Hermite Method Research Articles

Hermite collocation method for fractional order differential equations

This paper focuses on the approximate solutions of the higher order fractional differential equations with multi terms by the help of Hermite Collocation method (HCM). This new method is an adaptation of Taylor's collocation method in terms of truncated Hermite Series. With this method, the differential equation is transformed into an algebraic equation and the unknowns of the equation are the coefficients of the Hermite series solution of the problem. This method appears as a useful tool for solving fractional differential equations with variable coefficients. To show the pertinent feature of the proposed method, we test the accuracy of the method with some illustrative examples and check the error bounds for numerical calculations.

Zero-Coupon and Forward Yield Curves for Government of Ghana Bonds

This article seeks to provide a framework for modeling daily zero-coupon yield curve for Government of Ghana bonds based on secondary market daily trades. It also proposes method for modeling the forward yield curve. The current practice in Ghana is to produce yield curve for Government of Ghana bonds based on primary market weekly auctions. This article demonstrates the extraction and fitting of secondary market daily yield curves for Government of Ghana bonds, using bootstrapping and piecewise cubic hermite interpolation. The article also compares the piecewise cubic hermite method with the piecewise cubic spline method, the Nelson–Siegel–Svensson model, and the penalized smoothing spline method. Data used are the daily bond price data from the Ghana Fixed Income Market, accessible at the Central Securities Depository of Ghana. The results show yield curves that reflect the actual daily yield movements in the secondary bond market of Ghana. The results also show that the piecewise cubic hermite method fits the zero-coupon yield curve better than the other methods as far as the Ghanaian bond market is concerned. For the forward curves, we recommend that either or both of the piecewise cubic hermite method and the Nelson–Siegel–Svensson method could be used by the market participants.

A combined time-resolved infrared and density functional theory study of the lowest excited states of 9-fluorenone and 2-naphthaldehyde

A combined experimental and theoretical study of the infrared (IR) spectra of 2-naphthaldehyde and 9-fluorenone in their ground and first excited singlet and triplet electronic states is presented. IR studies have also been carried out using supercritical krypton (scKr) as a solvent to measure spectra in the ground and triplet excited states. This solvent provides a weakly interacting environment that is closer to the gas phase and allows a direct comparison with the calculated spectra for a single molecule. The IR spectra for the three different states are computed with Kohn-Sham density functional theory. For the first excited singlet state it is necessary to use an overlap procedure that allows the excited state to be studied by preventing variational collapse to the ground state. This allows the excited singlet state to be studied in an analogous approach to the ground and excited triplet state, in contrast to using time-dependent density functional theory. The good agreement between the calculated excited state spectra and the experiment provides insight into the nature of the excited states. For the ground and excited triplet state the anharmonic vibrational frequencies are computed using the transition optimized shifted Hermite method, and for these molecules the hybrid B97-1 functional is found to provide the closest agreement with experiment in the ground state.

A Comparative Study of Ten Different Methods on Numerical Solving of Point Reactor Neutron Kinetics Equations

Point reactor neutron kinetics equations describe the time-dependent neutron density variation in a nuclear reactor core. These equations are widely applied to nuclear system numerical simulation and nuclear power plant operational control. This paper analyzes the characteristics of ten different basic or normal methods to solve the point reactor neutron kinetics equations. The accuracy after introducing different kinds of reactivity, stiffness of methods, and computational efficiency are analyzed. The calculation results show that: considering both the accuracy and stiffness, implicit Runge–Kutta method and Hermite method are more suitable for solution on these given conditions. The explicit Euler method is the fastest, while the power series method spends the most computational time.

Hermite collocation method for solving Hammerstein integral equations

In this paper, we are presenting Hermite collocation method to solve numer- ically the Fredholm-Volterra-Hammerstein integral equations. We have clearly presented a theory to …nd ordinary derivatives. This method is based on replace- ment of the unknown function by truncated series of well known Hermite expan-sion of functions. The proposed method converts the equation to matrix equation which corresponding to system of algebraic equations with Hermite coe¢ cients. Thus, by solving the matrix equation, Hermite coe¢ cients are obtained. Some numerical examples are included to demonstrate the validity and applicability of the proposed technique.

Hermite Methods for the Scalar Wave Equation

Arbitrary order dissipative and conservative Hermite methods for the scalar wave equation are presented. Both methods use $(m+1)^d$ degrees of freedom per node for the displacement in $d$ dimension...

Flux-Conservative Hermite Methods for Simulation of Nonlinear Conservation Laws

A new class of Hermite methods for solving nonlinear conservation laws is presented. While preserving the high order spatial accuracy for smooth solutions in the existing Hermite methods, the new methods come with better stability properties. Artificial viscosity in the form of the entropy viscosity method is added to capture shocks.

A joint model for mixed and truncated longitudinal data and survival data, with application to HIV vaccine studies.

In HIV vaccine studies, a major research objective is to identify immune response biomarkers measured longitudinally that may be associated with risk of HIV infection. This objective can be assessed via joint modeling of longitudinal and survival data. Joint models for HIV vaccine data are complicated by the following issues: (i) left truncations of some longitudinal data due to lower limits of quantification; (ii) mixed types of longitudinal variables; (iii) measurement errors and missing values in longitudinal measurements; (iv) computational challenges associated with likelihood inference. In this article, we propose a joint model of complex longitudinal and survival data and a computationally efficient method for approximate likelihood inference to address the foregoing issues simultaneously. In particular, our model does not make unverifiable distributional assumptions for truncated values, which is different from methods commonly used in the literature. The parameters are estimated based on the h-likelihood method, which is computationally efficient and offers approximate likelihood inference. Moreover, we propose a new approach to estimate the standard errors of the h-likelihood based parameter estimates by using an adaptive Gauss-Hermite method. Simulation studies show that our methods perform well and are computationally efficient. A comprehensive data analysis is also presented.

Hermite analogs of the lowest order Raviart–Thomas mixed method for convection–diffusion equations

The Raviart–Thomas mixed finite-element method of the lowest order (Raviart and Thomas in mixed finite-element methods for second-order elliptic problems, Lecture Notes in mathematics, Springer, New York, 1977) commonly known as the $$RT_0$$ method, is a well-established and popular numerical tool to solve diffusion-like problems providing flux continuity across inter-element boundaries. Douglas and Roberts extended the method to the case of more general second-order boundary-value problems including the convection–diffusion equations (cf. this journal Douglas in Comput Appl Math 1:91–103; 1982). The main drawback of these methods, however, is the poor representation of the primal variable by piecewise constant functions. The Hermite analog of the $$RT_0$$ method for treating pure diffusion phenomena proposed in Ruas (J Comput Appl Math, 246:234–242; 2013) proved to be a valid alternative to attain higher order approximation of the primal variable while keeping intact the matrix structure and the quality of the discrete flux variable of the original $$RT_0$$ method. Non-trivial extensions of this method are studied here that can be viewed as Hermite analogs of the two Douglas and Roberts’ versions of the $$RT_0$$ method, to solve convection–diffusion equations. A detailed convergence study is carried out for one of the Hermite methods, and numerical results illustrate the performance of both of them, as compared to each other and to the corresponding mixed methods.

Variations on Hermite Methods for Wave Propagation

AbstractHermite methods, as introduced by Goodrich et al. in [15], combine Hermite interpolation and staggered (dual) grids to produce stable high order accurate schemes for the solution of hyperbolic PDEs. We introduce three variations of this Hermite method which do not involve time evolution on dual grids. Computational evidence is presented regarding stability, high order convergence, and dispersion/dissipation properties for each new method. Hermite methods may also be coupled to discontinuous Galerkin (DG) methods for additional geometric flexibility [4]. An example illustrates the simplification of this coupling for Hermite methods.

Hermite pesudospectral method and modified Hermite spectral method for long-short wave equations

Hermite pesudospectral method and modified Hermite spectral method for long-short wave equations

Hermite Interpolation with Triangular Fuzzy Numbers

In this paper the polynomial interpolation of triangular fuzzy number is discussed. First general form of the polynomial with fuzzy coefficients is proposed. The hermite interpolation method is studied with triangular fuzzy number an example is provided to illustrate the algorithm.

Efficient interpolation of precomputed kinetic data employing reduced multivariate Hermite Splines

Abstract Detailed surface kinetics can be efficiently implemented into complex reactor simulations by the use of precomputed solutions of the chemical rate equations. In this work we propose reduced Hermite splines as an interpolation method for the precomputed rate data. The reduced Hermite splines require significantly less storage space and less execution time than the tensor product splines that have been previously used for this task. Using previously published test cases on NH 3 oxidation, we demonstrate that despite of the reduced storage requirements and faster interpolation times, reduced Hermite interpolation achieves the same interpolation accuracy as conventional full tensor product splines. In many cases, the derivatives of the interpolated function with respect to the input quantities can be cheaply obtained during the computation of the function values. If these derivatives are provided as additional information, the accuracy of the reduced Hermite method even outperforms conventional tensor product splines, in our demonstration example by a factor of three. The advantage of the reduced Hermite schemes with respect to storage requirements and evaluation time strongly increase with the dimensionality of the interpolation problem. It is therefore expected that the reduced Hermite splines will allow to significantly extend the application range of solution mapping methods to higher dimensional problems (i.e. problems with a larger number of relevant chemical species).

Highly Accurate Pseudospectral Approximations of the Prolate Spheroidal Wave Equation for Any Bandwidth Parameter and Zonal Wavenumber

The prolate spheroidal wave equation (PSWE) is transformed, using suitable mappings, into three different canonical forms which resemble the Jacobi, Laguerre and the Hermite differential equations. The eigenpairs of the PSWE are approximated with the corresponding classical orthogonal polynomial as a basis set. It is observed that for any zonal wavenumber m the Jacobi type pseudospectral methods are well suited for small bandwidth parameters c whereas the Hermite and Laguerre pseudospectral methods are appropriate for very large c values. Moreover, Jacobi pseudospectral methods work well for any parameter values such that $$m\ge c$$mźc. Our numerical results confirm that for any values of m, the Jacobi $$\left[ (\alpha ,\beta )=(\pm 1/2,m)\right] $$(ź,β)=(±1/2,m) and the Laguerre $$({\upgamma }=\pm 1/2)$$(ź=±1/2) pseudospectral methods formulated in this article for the numerical solution of the PSWE with small and very large bandwidth parameters, respectively, are highly efficient both from the accuracy and fastness point of view.

Hermite interpolation method (HiM): compact surface potential MOSFET model based on the Hermite polynomial

An extremely accurate yet simple form of the charge-sheet model (CSM) is developed using the third-order Hermite interpolation polynomial to model the inversion charge in the channel. This new formulation of the drain current retains the same simplicity of the most advanced surface potential compact MOSFET models based on the symmetric linearisation method (SLM). However, unlike the SLM, it is developed without requiring the crude linearisation of the inversion charge, hence the asymmetry and the non-linearity in the channel are accurately accounted for. Nevertheless, the expression of the drain current can be worked out to be analytically equivalent to SLM. Its mathematic formulation is even easier with respect to SLM since it does not require the evaluation of the inversion charge at the surface potential midpoint, but at source and drain only, as the conventional CSM.

Likelihood analysis for a class of spatial geostatistical compositional models

We propose a model-based geostatistical approach to deal with regionalized compositions. We combine the additive-log-ratio transformation with multivariate geostatistical models whose covariance matrix is adapted to take into account the correlation induced by the compositional structure. Such specification allows the usage of standard likelihood methods for parameters estimation. For spatial prediction we combined a back-transformation with the Gauss–Hermite method to approximate the conditional expectation of the compositions. We analyse particle size fractions of the top layer of a soil for agronomic purposes which are typically expressed as proportions of sand, clay and silt. Additionally a simulation study assesses the small sample properties of the maximum likelihood estimator.

High-order central Hermite WENO schemes: Dimension-by-dimension moment-based reconstructions

In this paper, a class of high-order central finite volume schemes is proposed for solving one- and two-dimensional hyperbolic conservation laws. Formulated on staggered meshes, the methods involve Hermite WENO (HWENO) spatial reconstructions, and Lax–Wendroff type discretizations or the natural continuous extension of Runge–Kutta methods in time. Differently from the central Hermite WENO methods we developed previously in Tao et al. (2015) [34] , the spatial reconstructions, a core ingredient of the methods, are based on the zeroth-order and the first-order moments of the solution, and are implemented through a dimension-by-dimension strategy when the spatial dimension is higher than one. This leads to much simpler implementation of the methods in higher dimension and better cost efficiency. Meanwhile, the proposed methods have the attractive features of the general central Hermite WENO methods such as being compact in reconstruction and requiring neither flux splitting nor numerical fluxes, while being accurate and essentially non-oscillatory. A collection of one- and two-dimensional numerical examples is presented to demonstrate high resolution and robustness of the methods in capturing smooth and non-smooth solutions.

Hermite type radial basis function-based differential quadrature method for higher order equations

In the paper, the radial basis function-based differential quadrature method (RBF-DQM) that uses Hermite type interpolation is developed. The method is an extension of the known RBF-DQM, which is a meshless numerical technique for solving differential equations. According to this technique, derivatives in a governing equation are approximated by a linear weighted sum of the sought function values defined at scattered nodes. To allow the method to be applied in higher order equations, where more than one boundary condition is imposed at an edge, Hermite type interpolation for the radial basis functions is used and appropriate weighting coefficients for differential quadrature method are determined in the paper. As a numerical test the method is used to discretize the governing equation for the free vibration of thin plates with various boundary conditions. Different shaped plates with various boundary conditions are analyzed. The convergence tests carried out in the work confirm usefulness of the method as a truly meshless technique.

Multi-dimensional, fully-implicit, spectral method for the Vlasov–Maxwell equations with exact conservation laws in discrete form

A spectral method for the numerical solution of the multi-dimensional Vlasov–Maxwell equations is presented. The plasma distribution function is expanded in Fourier (for the spatial part) and Hermite (for the velocity part) basis functions, leading to a truncated system of ordinary differential equations for the expansion coefficients (moments) that is discretized with an implicit, second order accurate Crank–Nicolson time discretization. The discrete non-linear system is solved with a preconditioned Jacobian–Free Newton–Krylov method.It is shown analytically that the Fourier–Hermite method features exact conservation laws for total mass, momentum and energy in discrete form. Standard tests involving plasma waves and the whistler instability confirm the validity of the conservation laws numerically. The whistler instability test also shows that we can step over the fastest time scale in the system without incurring in numerical instabilities. Some preconditioning strategies are presented, showing that the number of linear iterations of the Krylov solver can be drastically reduced and a significant gain in performance can be obtained.

Symplectic integration for the collisional gravitationalN-body problem

We present a new symplectic integrator designed for collisional gravitational $N$-body problems which makes use of Kepler solvers. The integrator is also reversible and conserves 9 integrals of motion of the $N$-body problem to machine precision. The integrator is second order, but the order can easily be increased by the method of \citeauthor{yos90}. We use fixed time step in all tests studied in this paper to ensure preservation of symplecticity. We study small $N$ collisional problems and perform comparisons with typically used integrators. In particular, we find comparable or better performance when compared to the 4th order Hermite method and much better performance than adaptive time step symplectic integrators introduced previously. We find better performance compared to SAKURA, a non-symplectic, non-time-reversible integrator based on a different two-body decomposition of the $N$-body problem. The integrator is a promising tool in collisional gravitational dynamics.