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The purpose of this study is to establish recent inequalities based on double integrals of mappings whose higher-order partial derivatives in absolute value are convex on the co-ordinates on rectangle from the plane. Also, some special cases of results improved in this study are examined. [ABSTRACT FROM AUTHOR]

In this paper, we obtain the explicit solution of a class of singular impulsive fractional differential equation of order α ∈ (n − 1 , n) involving the Hadamard fractional derivatives. In particular, the result is given in the form of necessary and sufficient conditions about the explicit solution. The proof of the main theorem is obtained by converting boundary value problems into integral equations and applying analytical techniques. Finally, some examples explain the application of the main theorem in this paper. [ABSTRACT FROM AUTHOR]

The m-point non-local problem is considered for the partial differential equation of mixed-type with singular coefficients, namely fractional wave equation involving the right-hand side bi-ordinal Hilfer derivative and sub-diffusion equation with the regularized Caputo-like counterpart hyper-Bessel differential operator. The main technique of solving the problem is based on the method of separation variables. Also, the connection between the given data and the uniquely solvability of the problem is established and an explicit solution is represented by Fourier-Bessel series in the given domain. [ABSTRACT FROM AUTHOR]

This paper considers the problem of the semi-global asymptotic stabilisation of fractional-order (FO) linear systems subject to actuator saturation by output feedback. To solve the problem, a family of observer-based linear output feedback laws, parameterised in a positive scalar, is proposed by means of the low gain feedback design technique. The design applies to FO linear systems that are stabilisable and detectable, but not exponentially unstable. For such an FO system under the proposed observer-based linear low gain output feedback, the peak value of the control input for a given initial condition can be made arbitrarily small to avoid actuator saturation by decreasing the value of the parameter towards zero and thus semi-global asymptotic stabilisation is achieved. To obtain these results, we establish the properties of low gain feedback, derive asymptotic expansions and the bounds of high-order derivatives of the Mittag-Leffler (ML) functions to estimate the state responses of FO linear systems, and explicitly construct a Hermitian matrix to satisfy a linear matrix inequality (LMI) stability condition for FO linear systems. The results in this paper extend the corresponding results for integer-order (IO) linear systems. [ABSTRACT FROM AUTHOR]

The surface correction to the quadrupole source term of the Ffowcs Williams and Hawkings integral in the frequency domain suffers from the computation of high-order derivatives of Green's function. The far-field approximations to the derivatives of Green's function have been used without derivation and verification in previous work. In this work, we provide the detailed derivations of the far-field approximations to the derivatives of Green's function. The binomial expansions for the derivatives of Green's function and the far-field condition are employed during the derivations to circumvent the difficulties in computing the high-order derivatives. The approximations to the derivatives of Green's function are systemically verified by using the benchmarks two -dimensional convecting vortex and the co-rotating vortex pair. In addition, we provide the derivations of the approximations to the multiple integrals of Green's function by using the far-field approximations to the derivatives. [ABSTRACT FROM AUTHOR]

There are a good number of higher-order iterative methods for computing multiple zeros of nonlinear equations in the available literature. Most of them required first or higher-order derivatives of the involved function. No doubt, high-order derivative-free methods for multiple zeros are more difficult to obtain in comparison with simple zeros and with first order derivatives. This study presents an optimal family of fourth order derivative-free techniques for multiple zeros that requires just three evaluations of function ϕ , per iteration. The approximations of the derivative/s are based on symmetric divided differences. We also demonstrate the application of new algorithms on Van der Waals, Planck law radiation, Manning for isentropic supersonic flow and complex root problems. Numerical results reveal that the proposed derivative-free techniques are more efficient in comparison terms of CPU, residual error, computational order of convergence, number of iterations and the difference between two consecutive iterations with other existing methods. [ABSTRACT FROM AUTHOR]

In this paper, high-order compact-difference schemes involving a large number of mesh points in the computational stencils are used to numerically solve partial differential equations containing high-order derivatives. The test cases include a linear dispersive wave equation, the non-linear Korteweg–de Vries (KdV)-like equations, and the non-linear Kuramoto–Sivashinsky equation with known analytical solutions. It is shown that very high-order compact schemes, e.g., of 20th or 24th orders, cause a very fast drop in the L 2 norm error, which in some cases reaches a machine precision already on relatively coarse computational meshes. [ABSTRACT FROM AUTHOR]

The modified numerical inversion of the Laplace transform (dubbed NILT) has been recently proposed as a fast and provably stable numerical simulation for general circuits. Although it has enabled increasing the simulation time step, it highlighted the need for robust approach that can recover the full waveform in between the time points generated by NILT. This article presents a new approach that addresses this challenge. The proposed approach leverages the NILT framework from a high-order approximation paradigm that computes points on the circuit waveforms to a methodology that computes high-order derivatives of the waveforms at the same points. Using those derivatives, an interpolation approach based on Hermite interpolation is used to construct the circuit waveforms on dense points. Numerical experiments are presented to demonstrate the accuracy of both approaches. [ABSTRACT FROM AUTHOR]

The existence of the solutions with prescribed Lp-norm for a fractional p-Laplacian equation is investigated in this paper. The obtained result is suitable for all the order of the derivative 0 < s < 1 and p > 1, which extends the previous results for s = 1 or p = 2. In particular, to the best of our knowledge, as the Lp-subcritical or Lp-critical constrained minimization problem for fractional p-Laplacian equation, the critical exponent (pN + p²s)/N is properly established for the first time. On one hand, using Lions Vanishing Lemma and Brézis-Lieb Lemma, the compactness of minimizing sequences for the related constrained minimization problem is derived, then based on which the existence of constrained minimizers is achieved. On the other hand, the existence of weak solution and the nonexistence result are also provided. [ABSTRACT FROM AUTHOR]

We present the studies of the high-order harmonic generation of 802-nm, 64-fs pulses in the extended plasmas contained the carbon-based clusters (fullerenes, nanofibers, nanoparticles, and nanotubes). Our studies show that the 11th-19th harmonics in the 15-33 eV range may originate from the low-sized nanoparticles produced during ablation of large clusters and agglomerates. The conversion efficiency of the 11th harmonic generated from the 5-mm-long fullerene plasma was estimated to be 2 x 10-5. [ABSTRACT FROM AUTHOR]

In this paper, we consider the numerical solution of the time-dependent wave equation in a semi-infinite waveguide. We employ the Double Absorbing Boundary (DAB) method, by introducing two parallel artificial boundaries on the side where waves are outgoing. In contrast to the original implementation of the DAB, where the numerical solution involved either a low-order finite difference scheme or a low-order finite element scheme, here we incorporate the DAB into a high-order spectral element formulation, which provides us with very accurate solutions of wave problems in unbounded domains. This is demonstrated by numerical experiments. While the method is highly accurate, it suffers from long-time instability. We show how to postpone the onset of the instability by a prudent choice of the computational parameters. [ABSTRACT FROM AUTHOR]

In this paper, weight function and composition technique is utilized to speeds up the convergence order and increase the efficiency of an existing quadrature based iterative method. This results in the proposition of its improved form from a two-point quadrature based method of convergence order ρ = 3 with efficiency index EI = 1:3161 to a three-point method of convergence order ρ = 8 with EI = 1:5157 at the cost of one additional function evaluation. The method is used to approximate the solution of some nonlinear equations and the results generated are compared with that of some existing methods. Numerical results shows that method developed herein is very efficient in approximation of solution of nonlinear equations. [ABSTRACT FROM AUTHOR]

A class of high-order nonlinear filter schemes by Yee et al. (J Comput Phys 150:199–238, 1999), Sjögreen and Yee (J Comput Phys 225:910–934, 2007), and Kotov et al. (Commun Comput Phys 19:273–300, 2016; J Comput Phys 307:189–202, 2016) is examined for long-time integrations of computational aeroacoustics (CAA) turbulence applications. This class of schemes was designed for an improved nonlinear stability and accuracy for long-time integration of compressible direct numerical simulation and large eddy simulation computations for both shock-free turbulence and turbulence with shocks. They are based on the skew-symmetric splitting version of the high-order central base scheme in conjunction with adaptive low-dissipation control via a nonlinear filter step to help with stability and accuracy capturing at shock-free regions as well as in the vicinity of discontinuities. The central dispersion-relation-preserving schemes as well as classical central schemes of arbitrary orders fit into the framework of skew-symmetric splitting of the inviscid flux derivatives. Numerical experiments on CAA turbulence test cases are validated. [ABSTRACT FROM AUTHOR]

Differentiation arithmetic is a principal and accurate technique for the computational evaluation of derivatives of first and higher order. This article aims at recasting real differentiation arithmetic in a formalized theory of dyadic real differentiation numbers that provides a foundation for first and higher order automatic derivatives. After we set the stage by putting on a systematic basis certain fundamental notions of the algebra of differentiation numbers, we begin by setting up an axiomatic theory of real differentiation arithmetic, as a many-sorted extension of the theory of a continuously ordered field, and then establish the proofs for its consistency and categoricity. Next, we carefully construct the algebraic system of real differentiation arithmetic, deduce its fundamental properties, and prove that it constitutes a commutative unital ring. Furthermore, we describe briefly the extensionality of the system to an interval differentiation arithmetic and to an algebraically closed commutative ring of complex differentiation arithmetic. Finally, a word is said on machine realization of real differentiation arithmetic and its correctness, with an addendum on how to compute automatic derivatives of first and higher order. [ABSTRACT FROM AUTHOR]

Abstract There are a very small number of high quality derivative free methods with memory for solving a system of nonlinear equations numerically. Motivated and inspired by the fact, we propose a more efficient general class of Steffensen type multipoint methods with memory. This proposed class requires one divided difference and inverse of only one matrix per full iteration. We also demonstrate their applicability and illustrate that these methods produce approximations of greater accuracy and remarkably reduce the computational time for solving systems of nonlinear equations numerically. For quantitative comparison, we have also computed total number of convergent points and convergent percentage along with basins of attractions on a number of test problems to recommend the best quality algorithm. [ABSTRACT FROM AUTHOR]

The paper reconsiders the issue of the regularity of the Duhamel part of the solution to the L2-critical high-order NLS already studied by the authors in [4]. This model includes the mass critical 4D fourth order NLS. The improvement is due to the use of a more sophisticated space involving a nonlinear term and taking profit of a suitable trilinear Strichartz estimate. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we are concerned with the construction of a new high-order Absorbing Boundary Condition (ABC) for 2D-elastic scattering problems. It is defined by an approximate local Dirichlet-to-Neumann (DtN) map. First, we explain the derivation of this approximation. Next, a detailed analytical study in terms of Hankel functions in the circular case is addressed. The new ABC is compared with the standard low-order Lysmer–Kuhlemeyer ABC. Finally, its accuracy and efficiency are investigated for various numerical examples, particularly at high frequencies. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we study the distributed input-constrained consensus problem for continuous-time high-order multi-agent systems with time-delays. Each agent's control input is constrained to lie in a nonconvex set. A distributed control algorithm is proposed to achieve consensus of the agents. Finally, we illustrate the theoretical result by a numerical example. [ABSTRACT FROM AUTHOR]

Abstract In this paper, by using partial derivative formulas of generating functions for the multidimensional unification of the Bernstein basis functions and their functional equations, we derive derivative formulas and identities for these basis functions and their generating functions. We also give a conjecture and some open questions related to not only subdivision property of these basis functions, but also solutions of a higher-order special differential equations. Moreover, we provide an implementation for a real world problem of human facial expression recognition with the help of curvature of Bezier curves whose machine learning supported by statistical evaluations on feature vectors using in the aforementioned machine learning algorithm. [ABSTRACT FROM AUTHOR]

Abstract A three-dimensional high-order panel method based on Non-Uniform Rational B-Spline (NURBS) for predicting the ship–ship hydrodynamic interaction during meeting and overtaking in shallow water is developed. The NURBS surface is used to precisely represent the hull geometry. The velocity potential on the body surface is described by B-spline after the source density distribution on the boundary surface is determined. A collocation approach is applied to the boundary integral equation discretization, and the velocity potential is being solved at each time step. Under the low-speed assumption, the effect of free surface elevation is neglected in the numerical calculation, and the infinite image method is used to deal with the finite water depth effect. Two tankers in model tests are investigated, and the predicted hydrodynamic interaction forces and moments are compared with experimental measurements to clarify the validity of the proposed numerical method. Calculations are then conducted for different water depths, lateral ship–ship distances and ship speeds, and the detailed results are discussed to demonstrate the effects of these factors. Highlights • Ship–ship interaction in shallow water is investigated by a high-order panel method. • Interaction hydrodynamic forces during meeting and overtaking are predicted. • The method is validated by comparing the numerical results with experimental data. • Computations are conducted at different water depth, ship–ship distance and ship speed. • The influences of water depth, ship–ship distance and ship speed are demonstrated. [ABSTRACT FROM AUTHOR]

In this paper, we focus on developing locally conservative high order finite difference methods with provable total variation stability for solving one-dimensional scalar conservation laws. We introduce a new criterion for designing high order finite difference schemes with provable total variation stability by measuring the total variation of an expanded vector. This expanded vector is created from grid values at tn+1 and tn with ordering determined by upwinding information. Achievable local bounds for grid values at tn+1 are obtained to provide a sufficient condition for the total variation of the expanded vector not to be greater than total variation of the initial data. We apply the Flux-Corrected Transport type of bound preserving flux limiters to ensure that numerical values at tn+1 are within these local bounds. When compared with traditional total variation bounded high order methods, the new method does not depend on mesh-related parameters. Numerical results are produced to demonstrate: the total variation of the numerical solution is always bounded; the order of accuracy is not sacrificed. When the total variation bounded flux limiting method is applied to a third order finite difference scheme, we show that the third order of accuracy is maintained from the local truncation error point of view. [ABSTRACT FROM AUTHOR]

In this technical brief, we focus on solving trajectory optimization problems that have nonlinear system dynamics and that include high-order derivatives in the objective function. This type of problem comes up in robotics--for example, when computing minimum-snap reference trajectories for a quadrotor or computing minimum-jerk trajectories for a robot arm. DirCol5i is a transcription method that is specialized for solving this type of problem. It uses the fifth-order splines and analytic differentiation to compute higher-derivatives, rather than using a chain-integrator as would be required by traditional methods. We compare DirCol5i to traditional transcription methods. Although it is slower for some simple optimization problems, when solving problems with high-order derivatives DirCol5i is faster, more numerically robust, and does not require setting up a chain integrator. [ABSTRACT FROM AUTHOR]

Abstract Two families of order six for the solution of systems of nonlinear equations are developed and compared to existing schemes of order up to six. We have found that one of the methods in the literature has been rediscovered. The comparison is based on the total cost of an iteration and the performance on 14 examples of systems of dimensions 2–9. [ABSTRACT FROM AUTHOR]

A new family of three-stage two-step methods are presented in this paper. These methods are of algebraic order 12 and have an important P-stability property. To make these methods, vanishing phase-lag and some of its derivatives have been used. The main structure of these methods are multiderivative, and the combined phases have been applied for expanding stability interval and for achieving P-stability. The advantage of the new methods in comparison with similar methods, in terms of efficiency, accuracy, and stability, has been showed by the implementation of them in some important problems, including the radial time-independent Schrödinger equation during the resonance problems with the use of the Woods-Saxon potential, undamped Duffing equation, etc. [ABSTRACT FROM AUTHOR]

We propose a two-level implicit high order compact scheme for the one-dimensional unsteady biharmonic problem of first kind. The values of φ and are prescribed on the boundary. Using a combination of values of φ and at each grid point, the difference formula is derived for the unsteady biharmonic equation without discretizing the boundary conditions. The proposed method has second order time accuracy and fourth order space accuracy using just three grid points of a single compact stencil at every time level. The first order space derivative is also computed with same accuracy as a by-product of the method. Using the Von-Neumann analysis, the derived scheme is shown to be unconditionally stable. With a slight modification, the proposed method is applicable to solve singular problems. The performance of the proposed scheme is illustrated by numerical experiments done on a collection of test problems having physical significance including the nonlinear Kuramoto-Sivashinsky equation. [ABSTRACT FROM AUTHOR]

Our purpose in this paper is to investigate commutativity of a ring with involution (R , ∗) which admits a generalized derivation satisfying certain algebraic identities. Some well-known results characterizing commutativity of prime rings have been generalized. Moreover, we provide examples to show that the assumed restrictions cannot be relaxed. [ABSTRACT FROM AUTHOR]

Automatic differentiation (AD) is a technique for augmenting computer programs to compute derivatives. The essence of AD in its forward accumulation mode is to attach perturbations to each number, and propagate these through the computation by overloading the arithmetic operators. When derivatives are nested, the distinct derivative calculations, and their associated perturbations, must be distinguished. This is typically accomplished by creating a unique tag for each derivative calculation and tagging the perturbations. We exhibit a subtle bug, present in fielded implementations which support derivatives of higher-order functions, in which perturbations are confused despite the tagging machinery, leading to incorrect results. The essence of the bug is as follows: a unique tag is needed for each derivative calculation, but in existing implementations unique tags are created when taking the derivative of a function at a point. When taking derivatives of higher-order functions, these need not correspond! We exhibit a simple example: a higher-order function f whose derivative at a point x, namely f′(x), is itself a function which calculates a derivative. This situation arises naturally when taking derivatives of curried functions. Two potential solutions are presented, and their deficiencies discussed. One uses eta expansion to delay the creation of fresh tags in order to put them into one-to-one correspondence with derivative calculations. The other wraps outputs of derivative operators with tag substitution machinery. Both solutions seem very difficult to implement without violating the desirable complexity guarantees of forward AD. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we construct a family of high order compact symplectic (S-HOC) schemes for the Klein–Gordon–Schrödinger (KGS) equation. The KGS can be cast into a Hamiltonian form. At first, we discretize the Hamiltonian system in space by a high order compact method which has higher convergent rate than general finite difference methods. Then the semi-discretized system is approximated in time by the Euler midpoint scheme which preserves the symplectic structure of the original system. The conserved quantities of the scheme, including symplectic structure conservation law, charge conservation law and energy conservation law, are discussed. The local truncation error and global error of the numerical solvers are investigated. Finally, some numerical verifications are presented to numerically validate the theoretical analysis. The numerical results are persuasive and illustrate the theoretical analysis. [ABSTRACT FROM AUTHOR]

Abstract We propose a novel family of a posteriori sub-cell finite volume limiters for spatially high order accurate semi-implicit discontinuous Galerkin (DG) schemes on staggered Cartesian grids for the solution of the shallow water equations expressed in conservative form in one and two space dimensions. We start from the unlimited arbitrary high order accurate staggered semi-implicit DG scheme proposed by Dumbser and Casulli (2013). In this method, the continuity equation and the momentum equations are integrated using a discontinuous finite element strategy on staggered control volumes, where the discrete free surface elevation is defined on the main grid and the discrete momentum is defined on edge-based staggered dual control volumes. In the semi-implicit approach, pressure terms are discretized implicitly, while the nonlinear convective terms are discretized explicitly. Inserting the momentum equations into the discrete continuity equation leads to a well conditioned block penta-diagonal linear system for the free surface elevation which can be efficiently solved with modern iterative methods. However, according to Godunov's theorem, any unlimited high order scheme inevitably produces spurious oscillations in the vicinity of discontinuities and strong gradients. In the present paper, we therefore propose to extend the successful family of a posteriori subcell finite volume limiters recently introduced by Dumbser et al. (2014) for explicit DG schemes also to semi-implicit time discretizations. At time t n the unlimited DG scheme is run in order to produce a so-called candidate solution for time t n + 1. Then, the cells characterized by a non-admissible candidate solution are found by using physical and numerical detection criteria based on the positivity of the solution, the absence of floating point errors and the use of a relaxed discrete maximum principle (DMP) according to the MOOD strategy of Clain, Loubère and Diot (2013). In all the cells that are flagged as troubled control volumes a more robust semi-implicit finite volume (FV) method is then applied on a sub-grid composed of 2 P + 1 cells, where P denotes the polynomial degree used for approximating the discrete solution within the DG scheme. Then, after having identified the troubled cells, the linear system for the new free surface elevation is assembled and solved again, where unlimited cells use the high order semi-implicit DG scheme and limited cells are evolved via the more robust finite volume method. Finally, from the subcell finite volume averages a higher order DG polynomial is reconstructed and the scheme proceeds with the next time step. We apply the new semi-implicit staggered DG method with a posteriori subcell FV limiter to classical benchmarks such as Riemann problems in 1D and circular dam-break problems in 2D with shock waves, showing that the new subcell finite volume limiter is able to resolve shocks accurately without producing spurious oscillations. Moreover, if the solution is smooth, the detector does not find any troubled cells, as expected; consequently, the limiter is not activated and the method corresponds to the unlimited staggered semi-implicit DG scheme. In addition, we carry out numerical tests which show that the new scheme is well-balanced and able to deal with wet and dry fronts. [ABSTRACT FROM AUTHOR]

In this paper, some qualitative properties are discussed such as the boundedness, the periodicity and the global stability of the positive solutions of the nonlinear difference equation ..., where the coefficients A, αi, βi ∈ (0, ∞), i = 1, ..., 5, while the initial conditions y-5, y-4, y-3, y-2, y-1, y0 are arbitrary positive real numbers. Some numerical examples will be given to illustrate our results. [ABSTRACT FROM AUTHOR]

This paper is concerned with neutral type high-order cellular neural networks (HCNNs) involving proportional delays and D operators. Some criteria are established for the global exponential convergence of the addressed models by using differential inequality techniques. Moreover, an example and its numerical simulations are employed to illustrate the main results. [ABSTRACT FROM AUTHOR]

This paper discusses the mixture distribution-based data-driven robust chance constrained problem. We construct a data-driven mixture distribution-based uncertainty set from the perspective of simultaneously estimating higher-order moments. Then, we derive a reformulation of the data-driven robust chance constrained problem. As the reformulation is not a convex programming problem, we propose new and tight convex approximations based on the piecewise linear approximation method. We establish the theoretical foundation for these approximations. Finally, numerical results show that the proposed approximations are practical and efficient. [ABSTRACT FROM AUTHOR]

Highlights • Sparse representation produces the better prediction accuracy for SCIs. • A new blind scheme considering the relative gradient direction is designed. • HOG features as second-order derivatives represents the texture information well. Graphical abstract Abstract In this paper, an accurate blind metric evaluating screen content images (SCIs) considering texture feature via sparse representation is proposed. The existing theory on human vision tells us that human visual system (HVS) is sensitive to texture information of images. In addition, gradient direction has not been adequately explored as a predictive information source of SCIs. This method is first based on three kinds of gradient maps of SCIs—i.e., gradient magnitude map, relative gradient direction map, relative gradient magnitude map. Since the existing literatures about human neuroscience have revealed that image texture can be captured by high-order derivatives, texture information of SCIs can be represented by local histogram of oriented gradient (HOG) features extracted from the aforementioned gradient maps. Finally, it is composed of two important procedures: representing HOG features by means of sparse coding, then weighting subjective experimental quality scores via the sparse coding coefficients to predict objective perceptive quality scores of SCIs. Comparison experiments on the public SCI database demonstrate that the designed metric has the better superiority over existing the-state-of-art algorithms and delivers consistency and accuracy in relation with subjective evaluation. [ABSTRACT FROM AUTHOR]

Constructing mutants of order higher than first order is a key step in higher-order mutation testing. The majority of higher-order mutant generation techniques merge two (or more) first-order mutants (FOMs) to build a higher-order mutant. Unfortunately, these techniques suffer from the high cost due to the explosion in the number of higher-order mutants (HOMs). Consequently, developing techniques to find the minimum adequate and effective number of mutants are desired. Earlier work reduced the number of mutants by considering only a subset of mutants, a subset of operators, or selecting specific locations in the program (to be mutated) instead of the whole program. In this paper, we present three new techniques (SCWR, 2E2O, and 2E2OWR) to generate and reduce the overall number of HOMs and the equivalent ones as well. Each technique merges FOMs in more effective and simpler way than the previous techniques to find more effective HOMs. These techniques have been applied on a benchmark of programs, and the results have been compared to the results of some related work such as DiffOp, JudyDiffOp, and Last2First techniques. The results showed that SCWR, 2E2O, and 2E2OWR outperformed the related work and reduced the total number of mutants by 67.9, 16.0, and 60.1% comparing to approximately 50% for the related work and the number of equivalent mutants by 66.9, 79.6, and 65.0% comparing to 25.8 and 36.4% for the related work, respectively. [ABSTRACT FROM AUTHOR]

We put forward an algorithm for the issue on the partial eigenstructure assignment in a high‐order system, in which some certain eigenpairs, in a given system, can be assigned without changing others. Then a differential equation could be used to model the given systems, so that a multi‐input state feedback control deal with this assignment. Moreover, the algorithm requests the information of a few of the eigenpairs merely. As well some numerical cases are shown to prove the effect of our algorithm. [ABSTRACT FROM AUTHOR]

The purpose of this study was to explore how university students construct the relationship between the concepts of differential and derivative with the integration of GeoGebra and the ACODESA method. The participants in this study were 33 pre-service mathematics teachers. An open-ended questionnaire, knowledge test, tasks, and participants’ dynamic constructions were used as data collection tools. The analysis of participants’ products was based on descriptive analysis and Toulmin’s model. As a result of the analysis, it was found that the participants constructed the relationship between the concepts of differential and derivative as well as the concepts of tangent and slope by using them within the geometric framework. Due to the integration of GeoGebra and the ACODESA method, the participants explored the relationships among true change, estimated change, and error by using the geometric interpretation of the concept of differential. It was found that with this method, they deduced that Δx and dx were two different symbols for the same variable. [ABSTRACT FROM AUTHOR]

It is well known that for initial boundary value problems one may need to decrease the order of boundary closures to ensure strict stability (also called time stability) of finite difference schemes. For high-order schemes, this often reduces the global convergence rate significantly. To cure this issue, we derive high-order accurate boundary closures that satisfy a global conservation property. We show that the boundary closures cannot be both high-order accurate and conservative in the l 2 -inner product, if the grid is equidistant. Therefore, we derive particular sets of nonuniform solution points and quadrature weights near the domain boundaries. The new schemes are high-order accurate on the modified grids and conservative in the derived quadrature rules. Although we have no theoretical stability proof, numerical eigenvalue analysis suggests that the new schemes are strictly stable up to eleventh-order global accuracy for scalar linear advection equations. Numerical experiments with one- and two-dimensional linear equations corroborate the design orders of the schemes. [ABSTRACT FROM AUTHOR]

A one-way domain decomposition method (DDM) is considered for the solution of the time-harmonic electromagnetic scattering problem by inhomogeneous penetrable 3-D objects: the computational domain is partitioned into concentric subdomains and an integral representation (IR) of the electromagnetic fields on the outer boundary constitutes an exact radiation condition. The corresponding numerical code is efficiently parallelized and the full IR matrices are compressed in order to expedite the solution of very large problems. Exact and approximate high-order transmission conditions (HOTC) are obtained for the model problem of a multi-layer planar structure. Their application to real world objects is investigated in terms of well-posedness and numerical complexity. New well-posed low complexity HOTCs are proposed that speed up the convergence of the DDM algorithm. [ABSTRACT FROM AUTHOR]

A consistent, conservative and truly high order finite difference scheme for low Mach number turbulent variable density flow simulation is presented. In the present scheme, the scalar transport equation is discretized with the explicit weighted compact nonlinear scheme (WCNS) while the continuity and momentum equations are discretized with a conservative centered high order finite difference scheme on a staggered grid. Bounded hybrid high order upwind interpolation is proposed for the evaluation of density on the cell faces of the staggered grid. The use of explicit WCNS along with the bounded density interpolation resolves the issue of degraded order of accuracy in the existing high order finite difference schemes where the centered finite difference scheme for mass and momentum conservation is combined with a total variation stable scheme for scalar transport. With the same difference operator used for the continuity and scalar transport equations, the present scheme is consistent naturally, and resolves the inconsistency problem reported in Trisjono et al. (2016) [6] without employing any modification to a scheme. Such improvements are demonstrated for several test cases. [ABSTRACT FROM AUTHOR]

Strictly stable high-order accurate finite difference approximations are derived, for linear initial boundary value problems. The boundary closures are based on the diagonal-norm summation-by-parts framework and the boundary conditions are imposed using an improved projection method. The improved projection method removes some of the numerical issues with the previously derived (here referred to as traditional ) projection method. The finite difference approximations lead to fully explicit ODE systems. The accuracy and stability properties are demonstrated for linear hyperbolic and parabolic problems in 1D and 2D. In particular we study the problem where initial and boundary data are inconsistent, that causes the traditional projection method to fail. Comparisons are made also with weak (penalty) enforcement of boundary conditions. [ABSTRACT FROM AUTHOR]

In this paper, a fourth-order compact gas-kinetic scheme (GKS) is developed for the compressible Euler and Navier–Stokes equations under the framework of two-stage fourth-order temporal discretization and Hermite WENO (HWENO) reconstruction. Due to the high-order gas evolution model, the GKS provides a time dependent gas distribution function at a cell interface. This time evolution solution can be used not only for the flux evaluation across a cell interface and its time derivative, but also time accurate flow variables at a cell interface. As a result, besides updating the conservative flow variables inside each control volume, the cell averaged slopes inside each control volume through the differences of flow variables at the cell interfaces can be updated as well in GKS. So, with the updated flow variables and their slopes inside each cell, the HWENO techniques can be naturally implemented for the compact high-order reconstruction at the beginning of each time step. Therefore, a compact higher-order GKS, such as the two-stage fourth-order compact scheme can be constructed. This fourth-order compact GKS has the same stencil and as robust as the second-order scheme. In comparison with the fourth-order DG method, they have the same stencil. In order to get a fourth-order temporal accuracy, the GKS uses two stages and DG needs four stages in the time stepping methods. The CFL number used in GKS is on the order of 0.5 instead of 0.11 in the DG. This research concludes that beyond the first-order Riemann solver the use of high-order time evolution model at a cell interface is extremely helpful in the design of robust, accurate, and efficient higher-order compact schemes for the compressible flow simulations. [ABSTRACT FROM AUTHOR]

The derivation of new exponentially fitted (EF) modified two-step hybrid (MTSH) methods for the numerical integration of oscillatory second-order IVPs is analyzed. These methods are modifications of classical two-step hybrid methods so that they integrate exactly differential systems whose solutions can be expressed as linear combinations of the set of functions { exp ( λ t ) , exp ( − λ t ) } , λ ∈ C , or equivalently { sin ( ω t ) , cos ( ω t ) } when λ = i ω , ω ∈ R , where λ represents an approximation of the main frequency of the problem. The EF conditions and the conditions for this class of EF schemes to have algebraic order p (with p ≤ 8 ) are derived. With the help of these conditions we construct explicit EFMTSH methods with algebraic orders seven and eight which require five and six function evaluation per step, respectively. These new EFMTSH schemes are optimal among the two-step hybrid methods in the sense that they reach a certain order of accuracy with minimal computational cost per step. In order to show the efficiency of the new high order explicit EFMTSH methods in comparison to other EF and standard two-step hybrid codes from the literature some numerical experiments with several orbital and oscillatory problems are presented. [ABSTRACT FROM AUTHOR]

This paper uses polynomial interpolation to design a novel high-order algorithm for the numerical estimation of fractional differential equations. The Riemann–Liouville fractional derivative is expressed by using the Hadamard finite-part integral and the piecewise cubic interpolation polynomial is utilized to approximate the integral. The detailed error analysis is presented and it is established that the convergence order of the algorithm is O ( h 4 − α ) . Asymptotic expansion of the error for the presented algorithm is also investigated. Some numerical examples are provided and compared with the exact solution to show that the numerical results are in well agreement with the theoretical ones and also to illustrate the accuracy and efficiency of the proposed algorithm. [ABSTRACT FROM AUTHOR]

In this research, an efficient variant of mid-point iterative method is given for computing the sign of a square complex matrix having no pure imaginary eigenvalues. It is proven that the method is new and has global convergence with high order of convergence seven. To justify the effectiveness of the new scheme, several comparisons for matrices of different sizes are worked out to show that the new method is efficient. [ABSTRACT FROM AUTHOR]

Before using the deformable mirror (DM) for wavefront beam shaping, it is necessary to correct its surface profile to reduce the high-order aberrations, because the DM surface profile is different from the original set plane. In this paper, we demonstrate a simple method to correct the DM surface profile. The method is verified experimentally by using He-Ne laser. Results show that the DM surface profile is corrected to be near flat-topped by using the method. The output peak-to valley value changes from 2.38 λ to 0.21 λ of the He-Ne laser and the RMS value changes from 0.49 λ to 0.04 λ after correcting the DM surface. [ABSTRACT FROM AUTHOR]

A high-order discontinuous Galerkin (DG) method with nonconformal meshes is developed to accurately simulate large-scale poroelastic wave propagation in 3-D isotropic media. An exact upwind flux is succinctly derived to serve as an accurate coupling solver for the DG algorithm. Specifically, the eigenvalue problem in the Riemann solution is effectively reduced from the rank of 13 to 4. Furthermore, this new numerical flux gives more explicit physical insight, which indicates three-type waves in poroelastic media: two P waves and one S wave. Validations and verifications with analytical/semianalytical numerical solutions demonstrate the accuracy, robustness, and flexibility of the proposed solver. [ABSTRACT FROM AUTHOR]