Higher-order Problems Research Articles

An approximate analytical solution of the Navier\u2013Stokes equations within Caputo operator and Elzaki transform decomposition method

In this article, a hybrid technique of Elzaki transformation and decomposition method is used to solve the Navier–Stokes equations with a Caputo fractional derivative. The numerical simulations and examples are presented to show the validity of the suggested method. The solutions are determined for the problems of both fractional and integer orders by a simple and straightforward procedure. The obtained results are shown and explained through graphs and tables. It is observed that the derived results are very close to the actual solutions of the problems. The fractional solutions are of special interest and have a strong relation with the solution at the integer order of the problems. The numerical examples in this paper are nonlinear and thus handle its solutions in a sophisticated manner. It is believed that this work will make it easy to study the nonlinear dynamics, arising in different areas of research and innovation. Therefore, the current method can be extended for the solution of other higher-order nonlinear problems.

Comparison for the Approximate Solution of the Second-Order Fuzzy Nonlinear Differential Equation with Fuzzy Initial Conditions

This research focuses on the approximate solutions of second-order fuzzy differential equations with fuzzy initial condition with two different methods depending on the properties of the fuzzy set theory. The methods in this research based on the Optimum homotopy asymptotic method (OHAM) and homotopy analysis method (HAM) are used implemented and analyzed to obtain the approximate solution of second-order nonlinear fuzzy differential equation. The concept of topology homotopy is used in both methods to produce a convergent series solution for the propped problem. Nevertheless, in contrast to other destructive approaches, these methods do not rely upon tiny or large parameters. This way we can easily monitor the convergence of approximation series. Furthermore, these techniques do not require any discretization and linearization relative with numerical methods and thus decrease calculations more that can solve high order problems without reducing it into a first-order system of equations. The obtained results of the proposed problem are presented, followed by a comparative study of the two implemented methods. The use of the methods investigated and the validity and applicability of the methods in the fuzzy domain are illustrated by a numerical example. Finally, the convergence and accuracy of the proposed methods of the provided example are presented through the error estimates between the exact solutions displayed in the form of tables and figures.

Optimization of high-channel count, switch matrices for multinuclear, high-field MRI.

Modern magnetic resonance imaging systems are equipped with a large number of receive connectors in order to optimally support a large field-of-view and/or high acceleration in parallel imaging using high-channel count, phased array coils. Given that the MR system is equipped with a limited number of digitizing receivers and in order to support operation of multinuclear coil arrays, these connectors need to be flexibly routed to the receiver outside the RF shielded examination room. However, for a number of practical, economic and safety reasons, it is better to only route a subset of the connectors. This is usually accomplished with the use of switch matrices. These exist in a variety of topologies and differ in routing flexibility and technological implementation. A highly flexible implementation is a crossbar topology that allows to any one input to be routed to any one output and can use single PIN diodes as active elements. However, in this configuration, long open-ended transmission lines can potentially remain connected to the signal path leading to high transmission losses. Thus, especially for high-field systems compensation mechanisms are required to remove the effects of open-ended transmission line stubs. The selection of a limited number of lumped element reactance values to compensate for the for the effect of transmission line stubs in large-scale switch matrices capable of supporting multi-nuclear operation is non-trivial and is a combinatorial problem of high order. Here, we demonstrate the use of metaheuristic approaches to optimize the circuit design of these matrices that additionally carry out the optimization of distances between the parallel transmission lines. For a matrix with 128 inputs and 64 outputs a realization is proposed that displays a worst-case insertion loss of 3.8 dB.

Higher-order cluster consensus of a multi-agent network with continuous-time dynamics

This paper studies the higher-order cluster agreement problem for continuous-time networks evolving over any given directed graph. Necessary and sufficient conditions are derived to reach clusters that are not pre-determined as opposed to existing literature. The analysis shows that appropriate stabilizing choices of coupling strengths do not affect the number of clusters and the final values of the clusters. The results are verified by numerical examples.

An ultraweak-local discontinuous Galerkin method for PDEs with high order spatial derivatives

In this paper, we develop a new discontinuous Galerkin method for solving several types of partial differential equations (PDEs) with high order spatial derivatives. We combine the advantages of a local discontinuous Galerkin (LDG) method and the ultraweak discontinuous Galerkin (UWDG) method. First, we rewrite the PDEs with high order spatial derivatives into a lower order system, then apply the UWDG method to the system. We first consider the fourth order and fifth order nonlinear PDEs in one space dimension, and then extend our method to general high order problems and two space dimensions. The main advantage of our method over the LDG method is that we have introduced fewer auxiliary variables, thereby reducing memory and computational costs. The main advantage of our method over the UWDG method is that no internal penalty terms are necessary in order to ensure stability for both even and odd order PDEs. We prove the stability of our method in the general nonlinear case and provide optimal error estimates for linear PDEs for the solution itself as well as for the auxiliary variables approximating its derivatives. A key ingredient in the proof of the error estimates is the construction of the relationship between the derivative and the element interface jump of the numerical solution and the auxiliary variable solution of the solution derivative. With this relationship, we can then use the discrete Sobolev and Poincaré inequalities to obtain the optimal error estimates. The theoretical findings are confirmed by numerical experiments.

Higher–order nonlocal gradient elasticity: A consistent variational theory

A consistent variational theory of the higher–order nonlocal gradient elasticity is conceived to appropriately introduce the nonlocality to the higher-order strain gradient theory. The abstract variational approach, based on appropriate functional spaces of test fields, is applied to establish the higher–order nonlocal gradient mechanics of elastic beams in flexure. Two nonlocal and two gradient characteristic lengths are exploited to describe the size–dependent response of continua with nano–structures. Integral convolutions of the higher–order constitutive law are restored to the equivalent differential problem endowed with non–standard boundary conditions of constitutive–type. The higher–order strain gradient theory, higher–order nonlocal elasticity and modified nonlocal strain gradient theory, extensively adopted in the community of Engineering Science, are demonstrated to be particular cases of the introduced higher-order nonlocal gradient theory. The well-posedness of the developed higher-order nonlocal gradient problem is revealed by studying the flexural response of structures with wide-ranging applications in nano-engineering. Exact analytical solution for elastostatic deflections of nano-beams is derived and new benchmark examples of nano-mechanics interest are detected. The higher-order nonlocal gradient elasticity theory can effectively characterize nanoscopic phenomena in advanced nano–composites and nano–structures.

A nonlocal continuum mechanics-based asymptotic theory for the buckling analysis of SWCNTs embedded in an elastic medium subjected to combined hydrostatic pressure and axial compression

Based on nonlocal continuum mechanics, the authors develop an asymptotic theory by which to examine the buckling behavior of simply supported, single-walled carbon nanotubes (SWCNTs) embedded in an elastic medium under combined hydrostatic pressure and axial compression using the perturbation method. In the formulation, the Eringen nonlocal constitutive relations are used to account for the small length scale effect; the interactions between the SWCNT and its surrounding medium are modelled as a Winkler foundation model, and the half-thickness- to- the mid-surface radius ratio is selected as the perturbation parameter. After performing some mathematical processes, including non-dimensionalization, asymptotic expansion, and successive integration, among others, the authors separate the three-dimensional (3D) nonlocal elasticity equations into recursive sets of governing equations (GEs) based on the Donnell-type nonlocal classical shell theory (CST) for various order problems. The Donnell-type nonlocal CST is derived as a first-order approximation of the 3D nonlocal elasticity problem, where the GEs for higher-order problems retain the same differential operators as those of the Donnell-type nonlocal CST although with different nonhomogeneous terms. The current asymptotic solutions for the critical load parameters of the embedded SWCNT are obtained in order to assess the accuracy of various nonlocal CSTs and nonlocal beam theories available in the literature.

Three-dimensional asymptotic nonlocal elasticity theory for the free vibration analysis of embedded single-walled carbon nanotubes

Within the framework of three-dimensional (3D) nonlocal elasticity theory, the authors develop an asymptotic theory to investigate the free vibration characteristics of simply supported, single-walled carbon nanotubes (SWCNTs) non-embedded or embedded in an elastic medium using the multiple time scale method. Eringen’s nonlocal constitutive relations are adopted to account for the small length scale effect in the formulation. The interactions between the SWCNT and its surrounding medium are modeled as a two-parameter Pasternak foundation model. After performing a series of mathematical processes, including nondimensionalization, asymptotic expansion, and successive integration, etc., the authors obtain recurrent sets of motion equations for various order problems. The nonlocal classical shell theory (CST) is derived as a first-order approximation of the current 3D nonlocal elasticity problem, and the equations of motion for higher-order problems retain the same differential operators as those of the nonlocal CST, although with different nonhomogeneous terms. The current asymptotic solutions for the natural frequency parameters of non-embedded or embedded SWCNTs and their corresponding through-thickness modal stress and displacement component distributions are obtained to assess the accuracy of various nonlocal shell and beam theories available in the literature.

A note on the numerical resolution of Heston PDEs

In this paper we aim to compare a popular numerical method with a new, recently proposed meshless approach for Heston PDE resolution. In finance, most famous models can be reformulated as PDEs, which are solved by finite difference and Monte Carlo methods. In particular, we focus on Heston model PDE and we solve it via radial basis functions (RBF) methods and alternating direction implicit. RBFs have become quite popular in engineering as meshless methods: they are less computationally heavy than finite differences and can be applied for high-order problems.

An Approximation Method for Time-Dependent Problems in High Energy Density Thermal Radiative Transfer

We analyze a computational method for solving multidimensional thermal radiative transfer (TRT) problems based on the radiative transfer equation (RTE) with approximate time evolution operator. The variation in the specific intensity of radiation over each time interval is approximated by an exponential function. This approximation reduces the RTE to an equation of steady-state form with time-dependent coefficients and a modified collision rate density term. This enables one to avoid storage of the high-dimensional solution from the previous time level. The numerical method for TRT problems applies the RTE with the time-dependent approximation as a high-order problem in multilevel system of low-order quasidiffusion (LOQD) equations. The change rate in the intensity is evaluated by the solution of the LOQD equations for the moments of the specific intensity. We study the accuracy of the approximate method in modeling the evolution of temperature and radiation waves in TRT problems.

Algorithm on Higher-Order Derivative Based on Ternary Optical Computer

As an important tool in the field of mathematics, higher-order derivation problems are widely used in differentials, quantum mechanics, and engineering applications. However, in the electronic computer (EC), due to the existence of the carry in the calculation, the computational efficiency is low when solving the higher-order derivation problem. In response to this problem, the ternary optical computer (TOC) has the advantages of no carry-in and the characteristics of numerous data bits, reconfigurable processors and parallel computing. Solve the higher-order derivation problems with complex operations by constructing multipliers and adders on the TOC platform, and copying multiple composite operator units (COUs). This article introduces the design of the higher-order derivative algorithm based on TOC in detail, the reconfiguration process of the multiplier and adder, and the number of bits of the multiplier and adder required in the implementation is given. Besides, the hardware resources and clock cycles in the operation are analyzed. The feasibility of the implementation scheme is verified by experiments. Compared with the traditional higher-order derivative, the higher-order derivative based on the TOC is superior in time performance, computational efficiency, and processing of complex operations. Due to the limitation of the research stage, the algorithm is only applicable to the function of polynomials, which lays a foundation for the further research of higher-order derivative algorithms, and has certain application significance.

Parametric Modeling of EM Behavior of Microwave Components Using Combined Neural Networks and Hybrid-Based Transfer Functions

Neuro-transfer function (neuro-TF) approaches have become more and more popular in parametric modeling for electromagnetic (EM) behavior of microwave components. Existing pole-residue-based neuro-TF approach has better capability of dealing with high-order problem than the rational-based neuro-TF approach, but has the discontinuity issue and the associated non-smoothness issue of the poles/residues when the geometrical variations become large while the rational-based neuro-TF approach does not have. This paper addresses this situation and presents a novel hybrid-based neuro-TF technique which systematically combines both pole-residue and rational formats of the transfer functions. Starting with the pole-residue-based transfer functions, we propose a novel technique to automatically identify the poles/residues that are smooth-continuous and the poles/residues that have the discontinuity and non-smoothness issues. The proposed technique converts the poles/residues that have those issues into the coefficients of the rational-based transfer function to solve the discontinuity and non-smoothness issues in the existing pole-residue-based neuro-TF approach. The proposed technique remains the smooth-continuous poles/residues in the pole-residue format of the transfer function to maintain the capability of handling high-order problem. Compared with the existing neuro-TF modeling methods, the proposed technique can obtain better accuracy in challenging applications of large geometrical variations and high order. The proposed technique is illustrated by two examples of parametric modeling of microwave components.

Categorizing student as a Convergent and Divergent Thinker in Problem-solving using Learning Analytics Framework

Abstract Convergent and divergent thinking is an essential skill required to think critically and solve a Higher-Order level problem. The practice in improving both convergent and divergent thinking during various phases of problem-solving can enable students to produce a better solution. This paper presents a Learning Analytics Framework used for categorizing a student as a convergent and divergent thinker based on problem-solving skills that focus on identifying student’s problem-solving abilities through Moodle tools and plugins. The core of this framework is a Moodle platform, that is acquiescent with the Learning Management System, and that has the ability of monitoring and recording the events generated by students. Based on these event logs and activities, the tools extract the process in which a student solves a problem of the course, allowing teachers to rethink and introduce changes in the problem-solving methods of the student. A study was conducted on third-year students from Computer Science Engineering to investigate the effects of the intervention. The results show a gradual development in learners’ thoughts in both convergent and divergent thinking during problem-solving.

Asynchronous Consensus Dynamics for Group of High-Order Agents Under Switching Topologies and Time-Varying Delays

This paper investigates the consensus problem of high-order multi-agent systems (MASs) under multiple communication constraints, including asynchronous communication, arbitrary switching topologies and time-varying delays. In the scenario of asynchronous communication, each agent only receives its neighbors’ states information at the time instants when it needs to adjust the state. For each agent, an distributed control protocol involving its own all-order state information and its neighbors’ first-order delayed state information is designed. With the help of state vector transformation technology, the high-order consensus problem under multiple communication constraints can be equivalently transformed into a convergence problem of products of nonnegative matrices. The related knowledge of nonnegative matrix theory is used to solve this convergence problem, and further establish a sufficient algebraic condition of achieving high-order consensus. Finally, the achievement of high-order consensus is demonstrated through a numerical example.

Uniform bounds for higher-order semilinear problems in conformal dimension

We establish uniform a-priori estimates for solutions of the semilinear Dirichlet problem (−Δ)mu=h(x,u) in Ω,u=∂nu=⋯=∂nm−1u=0 on ∂Ω,where h is a positive superlinear and subcritical nonlinearity in the sense of the Trudinger–Moser–Adams inequality, either when Ω is a ball or, provided an energy control on solutions is prescribed, when Ω is a smooth bounded domain. Our results are sharp within the class of distributional solutions. The analogous problem with Navier boundary conditions is also studied. Finally, as a consequence of our results, existence of a positive solution is shown by degree theory.

A high-order three-scale reduced asymptotic approach for thermo-mechanical problems of nonlinear heterogeneous materials with multiple spatial scales

An effective high-order three-scale reduced asymptotic (HTRA) approach is developed to simulate the coupled thermo-mechanical problems of nonlinear heterogeneous materials with multiple spatial scales. In this paper, the heterogeneous structures are constructed by periodical layout of unit cells on microscopic and mesoscopic scale. At first, the linear and nonlinear higher-order local cell functions defined at microscale and mesoscale are derived by multiscale asymptotic expansion approach. Further, two types of homogenized parameters are introduced, and related nonlinear homogenized solutions are obtained on the macroscopic domain. Then, the high-order three-scale temperature and displacement solutions are established by assembling the various multiscale local solutions and homogenization solutions. The key characteristic of the presented methods is an efficient reduced form for evaluating higher-order local cell problems, and hence greatly reducing the calculating amount in comparison to direct numerical simulations. Also, the new high-order nonlinear homogenization solutions that do not require higher-order continuity of the macroscale solutions. Finally, the effectiveness and accuracy of the multiscale algorithms are verified by some typical numerical examples.

A Sturm–Liouville theorem for quadratic operator pencils

We establish a Sturm–Liouville theorem for quadratic operator pencils with matrix-valued potentials counting their unstable real roots, with applications to stability of waves. Such pencils arise, for example, in reduction of eigenvalue systems to higher-order scalar problems.

A perturbation-based approach for solving fractional-order Volterra–Fredholm integro differential equations and its convergence analysis

The present work considers the approximation of solutions of a type of fractional-order Volterra–Fredholm integro-differential equations, where the fractional derivative is introduced in Caputo sense. In addition, we also present several applications of the fractional-order differential equations and integral equations. Here, we provide a sufficient condition for existence and uniqueness of the solution and also obtain an a priori bound of the solution of the present problem. Then, we discuss about the higher-order model equation which can be written as a system of equations whose orders are less than or equal to one. Next, we present an approximation of the solution of this problem by means of a perturbation approach based on homotopy analysis. Also, we discuss the convergence analysis of the method. It is observed through different examples that the adopted strategy is a very effective one for good approximation of the solution, even for higher-order problems. It is shown that the approximate solutions converge to the exact solution, even for higher-order fractional differential equations. In addition, we show that the present method is highly effective compared to the existed method and produces less error.

Entire nodal solutions to the critical Lane-Emden system

We establish the existence of finitely many sign-changing solutions to the Lane-Emden systemwhere the exponents p and q lie on the critical hyperbola These solutions are nonradial and arise as limit profiles of symmetric sign-changing minimizing sequences for a critical higher-order problem in a bounded domain.

Using Parallel Genetic Algorithms for Estimating Model Parameters in Complex Reactive Transport Problems

In this study, we present the details of an optimization method for parameter estimation of one-dimensional groundwater reactive transport problems using a parallel genetic algorithm (PGA). The performance of the PGA was tested with two problems that had published analytical solutions and two problems with published numerical solutions. The optimization model was provided with the published experimental results and reasonable bounds for the unknown kinetic reaction parameters as inputs. Benchmarking results indicate that the PGA estimated parameters that are close to the published parameters and it also predicted the observed trends well for all four problems. Also, OpenMP FORTRAN parallel constructs were used to demonstrate the speedup of the code on an Intel quad-core desktop computer. The parallel code showed a linear speedup with an increasing number of processors. Furthermore, the performance of the underlying optimization algorithm was tested to evaluate its sensitivity to the various genetic algorithm (GA) parameters, including initial population size, number of generations, and parameter bounds. The PGA used in this study is generic and can be easily scaled to higher-order water quality modeling problems involving real-world applications.