Constant Coefficient Differential Equations Research Articles

Three-wave lump solutions and their dynamic behaviors for the (3+1)-dimensional constant-coefficient and variable-coeffcient differential equations

In this paper, we propose two theorems to illustrate the types of equations that can be solved using the quadratic function method to derive the lump solutions localized in the whole plane, which are called three-wave lump solutions, and provide two constant-coefficient equations to illustrate. We further extend the quadratic function method to the variable-coefficient differential equations and obtain the three-wave lump solutions for two (3+1)-dimensional variable-coefficient equations. Moreover, the amplitudes of these lump waves and the distances between the two valleys of each lump are also obtained. Meanwhile, the motion trails, displacements and the velocities of these lump waves are analyzed in detail by virtue of numerical simulation. The study can be used to describe the motion of nonlinear waves in shallow water under the influence of time, and the results can enrich the types of solutions for the KdV-type equations. In addition, the 3d plots and corresponding density plots of the lump waves are displayed to show their spatial structures.

Integral Transforms and the Hyers–Ulam Stability of Linear Differential Equations with Constant Coefficients

Integral transform methods are a common tool employed to study the Hyers–Ulam stability of differential equations, including Laplace, Kamal, Tarig, Aboodh, Mahgoub, Sawi, Fourier, Shehu, and Elzaki integral transforms. This work provides improved techniques for integral transforms in relation to establishing the Hyers–Ulam stability of differential equations with constant coefficients, utilizing the Kamal transform, where we focus on first- and second-order linear equations. In particular, in this work, we employ the Kamal transform to determine the Hyers–Ulam stability and Hyers–Ulam stability constants for first-order complex constant coefficient differential equations and, for second-order real constant coefficient differential equations, improving previous results obtained by using the Kamal transform. In a section of examples, we compare and contrast our results favorably with those established in the literature using means other than the Kamal transform.

Heat transfer comparison investigation of the permanent magnet synchronous motor for electric vehicles based on the BEM and the FEM

In the field of heat transfer in permanent magnet synchronous motors (PMSM) for electric vehicles, the boundary element method (BEM) has been applied for the first time to calculate the steady-state temperature of the PMSM with a spiral water-cooled system. In this investigation, the boundary-integration equation for the steady-state heat transfer problem of a water-cooled PMSM is first derived on the basis of thermodynamic theory, and the system of constant coefficient differential equations is obtained by discretizing its boundaries, while the temperature results obtained from the BEM are compared with the finite element method (FEM) results. Furthermore, the temperature distribution and heat transfer characteristics obtained from the FEM and BEM were verified twice using the PMSM prototype and test platform. The results show that the maximum relative error between the temperature calculation results of FEM and BEM is 1.97%, and the maximum relative error between the results of BEM and the test does not exceed 3%, which finally verifies the validity and accuracy of BEM in solving the heat transfer problems of water-cooled PMSM.

Modeling a building during an earthquake to demonstrate resonance

ABSTRACT Spring-mass systems are presented as an application of second-order, constant coefficient differential equations in many differential equations textbooks. The phenomenon of resonance can then be analysed after nonhomogeneous differential equations are introduced. With some simplifying assumptions, the movement of the roof of a one-story building can be modeled as a spring-mass system and ground vibrations, such as those from an earthquake, can then be viewed as an external force acting on this system. In this paper, I will discuss a project I wrote and use in which students analyse the movement of a building resulting from an earthquake. This requires students to think in terms of amplitude/displacement (rather than position) and observe how this is affected by the frequency of an external force.

Linear Quadratic Regulator for Delayed Systems Using the Hamiltonian Approach and Exact Closed-Loop Poles for First-Order Systems

Abstract We consider the linear quadratic regulator (LQR) for linear constant-coefficient delay differential equations (DDEs) with multiple delays. The Hamiltonian approach is used instead of an algebraic Riccati partial differential equation. Two coupled DDEs governing the state and control input are derived using the calculus of variations. This coupled system, with infinitely many roots in both left and right half-planes, defines a boundary value problem. Its left half-plane roots are the exact closed-loop poles of the controlled system. These closed-loop poles have not been used to compute the optimal feedback before. Here, the distributed delay kernel that yields exactly those poles is first computed using an eigenfunction expansion. Increasing the number of terms in the truncated expansion yields a highly oscillatory kernel. However, the oscillatory kernel's antiderivative converges to a piecewise smooth function on the delay interval plus a Dirac delta function at zero. Discontinuities in the kernel coincide with discrete delay values in the original DDE. Using this insight, a fitted piecewise polynomial kernel matches the exact closed-loop poles very well. The twofold contribution of the Hamiltonian approach is thus clarity on the form of the feedback kernel as well as the exact closed-loop poles. Subsequently, the fitted piecewise polynomial kernel can be used for a much simpler control calculation. The polynomial coefficients can be fitted by solving a few simultaneous linear equations. Two detailed numerical examples of the LQR for DDEs, one with two delays and one with three delays, show excellent results.

Research on optimization model of wave energy transmission device

As a pollution-free renewable resource with great energy conversion potential, wave energy is attracting more and more attention from the society. There is an existing wave energy power generation energy output device. Considering the motion stability of the device in the ocean and the energy conversion problem, this paper will introduce the principle and establish a mathematical model around two main problems. Under the condition that only the heave motion is considered, this paper establishes the motion period model of the float and the vibrator based on the linear wave principle, analyzes the force on the float and the vibrator respectively, mainly based on the second-order constant coefficient differential equation of time t established by Newton's second law, and gives the displacement and velocity of the float and the vibrator in heave motion at,,, and under different damping coefficient conditions according to the condition solution. When the damping coefficient is constant, the damping coefficient at the maximum average output power is obtained. In this paper, based on the differential equation obtained from the motion state of the float and vibrator, an optimization model with the average output power as the objective function is established. The damping coefficient is taken as the decision variable, and the decision variable is optimized through genetic algorithm. When the damping coefficient is constant, the damping coefficient is, and the maximum power is. When there is a power exponential relationship between the damping coefficient and the relative velocity, it is found that when the damping coefficient is, the maximum power is.

Design of discrete-time matrix all-pass filters using subspace Nevanlinna pick interpolation

Unitary matrix-valued functions of frequency are matrix all-pass systems, since they preserve the norm of the input vector signals. Typically, such systems are represented and analyzed using their unitary-matrix valued frequency domain characteristics, although obtaining rational realizations for matrix all-pass systems enables compact representations and efficient implementations. However, an approach to obtain matrix all-pass filters that satisfy phase constraints at certain frequencies was hitherto unknown. In this paper, we present an interpolation strategy to obtain a rational matrix-valued transfer function from frequency domain constraints for discrete-time matrix all-pass systems. Using an extension of the Subspace Nevanlinna Pick Interpolation Problem (SNIP), we design a construction for discrete-time matrix all-pass systems that satisfy the desired phase characteristics. An innovation that enables this is the extension of the SNIP to the boundary case to obtain efficient time-domain implementations of matrix all-pass filters as matrix linear constant coefficient difference equations, facilitated by a rational (realizable) matrix transfer function. We also show that the derivative of matrix phase constraints, related to the group delay at the interpolating points, can be optimized to control the all-pass transfer matrices at the unspecified frequencies. Simulations show that the proposed technique for unitary matrix filter design performs as well as traditional DFT based interpolation approaches, including Geodesic interpolation and the popular Givens rotation based matrix parameterization.

A Unified Lattice Boltzmann Model for Fourth Order Partial Differential Equations with Variable Coefficients.

In this work, a unified lattice Boltzmann model is proposed for the fourth order partial differential equation with time-dependent variable coefficients, which has the form . A compensation function is added to the evolution equation to recover the macroscopic equation. Applying Chapman-Enskog expansion and the Taylor expansion method, we recover the macroscopic equation correctly. Through analyzing the error, our model reaches second-order accuracy in time. A series of constant-coefficient and variable-coefficient partial differential equations are successfully simulated, which tests the effectiveness and stability of the present model.

On a variant of multivariate Mittag-Leffler's function arising in the Laplace transform method

By using the Laplace transform method, we revisit the multivariate Mittag-Leffler function as an effective tool to construct a solution for some classes of fractional differential equations with constant coefficients. To support our results, we discuss several particular cases related to classical fractional differential operators. The techniques are not only restricted to fractional derivative operators but also can be applied to general constant coefficient differential equations, including high-order ordinary differential equations.

Hyers–Ulam Stability for Differential Systems with $$2\times 2$$ Constant Coefficient Matrix

We explore the Hyers–Ulam stability of perturbations for a homogeneous linear differential system with \(2\times 2\) constant coefficient matrix. New necessary and sufficient conditions for the linear system to be Hyers–Ulam stable are proven, and for the first time, the best (minimal) Hyers–Ulam constant for systems is found in some cases. Several examples are provided. Obtaining the best Hyers–Ulam constant for second-order constant coefficient differential equations illustrates the applicability of the strong results.

Differential equations defined by (convergent) Laurent series

The class of ordinary linear constant coefficient differential equations is naturally embedded into a wider class by associating differential equations to (convergent) Laurent series. It is thought that these more general differential equations can be used as an alternative description of some special functions.

Investigation of the Free Vibrations of Radial Functionally Graded Circular Cylindrical Beams Based on Differential Quadrature Method

In the current research, an effective differential quadrature method (DQM) has been developed to solve natural frequency and vibration modal functions of circular section beams along radial functional gradient. Based on the high-order theory of transverse vibration of circular cross-section beams, lateral displacement equation was reconstructed neglecting circumferential shear stress. Two equations coupled with deflection and rotation angles were derived based on elastic mechanics theory and further simplified into a constant coefficient differential equation with natural frequency as eigenvalue. Then, differential quadrature method was applied to transform the eigenvalue problem of the derived differential equation into a set of algebraic equation eigenvalue problems. Natural frequencies of the free vibrations of cylindrical beams with circular cross-sections were calculated at one time, and corresponding modal functions were solved together. The obtained numerical results indicated that the natural frequencies of functionally graded (FG) circular cylindrical beams obtained using differential quadrature method agreed with the results reported in related literatures. In addition, influences of varying gradient parameters on the modal shapes of circular cylindrical beams were found to be strongly consistent with previous studies. Numerical results further validated the feasibility and accuracy of the developed differential quadrature method in solving the transverse vibration of FG circular cross-section beams.

A Semi-Analytical Solution Approach for Solving Constant-Coefficient First-Order Partial Differential Equations

Simulation and control of many dynamic systems involve solving partial differential equations (PDE). This letter proposes a semi-analytical solution (SAS) approach for fast and high-quality solution of first-order PDEs. The region of interest of the studied PDE is divided into a grid, and an SAS is derived for each grid cell in the form of the multivariate polynomials, of which the coefficients are identified using initial value and boundary value conditions. The solutions are solved in a “time-stepping” manner, i.e., within one time step, the coefficients of the SAS are identified and the initial value of the next time step is evaluated. This approach achieves a significantly larger grid cell than the widely used finite difference method, and thus enhances the computational efficiency significantly. The simulation result on the natural gas pipeline model demonstrates the advantages of SAS in accuracy and computational efficiency.

A Generalized Method of Undetermined Coefficients

The method of undetermined coefficients is used to solve constant coefficient nonhomogeneous differential equations whose forcing function is itself the solution of a homogeneous constant coefficient differential equation. In this paper, we show that the classical methods for tackling constant coefficient equations, including the method of undetermined coefficients, generalize to much wider class linear differential equations which, for example, include Cauchy-Euler type equations. This general method includes an explicit construction of the fundamental solution sets of such equations. We also briefly consider where this method can be applied by producing the most general second and third order differential equations that are polynomial in a first order differential operator. In addition, we provide a number of constant coefficient, Cauchy-Euler, and novel examples.

Improved equivalent transformation method for reduction NLPDEs with time-dependent variables

For dealing with exact solutions and other properties, the variable-coefficient equations are far more complicated than its constant-coefficient counterparts. So far, there are few systematic and effective methods to deal with this problem, in this paper, we develop an effective approach to doing this. The present paper is concerned with the nonlinear partial differential equations (NLPDEs) with time-dependent. By the improved equivalent transformation (ET) method, the variable-coefficient NLPDEs (vc-NLPDEs) are transformed into different constant-coefficient partial differential equations (cc-PDEs) under different conditions, one of the cc-PDEs is simpler, so it is easy to solve. Moreover, we give the vc-PDE which can be transformed into the corresponding cc-PDE based on the condition. Thus, the exact solutions to vc-PDEs are obtained by exact solutions to the cc-PDEs. In general, this method can be applied to reduction all time-dependent coefficients NLPDEs.

Numerical-analytical method for solving boundary value problem for the generalized moisture transport equation

The paper studies qualitatively new equations of moisture transfer, which generalize the Aller and Aller-Lykov equations. The generalization contributes to revealing in the original equations the specific features of the studied massifs, their structure, physical properties, processes occurring in them through the introduction of the notion of the rates of change of the fractal dimension. We have obtained solutions to the constant coefficient difference equations as a system arising when using the method of lines for the equations with a Riemann-Liouville time fractional derivative with boundary conditions of the first kind. A priori estimates are obtained that imply convergence of the obtained solutions to systems of ordinary differential equations with variable fractional coefficients. Numerical tests have been carried out to confirm theoretical results of the study.

Nonuniform Sampling Rate Conversion:An Efficient Approach

We present a discrete-time algorithm for nonuniform sampling rate conversion that presents low computational complexity and memory requirements. It generalizes arbitrary sampling rate conversion by accommodating time-varying conversion ratios, i.e., it can efficiently adapt to instantaneous changes of the input and output sampling rates. This approach is based on appropriately factorizing the time-varying discrete-time filter used for the conversion. Common filters that satisfy this factorization property are those where the underlying continuous-time filter consists of linear combinations of exponentials, e.g., those described by linear constant-coefficient differential equations. This factorization separates the computation into two parts: one consisting of a factor solely depending on the output sampling instants and the other consists of a summation -- that can be computed recursively -- whose terms depend solely on the input sampling instants and its number of terms is given by a relationship between input and output sampling instants. Thus, nonuniform sampling rates can be accommodated by updating the factors involved and adjusting the number of terms added. When the impulse response consists of exponentials, computing the factors can be done recursively in an efficient manner.

Neighborhood global learning based flower pollination algorithm and its application to unmanned aerial vehicle path planning

Flower pollination algorithm (FPA) is a meta-heuristic optimization algorithm that imitates the pollination phenomenon of flowering plants in nature. Due to this algorithm is prone to premature convergence when solving complex optimization problems. So this paper introduces a neighborhood global learning based flower pollination algorithm(NGFPA). Firstly, we analyze the FPA using the constant coefficient differential equation and change the FPA’s global equation. Secondly, we build a neighborhood global learning to enhance population diversity. Finally, the population reconstruction mechanism is added to inhibit the population premature convergence. The convergence of NGFPA is proven using the knowledge of differential equations and stochastic function analysis. We test the performance of NGFPA by optimizing CEC2017. Experiment results show that NGFPA has better performance in comparison with other swarm intelligence algorithms. Furthermore, NGFPA is used to solve the problem of unmanned aerial vehicle (UAV) path planning. Simulation results indicate that NGFPA can obtain smoother paths in different obstacle environments. Therefore, NGFPA is effective and valuable.

On computing the analytic-signal backbone of the unforced harmonic oscillator

Backbone curves are usually obtained by analyzing nonlinear oscillators (e.g., the Duffing equation) whereas the backbones of linear oscillators are incidental by-products of such studies. Nevertheless, we focus on the harmonic oscillator, a linear, homogeneous constant-coefficient second-order ordinary differential equation. The backbone of the sinusoidally forced harmonic oscillator is well known, but the backbone of the unforced harmonic oscillator seems to have received essentially no attention. Since the 1980s, various computerized methods of backbone extraction have been under development, one of which is based on analytic-signal theory, a subfield of digital signal processing that involves Hilbert transform computations. By focusing on linear-theory backbones, a prerequisite to nonlinear analysis, we aim to pinpoint some of the theoretical and computational issues that arise when dealing with analytic-signal backbones. This paper examines the Mirror Periodic Method (MPM) as a practical way to reduce end effects in Hilbert transform computations and approximate analytic-signal backbones of unforced harmonic oscillators with light damping (less than 10 percent of critical). Using numerical examples, we demonstrate that MPM works well with both displacement and acceleration data, even for short signals, meaning just one or two damped cycles of oscillation. We then apply MPM techniques to accelerometer data that were recorded as part of earthquake-engineering teaching demonstrations involving a small-scale laboratory model of a linear single-degree-of-freedom system.

Equivalent transformations, bifurcations and exact solutions to a class of variable-coefficient PDE

In this paper, the equivalent transformations (ETs) of the generalized variable-coefficient KdV (vc-KdV) types of equations are constructed, so these variable-coefficient partial differential equations (vc-PDEs) are transformed into constant-coefficient partial differential equations (cc-PDEs) under some conditions, the vc-KdV and vc-mKdV equations are transformed into classical KdV and mKdV equations accordingly. Furthermore, the dynamical behavior is provided by the bifurcation analysis method, the exact parametric representations are investigated. Then the explicit solutions to the variable-coefficient equations are presented in terms of the ETs and parametric representations.