results that as good as, if not better than published results determined using the other methods. The main advantage of the new approach is that it offers better accuracy on coarser grids which significantly improves the computational speed of the method. The technique also leads to faster convergence to the required solution.

Bivariate Lagrange interpolation at the node points of non-degenerate Lissajous curves

Motivated by an application in Magnetic Particle Imaging, we study bivariate Lagrange interpolation at the node points of Lissajous curves. The resulting theory is a generalization of the polynomial interpolation theory developed for a node set known as Padua points. With appropriately defined polynomial spaces, we will show that the node points of non-degenerate Lissajous curves allow unique interpolation and can be used for quadrature rules in the bivariate setting. An explicit formula for the Lagrange polynomials allows to compute the interpolating polynomial with a simple algorithmic scheme. Compared to the already established schemes of the Padua and Xu points, the numerical results for the proposed scheme show similar approximation errors and a similar growth of the Lebesgue constant.

Numerical Investigation of the Effect of Unsteadiness on Three-Dimensional Flow of an Oldroyb-B Fluid.

A spectral relaxation method used with bivariate Lagrange interpolation is used to find numerical solutions for the unsteady three-dimensional flow problem of an Oldroyd-B fluid with variable thermal conductivity and heat generation. The problem is governed by a set of three highly coupled nonlinear partial differential equations. The method, originally used for solutions of systems of ordinary differential equations is extended to solutions of systems of nonlinear partial differential equations. The modified approach involves seeking solutions that are expressed as bivariate Lagrange interpolating polynomials and applying pseudo-spectral collocation in both independent variables of the governing PDEs. Numerical simulations were carried out to generate results for some of the important flow properties such as the local skin friction and the heat transfer rate. Numerical analysis of the error and convergence properties of the method are also discussed. One of the benefits of the proposed method is that it is computationally fast and gives very accurate results after only a few iterations using very few grid points in the numerical discretization process.

The quality control method for remanufacturing assembly based on the Jacobian-torsor model

To improve the assembly accuracy of remanufactured parts with multiple heterogeneity and properties, and to guarantee the quality of remanufacturing product no worse than the original one, this paper takes the air tightness of remanufactured engines’ cylinder block and head as the research target and proposes a quality control method based on the Jacobian-torsor model for remanufacturing assembly. First of all, three different Jacobian-torsor models for assembly tolerance are structured, and according to that, the defined torsor difference (∆FR) of remanufacturing products is calculated by considering the geometric error of remanufactured related parts assembly influenced by the temperature field and force field. Then, the optimal tightening torque of cylinder heads’ main bolts corresponding to ∆FR is achieved by applying the bivariate Lagrange interpolation method. Finally, taking the assembly of remanufactured engines’ cylinder block and head as a case study to verify the proposed methods is feasible and effective.

Bivariate uniform approximation via bivariate Lagrange interpolation polynomials

In the present note, we extend some univariate uniform approximation results by means of Lagrange interpolating polynomials [Ivan, M., Elements of Interpolation Theory, Mediamira Science Publisher, Cluj-Napoca (2004)] to the bivariate case. It is well known that generally, in the univariate case, the sequence of Lagrange interpolation polynomials does’t converges to the approximated function. This fact was first observed by G. Faber (see [9]), which constructed an example when the sequence of Lagrange interpolation polynomials diverges. The result of G. Faber was more generalized by I. Muntean (see [12]). M. Ivan established first sufficient conditions for the uniform convergence of the sequence of Lagrange interpolation polynomials associated to a univariate real valued function. First, we represent the remainder term of bivariate Lagrange interpolation formula in terms of bivariate divided difference. Using this representation we establish sufficient conditions for the uniform convergence of the sequence of bivariate Lagrange interpolation polynomials to the approximated function.

A bivariate Chebyshev spectral collocation quasilinearization method for nonlinear evolution parabolic equations.

This paper presents a new method for solving higher order nonlinear evolution partial differential equations (NPDEs). The method combines quasilinearisation, the Chebyshev spectral collocation method, and bivariate Lagrange interpolation. In this paper, we use the method to solve several nonlinear evolution equations, such as the modified KdV-Burgers equation, highly nonlinear modified KdV equation, Fisher's equation, Burgers-Fisher equation, Burgers-Huxley equation, and the Fitzhugh-Nagumo equation. The results are compared with known exact analytical solutions from literature to confirm accuracy, convergence, and effectiveness of the method. There is congruence between the numerical results and the exact solutions to a high order of accuracy. Tables were generated to present the order of accuracy of the method; convergence graphs to verify convergence of the method and error graphs are presented to show the excellent agreement between the results from this study and the known results from literature.

On an Interpolation Based Spectral Homotopy Analysis Method for PDE Based Unsteady Boundary Layer Flows

This work presents a new approach to the application of the spectral homotopy analysis method (SHAM) in solving non-linear partial differential equations (PDEs). The proposed approach is based on an innovative idea of seeking solutions that obey a rule of solution expression that is defined in terms of bivariate Lagrange interpolation polynomials. The applicability and effectiveness of the expanded SHAM approach are tested on a non-linear PDE that models the problem of unsteady boundary layer flow caused by an impulsively stretching plate. Numerical simulations are conducted to generate results for the important flow properties such as the local skin friction. The accuracy of the present results is validated against existing results from the literature and against results generated using the Keller-box method. The preliminary results from the proposed study indicate that the present method is more accurate and computationally efficient than more traditional methods used for solving PDEs that describe nonsimilar boundary layer flow.

On the Bivariate Spectral Homotopy Analysis Method Approach for Solving Nonlinear Evolution Partial Differential Equations

This paper presents a new application of the homotopy analysis method (HAM) for solving evolution equations described in terms of nonlinear partial differential equations (PDEs). The new approach, termed bivariate spectral homotopy analysis method (BISHAM), is based on the use of bivariate Lagrange interpolation in the so-called rule of solution expression of the HAM algorithm. The applicability of the new approach has been demonstrated by application on several examples of nonlinear evolution PDEs, namely, Fisher’s, Burgers-Fisher’s, Burger-Huxley’s, and Fitzhugh-Nagumo’s equations. Comparison with known exact results from literature has been used to confirm accuracy and effectiveness of the proposed method.

Bivariate Lagrange interpolation at the Chebyshev nodes

We discuss Lagrange interpolation on two sets of nodes in two dimensions where the coordinates of the nodes are Chebyshev points having either the same or opposite parity. We use a formula of Xu for Lagrange polynomials to obtain a general interpolation theorem for bivariate polynomials at either set of Chebyshev nodes. An extra term must be added to the interpolation formula to handle all polynomials with the same degree as the Lagrange polynomials. We express this term as a specifically determined linear combination of canonical polynomials that vanish on the set of Chebyshev nodes being considered. As an application we deduce in an elementary way known minimal and near minimal cubature formulas applying to both the even and the odd Chebyshev nodes. Finally, we restrict to triangular subsets of the Chebyshev nodes to show unisolvence and deduce a Lagrange interpolation formula for bivariate symmetric and skew-symmetric polynomials. This result leads to another proof of the interpolation formula.

Padua2DM: fast interpolation and cubature at the Padua points in Matlab/Octave

We have implemented in Matlab/Octave two fast algorithms for bivariate Lagrange interpolation at the so-called Padua points on rectangles, and the corresponding versions for algebraic cubature.

A Note on the Bivariate Lagrange Interpolation Polynomials

The focus of the research is to obtain reccurence formulas for the bivariate Lagrange interpolation polynomials, similar to the reccurence formulas verified by the univariate Lagrange interpolation polynomials.

Exact order of the Lebesgue constants for bivariate Lagrange interpolation at certain node-systems

The authors investigate two sets of nodes for bivariate Lagrange interpolation in [−1, 1] 2 and prove that the order of the corresponding Lebesgue constants is (log n ) 2 .

Multipartite Secret Sharing by Bivariate Interpolation

Given a set of participants that is partitioned into distinct compartments, a multipartite access structure is an access structure that does not distinguish between participants belonging to the same compartment. We examine here three types of such access structures: two that were studied before, compartmented access structures and hierarchical threshold access structures, and a new type of compartmented access structures that we present herein. We design ideal perfect secret sharing schemes for these types of access structures that are based on bivariate interpolation. The secret sharing schemes for the two types of compartmented access structures are based on bivariate Lagrange interpolation with data on parallel lines. The secret sharing scheme for the hierarchical threshold access structures is based on bivariate Lagrange interpolation with data on lines in general position. The main novelty of this paper is the introduction of bivariate Lagrange interpolation and its potential power in designing schemes for multipartite settings, as different compartments may be associated with different lines or curves in the plane. In particular, we show that the introduction of a second dimension may create the same hierarchical effect as polynomial derivatives and Birkhoff interpolation were shown to do in Tassa (J. Cryptol. 20:237–264, 2007).