Class Of Differential Equations Research Articles

An example of quaternion parameterization for dynamical simulations

The dynamical simulation of rigid bodies can be gathered from the classical Newton-Euler differential equations, which commonly make use of the Euler angles parametrization. In this work, the initial value problem associated with motion is presented in terms of quaternion formulation instead of the Euler one. The reason why the quaternion parametrization is proposed lies on the possibility of avoiding singularities that can occur by considering Euler angles. Moreover, the strength of quaternions is represented by the linearity of their formulation, the easiness of their algebraic structure and, overall, on their stability and efficiency. Our proposed application is the mathematical modelling of a small Unmanned Aerial Vehicle dynamics. In particular a multirotor with six blades has been taken into account, its mathematical model is deduced and a comparison between the results obtained by implementing our formulation and the classical one is produced.

A fast and faithful collocation method with efficient matrix assembly for a two-dimensional nonlocal diffusion model

The peridynamic theory provides an appropriate description of the deformation of a continuous body involving discontinuities or other singularities, which cannot be described properly by the classical theory of solid mechanics. However, the operator in the peridynamic theory is nonlocal, so the resulting numerical methods generate dense or full coefficient matrices which require O(N2) of memory where N is the number of unknowns in the discretized system. Gaussian types of direct solvers, which were traditionally used to solve these problems, require O(N3) of operations. Furthermore, due to the singularity of the kernel in the peridynamic model, the evaluation and assembly of the coefficient matrix can be very expensive. Numerous numerical experiments have shown that in many practical simulations the evaluation and assembly of the coefficient matrix often constitute the main computational cost! The significantly increased computational work and memory requirement of the peridynamic model over those for the classical partial differential equation models severely limit their applications, especially in multiple space dimensions.We develop a fast and faithful collocation method for a two-dimensional nonlocal diffusion model, which can be viewed as a scalar-valued version of a peridynamic model, without using any lossy compression, but rather, by exploiting the structure of the coefficient matrix. The new method reduces the evaluation and assembly of the coefficient matrix by O(N), reduces the computational work from O(N3) required by the traditional methods to O(Nlog2N) and the memory requirement from O(N2) to O(N). Numerical results are presented to show the utility of the fast method.

On the Deflexion of Anisotropic Structural Composite Aerodynamic Components

This paper presents closed form solutions to the classical beam elasticity differential equation in order to effectively model the displacement of standard aerodynamic geometries used throughout a number of industries. The models assume that the components are constructed from in-plane generally anisotropic (though shown to be quasi-isotropic) composite materials. Exact solutions for the displacement and strains for elliptical and FX66-S-196 and NACA 63-621 aerofoil approximations thin wall composite material shell structures, with and without a stiffening rib (shear-web), are presented for the first time. Each of the models developed is rigorously validated via numerical (Runge-Kutta) solutions of an identical differential equation used to derive the analytical models presented. The resulting calculated displacement and material strain fields are shown to be in excellent agreement with simulations using the ANSYS and CATIA commercial finite element (FE) codes as well as experimental data evident in the literature. One major implication of the theoretical treatment is that these solutions can now be used in design codes to limit the required displacement and strains in similar components used in the aerospace and most notably renewable energy sectors.

Application of viability theory for road vehicle active safety during cornering manoeuvres

Viability theory proposes geometric metaphors in addition to classical ordinary differential equation analysis. In this paper, advantages of applying viability theory to road safety domain are presented. The exact issue is to determine if, from an initial state of a vehicle/road/driver system, a soft controls strategy is compatible with a safe driving sequence. The case of a car negotiating a curve is considered. The application of the viability theory to this issue offers the advantage to avoid classical full computing of the system. Instead of that, it consists on verifying that the states and the controls belong to a subset called the viability kernel. The construction and the use of the viability kernel for a vehicle system dynamic is proposed by using support vector machines algorithm. Then, the applicability of this theory is demonstrated through experimental tests. This innovative application of the viability theory to vehicle dynamics with road safety concerns could benefit to robust embedded warning systems.

A novel diffusion system for impulse noise removal based on a robust diffusion tensor

The removal of impulse noise is a prerequisite step in image analysis. The classic partial differential equation (PDE) has achieved a great success in suppressing Gaussian noise, but its performance in reducing impulse noise is less satisfactory. The main difficulty arises from finding a nice diffusion function. To tackle this problem, the paper develops a novel diffusion system to suppress impulse noise. The proposed diffusion system consists of two phases. In the first phase, an effective image filter called Clean Pixel Excluder (CPE) is designed to identify clean pixels from the noisy ones. In the second phase, a robust diffusion model is reformulated by developing a novel diffusion tensor to control the smoothing on both direction and strength adaptively. A numerical scheme based on the multi-scale technique is provided. Extensive experiments on both synthetic and real images show that the proposed system achieves a superior performance over several standard methods in terms of noise suppression and detail preservation.

Diffusion through a half space: equivalence between different formulations of the unique solution

Diffusion through a half space involves a classical parabolic partial differential equation that is encountered in many fields of physics and has significant engineering applications, concerning particularly heat and mass transfer. However, in the specialized literature, the solution is usually achieved restricting the problem to particular cases and attaining apparently different formulations, thus a comprehensive overview is hindered. In this paper, the solution of the diffusion equation in a half space with a boundary condition of the first kind is worked out by means of the Fourier’s Transform, the Green’s function and the similarity variable, with a proof of equivalence – not found elsewhere – of these different approaches. The keystone of the proof rests on the square completion method applied to Gaussian-like integrals, widely used in Quantum Field Theory.

On Finite-Order Meromorphic Solutions of a Class Containing a Difference Painlevé I Equation

In this article, we investigate the growth and value distribution of finite-order transcendental meromorphic solutions for some non-linear difference equations. For instance, we show that demanding the existence of a transcendental meromorphic solution of a large class of difference equations containing difference Painleve I reduces the class either to a smaller one which still contains the difference Painleve I, or to a class of first-order equations. In addition, for certain classes of equations, we prove the existence of rational solutions and give their forms.

Two-dimensional wave equations with fractal boundaries

Wave equations are one of the three types of classical partial differential equations of second order. These equations arise in mathematical modelling of the motion of vibrating strings and membranes, and are typical in studying partial differential equations of second order. Wave equations in the sense of classical derivative with smooth boundaries have been investigated extensively. However, few studies on membranes with fractal boundaries have appeared in the literature. This article is devoted to two types of two-dimensional wave equations with fractal boundaries.

Polynomial Chaos for random fractional order differential equations

Fractional order models provide a powerful instrument for description of memory and hereditary properties of systems in comparison to integer order models, where such effects are difficult to incorporate and often neglected. Also, many physical real world problems that involve uncertainties and errors can be best modeled with random differential equations. Thus, to be able to deal with many real life problems, it is important to develop mathematical methodologies to solve systems that include both memory effects and uncertainty. The aim of this paper is to study the application of the generalized Polynomial Chaos (gPC) to random fractional ordinary differential equations. The method of Polynomial Chaos has played an increasingly important role when dealing with uncertainties. The main idea of the method is the projection of the random parameters and stochastic processes in the system onto the space of polynomial chaoses. However, to apply Polynomial Chaos to random fractional differential equations requires careful attention due to memory effects and the increasing of the computation time in respect to the classic random differential equations. In order to avoid more complex numerical computations and obtain accurate solutions we rely on Richardson extrapolation. It is shown that the application of generalized Polynomial Chaos method in conjunction with Richardson extrapolation is a reliable and accurate method to numerically solve random fractional ordinary differential equations.

Multiplicity results on discrete boundary value problems with double resonance via variational methods

The existence of solutions for a class of difference equations with double resonance is studied via variational methods, and multiplicity results are derived.

Stress analysis in mitre bends embedded in rock – new derivation and extension of existing formulae for use in penstocks

AbstractIn the construction of pressurized conduits for hydropower plants, segmented bends (mitre bends) are widely used where ducts change direction. In practical applications, different approaches are used to consider the effects for exposed mitre bends in structural steel design. However, formulae for mitre bends embedded in rock or concrete for penstocks in underground installations have not been available so far.In view of these restrictions on the calculation of underground mitre bends, the authors have developed analytical formulae for mitre bends embedded in rock by introducing terms for the elastic foundation of the rock mass to extend the classical differential equation for cylindrical shells. These formulae allow the whole local stress state in the bend to be calculated, in particular for the stress concentrations at the intersection of the mitre. The formulae were then validated by comparing them with the results of non‐linear FE analyses.

Numerical approach to the Caputo derivative of the unknown function

Abstract If a function can be explicitly expressed, then one can easily compute its Caputo derivative by the known methods. If a function cannot be explicitly expressed but it satisfies a differential equation, how to seek Caputo derivative of such a function has not yet been investigated. In this paper, we propose a numerical algorithm for computing the Caputo derivative of a function defined by a classical (integer-order) differential equation. By the properties of Caputo derivative derived in this paper, we can change the original typical differential system into an equivalent Caputo-type differential system. Numerical examples are given to support the derived numerical method.

Wavelet operational matrix method for solving the Riccati differential equation

A Haar wavelet operational matrix method (HWOMM) was derived to solve the Riccati differential equations. As a result, the computation of the nonlinear term was simplified by using the Block pulse function to expand the Haar wavelet one. The proposed method can be used to solve not only the classical Riccati differential equations but also the fractional ones. The capability and the simplicity of the proposed method was demonstrated by some examples and comparison with other methods.

Nonlinear Coordinated Control of STATCOM and Generator Excitation Based on Improved Dynamic Surface Control Method

For the single machine connected to infinite bus power system with uncertainties, one nonlinear coordinated control scheme for Static Synchronous Compensator (STATCOM) and excitation is proposed in this paper. Firstly, in order to avoid solving the differential algebraic equations (DAEs) model of system, we simplify the DAEs into the classical differential equations, and then a new nonlinear parameter strict feedback model is given. Secondly, in order to make the system achieve the desired results, the controller is designed in two parts based on improved dynamic surface control method (IDSC) and passive control techniques. The theoretical analysis shows that the derived controller can not only attenuate the influences of external disturbances, but also has strong robustness for system parameters variety. The control law obtained is more effective and the system globally and uniformly ultimately bounded can be achieved using full nature of nonlinear dynamic. Lastly, the further simulation results indicate that the proposed controller can ensure transient stability of the power system under large sudden fault.

Verifying the Hanging Chain Model

The wave equation with variable tension is a classic partial differential equation that can be used to describe the horizontal displacements of a vertical hanging chain with one end fixed and the other end free to move. Using a web camera and Tracker software to record displacement data from a vibrating hanging chain, we verify a modified version of the wave equation with variable tension that accounts for damping. Supplemental materials are available for this article. Go to the publisher's online edition of PRIMUS to view the supplemental file.

Global behaviour of solutions to a class of second-order rational difference equations when prime period-two solutions exist

For non-negative parameters α, β, γ, A, B and C such that A+B+C>0, consider the class of difference equations where either if A = 0, or R = [0,∞)2 if A>0. This class consists of 49 non-trivial second-order rational difference equations. We give a complete qualitative description of the global behaviour of solutions including basins of attraction of equilibria and prime period-two (PP2) solutions for all nonlinear difference equations (E) for which PP2 solutions exist. In particular, we show that every solution of (E) converges to an equilibrium or to a PP2 solution.

Elastic Soft-Core Sandwich Plates: Critical Loads and Energy Errors in Commercial Codes Due to Choice of Objective Stress Rate

Most commercial finite element codes, such as ABAQUS, LS-DYNA, ANSYS and NASTRAN, use as the objective stress rate the Jaumann rate of Cauchy (or true) stress, which has two flaws: It does not conserve energy since it is not work-conjugate to any finite strain tensor and, as previously shown for the case of sandwich columns, does not give a correct expression for the work of in-plane forces during buckling. This causes no appreciable errors when the skins and the core are subdivided by several layers of finite elements. However, in spite of a linear elastic behavior of the core and skins, the errors are found to be large when either the sandwich plate theory with the normals of the core remaining straight or the classical equivalent homogenization as an orthotropic plate with the normals remaining straight is used. Numerical analysis of a plate intended for the cladding of the hull of a light long ship shows errors up to 40%. It is shown that a previously derived stress-dependent transformation of the tangential moduli eliminates the energy error caused by Jaumann rate of Cauchy stress and yields the correct critical buckling load. This load corresponds to the Truesdell objective stress rate, which is work-conjugate to the Green–Lagrangian finite strain tensor. The commercial codes should switch to this rate. The classical differential equations for buckling of elastic soft-core sandwich plates with a constant shear modulus of the core are shown to have a form that corresponds to the Truesdell rate and Green–Lagrangian tensor. The critical in-plane load is solved analytically from these differential equations with typical boundary conditions, and is found to agree perfectly with the finite element solution based on the Truesdell rate. Comparisons of the errors of various approaches are tabulated.

A solution form of a class of higher-order rational difference equations

Abstract We obtain in this paper the expressions of solutions of the following class of difference equations x n + 1 = x n - 2 k + 1 ± 1 ± Π i = 1 k x n - 2 i + 1 , n = 0 , 1 , 2 , … with conditions posed on the initial values x−j, j = 0, 1, 2, … , 2k − 1, where k ∈ {1, 2, …}.

Loop quantum Brans-Dicke cosmology

The spatially flat and isotropic cosmological model of Brans-Dicke theory with coupling parameter $\ensuremath{\omega}\ensuremath{\ne}\ensuremath{-}\frac{3}{2}$ is quantized by the approach of loop quantum cosmology. An interesting feature of this model is that although the Brans-Dicke scalar field is nonminimally coupled with curvature, it can still play the role of an emergent time variable. In the quantum theory, the classical differential equation which represents cosmological evolution is replaced by a quantum difference equation. The effective Hamiltonian and modified dynamical equations of loop quantum Brans-Dicke cosmology are also obtained, which lay a foundation for the phenomenological investigation to possible quantum gravity effects in cosmology. The effective equations indicate that the classical big bang singularity is again replaced by a quantum bounce in loop quantum Brans-Dicke cosmology.

Regularization of differential variational inequalities with locally prox-regular sets

This paper studies, for a differential variational inequality involving a locally prox-regular set, a regularization process with a family of classical differential equations whose solutions converge to the solution of the differential variational inequality. The concept of local prox-regularity will be termed in a quantified way, as \((r,\alpha )\)-prox-regularity.