Fourth-order Differential Equation Research Articles

Extending Density Phase-Field Simulations to Dynamic Regimes

Density-based phase-field (DPF) methods have emerged as a technique for simulating grain boundary thermodynamics and kinetics. Compared to the classical phase-field, DPF gives a more physical description of the grain boundary structure and chemistry, bridging CALPHAD databases and atomistic simulations, with broad applications to grain boundary and segregation engineering. Notwithstanding their notable progress, further advancements are still warranted in DPF methods. Chief among these are the requirements to resolve its performance constraints associated with solving fourth-order partial differential equations (PDEs) and to enable the DPF methods for simulating moving grain boundaries. Presented in this work is a means by which the aforementioned problems are addressed by expressing the density field of a DPF simulation in terms of a traditional order parameter field. A generic DPF free energy functional is derived and used to carry out a series of equilibrium and dynamic simulations of grain boundaries in order to generate trends such as grain boundary width vs. gradient energy coefficient, grain boundary velocity vs. applied driving force, and spherical grain radius vs. time. These trends are compared with analytical solutions and the behavior of physical grain boundaries in order to ascertain the validity of the coupled DPF model. All tested quantities were found to agree with established theories of grain boundary behavior. In addition, the resulting simulations allow for DPF simulations to be carried out by existing phase-field solvers.

Alcubierre warp drive in Bohmian Quantum Gravity

In the realm of theoretical physics, the Alcubierre warp drive metric has long been a topic of fascination. However, the feasibility of this seemingly miraculous method of faster-than-light travel has been limited by the requirement of exotic matter, which has proven elusive to obtain. This research has revealed a potential breakthrough in the quest for exotic matter, through the coupling of the Alcubierre warp drive metric to a quantum mechanical scalar matter field. By mapping the requirement of exotic matter into a conformal wave equation, it becomes possible to express the problem as a fourth-order partial differential equation in terms of the quantum mechanical density. This research reveals that the key to solving the exotic matter problem lies in finding a proper quantum mechanical density that satisfies the proposed partial differential equation.

Painlevé Test, Phase Plane Analysis and Analytical Solutions of the Chavy–Waddy–Kolokolnikov Model for the Description of Bacterial Colonies

The Chavy–Waddy–Kolokolnikov model for the description of bacterial colonies is considered. In order to establish if the mathematical model is integrable, the Painlevé test is conducted for the nonlinear ordinary differential equation which corresponds to the fourth-order partial differential equation. The restrictions on the mathematical model parameters for ordinary differential equations to pass the Painlevé test are obtained. It is determined that the method of the inverse scattering transform does not solve the Cauchy problem for the original mathematical model, since the corresponding nonlinear ordinary differential equation passes the Painlevé test only when its solution is stationary. In the case of the stationary solution, the first integral of the equation is obtained, which makes it possible to represent the general solution in the quadrature form. The stability of the stationary points of the investigated mathematical model is carried out and their classification is proposed. Periodic and solitary stationary solutions of the Chavy–Waddy–Kolokolnikov model are constructed for various parameter values. To build analytical solutions, the method of the simplest equations is also used. The solutions, obtained in the form of a truncated expansion in powers of the logistic function, are represented as a closed formula using the formula for the Newton binomial.

Fourth-Order Emden–Fowler Neutral Differential Equations: Investigating Some Qualitative Properties of Solutions

In this article, we investigate some of the qualitative properties of a class of fourth-order neutral differential equations. We start by obtaining new inequalities and relations between the solution and its corresponding function, as well as with its derivatives. The new relations allow us to improve the monotonic and asymptotic properties of the positive solutions of the studied equation. Then, using an improved approach, we establish new criteria that test the oscillation of all solutions. We also rely on the principle of symmetry between positive and negative solutions to obtain the new criteria. The paper provides illustrative examples that highlight the significance of our findings.

Unified solution of some problems of rectangular plates with four free edges based on symplectic superposition method

PurposeA unified framework for solving the bending, buckling and vibration problems of rectangular thin plates (RTPs) with four free edges (FFFF), including isotropic RTPs, orthotropic rectangular thin plates (ORTPs) and nano-rectangular plates, is established by using the symplectic superposition method (SSM).Design/methodology/approachThe original fourth-order partial differential equation is first rewritten into Hamiltonian system. The class of boundary value problems of the original equation is decomposed into three subproblems, and each subproblem is given the corresponding symplectic eigenvalues and symplectic eigenvectors by using the separation variable method in Hamiltonian system. The symplectic orthogonality and completeness of symplectic eigen-vectors are proved. Then, the symplectic eigenvector expansion method is applied to solve the each subproblem. Then, the symplectic superposition solution of the boundary value problem of the original fourth-order partial differential equation is given through superposing analytical solutions of three foundation plates.FindingsThe bending, vibration and buckling problems of the rectangular nano-plate/isotropic rectangular thin plate/orthotropic rectangular thin plate with FFFF can be solved by the unified symplectic superposition solution respectively.Originality/valueThe symplectic superposition solution obtained is a reference solution to verify the feasibility of other methods. At the same time, it can be used for parameter analysis to deeply understand the mechanical behavior of related RTPs. The advantages of this method are as follows: (1) It provides a systematic framework for solving the boundary value problem of a class of fourth-order partial differential equations. It is expected to solve more complicated boundary value problems of partial differential equations. (2) SSM uses series expansion of symplectic eigenvectors to accurately describe the solution. Moreover, symplectic eigenvectors are orthogonal and directly reflect the orthogonal relationship of vibration modes. (3) The SSM can be carried to bending, buckling and free vibration problems of the same plate with other boundary conditions.

An Effective Approach Based on Generalized Bernstein Basis Functions for the System of Fourth-Order Initial Value Problems for an Arbitrary Interval

The system of ordinary differential equations has many uses in contemporary mathematics and engineering. Finding the numerical solution to a system of ordinary differential equations for any arbitrary interval is very appealing to researchers. The numerical solution of a system of fourth-order ordinary differential equations on any finite interval [a,b] is found in this work using a symmetric Bernstein approximation. This technique is based on the operational matrices of Bernstein polynomials for solving the system of fourth-order ODEs. First, using Chebyshev collocation nodes, a generalised approximation of the system of ordinary differential equations is discretized into a system of linear algebraic equations that can be solved using any standard rule, such as Gaussian elimination. We obtain the numerical solution in the form of a polynomial after obtaining the unknowns. The Hyers–Ulam and Hyers–Ulam–Rassias stability analyses are provided to demonstrate that the proposed technique is stable under certain conditions. The results of numerical experiments using the proposed technique are plotted in figures to demonstrate the accuracy of the specified approach. The results show that the suggested Bernstein approximation method for any interval is quick and effective.

Approximate Solution to Fractional Order Models Using a New Fractional Analytical Scheme

In the present work, a new fractional analytical scheme (NFAS) is developed to obtain the approximate results of fourth-order parabolic fractional partial differential equations (FPDEs). The fractional derivatives are considered in the Caputo sense. In this scheme, we show that a Taylor series destructs the recurrence relation and minimizes the heavy computational work. This approach presents the results in the sense of convergent series. In addition, we provide the convergence theorem that shows the authenticity of this scheme. The proposed strategy is very simple and straightforward for obtaining the series solution of the fractional models. We take some differential problems of fractional orders to present the robustness and effectiveness of this developed scheme. The significance of NFAS is also shown by graphical and tabular expressions.

Coherent hybrid block method for approximating fourth-order ordinary differential equations

Conventionally, the most used method of solving fourth-order initial value problems of ordinary differential is to first reduce to a system of first-order differential equations. This approach affects the effectiveness and convergence of the numerical method due to the transformation. This paper comprises the derivation, analysis, and implementation of a new hybrid block method for direct solution of fourth-order equations. The method is derived by collocation and interpolation of an assumed basis function. The basic properties of the block method, including zero stability, error constants, consistency, order, and convergence, were analyzed. From the analysis, the block method derived was found to be zero-stable, consistent, and convergent. Errors were computed for the proposed method, and they were proven to produce approximations that agree with exact solutions and as such this shows improvement with those of existing works.

Generalized and multi-oscillation solitons in the nonlinear Schrödinger equation with quartic dispersion.

We study different types of solitons of a generalized nonlinear Schrödinger equation (GNLSE) that models optical pulses traveling down an optical waveguide with quadratic as well as quartic dispersion. A traveling-wave ansatz transforms this partial differential equation into a fourth-order nonlinear ordinary differential equation (ODE) that is Hamiltonian and has two reversible symmetries. Homoclinic orbits of the ODE that connect the origin to itself represent solitons of the GNLSE, and this allows one to study the existence and organization of solitons with advanced numerical tools for the detection and continuation of connecting orbits. In this paper, we establish the existence of new types of connecting orbits, namely, PtoP connections from one periodic orbit to another. As we show, these global objects provide a general mechanism that generates additional families of two types of solitons in the GNLSE. First, we find generalized solitons with oscillating tails whose amplitude does not decay but reaches a nonzero limit. Second, PtoP connections in the zero energy level can be combined with EtoP connections from the origin to a selected periodic orbit to create multi-oscillation solitons; their characterizing property is to feature several episodes of different oscillations in between decaying tails. As is the case for solitons that were known previously, generalized solitons and multi-oscillation solitons are shown to be an integral part of the phenomenon of truncated homoclinic snaking.

Asymptotic Properties of Solutions to Fourth-Order Difference Equations on Time Scales

Abstract We provide sufficient criteria for the existence of solutions for fourth-order nonlinear dynamic equations on time scales ( a ( t ) x Δ 2 ( t ) ) Δ 2 = b ( t ) f ( x ( t ) ) + c ( t ) , {\left( {a\left( t \right){x^{{\Delta ^2}}}\left( t \right)} \right)^{{\Delta ^2}}} = b\left( t \right)f\left( {x\left( t \right)} \right) + c\left( t \right), such that for a given function y : 𝕋 → ℝ there exists a solution x : 𝕋 → ℝ to considered equation with asymptotic behaviour x ( t ) = y ( t ) + o ( 1 t β ) x\left( t \right) = y\left( t \right) + o\left( {{1 \over {{t^\beta }}}} \right) . The presented result is applied to the study of solutions to the classical Euler–Bernoulli beam equation, which means that it covers the case 𝕋 = ℝ.

General solutions’ laws of linear partial differential equations II

This paper uses Z transformations to obtain the general solutions of a large number of second-order, third-order and fourth-order linear partial differential equations for the first time, including one-dimensional inhomogeneous wave equation. Using general solutions, we have obtained plenty of exact solutions of the typical definite solution problems, which proves the important value of general solutions. We propose the Z4 transformation for the first time and initially use it to solve a specific case. We successfully obtain the Fourier series solution using the series general solution of the one-dimensional homogeneous wave equation, which successfully solves a famous unresolved debate in the history of mathematics.

Existence of solutions to a doubly degenerate fourth-order partial differential equation with a degenerate diffusion

A fourth-order partial differential equation with boundary degeneracy is considered in two-dimensional space. Both the fourth-order part and the second-order diffusion term have the double degenerate forms, which can yield the corresponding research difficulties in dealing with the degeneracy effects. We transform the parabolic problem into the corresponding semi-discrete steady-state problems when the coefficient is non-degenerate, and the corresponding existence and uniqueness are given by variational method. The energy method is applied to show the existence and uniqueness for the corresponding evolution problem with non-degenerate coefficient. Finally, by establishing the uniform estimates with respect to the regularization parameter, we perform the corresponding asymptotic limits for approximation solutions of the parabolic problem with boundary-degenerate case.

Hermitian numerical manifold method for large deflection of irregular Föppl-von Kármán plates

We proposed the Hermitian manifold numerical method (HNMM) fitted for finite strain analysis of thin plates with irregular domains. The large deflection of elastic thin plates is generally characterized by Föppl-von Kármán (FvK) equations, coupled with nonlinear fourth-order partial differential equations. The corresponding primal variational formulation requires the solution function to be globally C1 continuous but piecewise C2 continuous (namely H2 regular). Hermitian numerical manifold method (HNMM) can easily construct an approximation to solutions that satisfy the H2 regular requirements with structured meshes. Furthermore, those classical plate elements in finite element history can be embedded in the framework of HNMM and be more flexible and adaptable to solve plates with irregular domains. Thus, HNMM is formulated based on the Föppl-von Kármán (FvK) model using the dual cover system and a triplet attributes group on the physical patches. The numerical results demonstrate the accuracy of HNMM in large deflection of Föppl-von Kármán plate with complex domain shape.

Existence of Solutions to Nonlinear Fourth-Order Beam Equation

This paper studies the boundary value problem for a fourth-order difference equation with three quasidifferences. The new existence criterion of at least one solution to the issues considered is obtained using the theory of variational methods. The main result is illustrated in some examples.

Solving Differential Equation via Orthogonal Branciari Metric Spaces

In this paper, we investigate an orthogonal L ⋆ -contraction map concept and prove the fixed-point theorem in an orthogonal complete Branciari metric space (OCBMS). We also provide illustrative examples to support our theorems. We demonstrated the existence of a uniqueness solution to the fourth-order differential equation using a more orthogonal L ⋆ contraction operator in OCBMS as an application of the main results.

Solvability of a Nonlocal Inverse Problem for a Fourth-Order Equation

В данной работе для уравнения в частных производных четвертого порядка в прямоугольной области рассмотрена нелокальная обратная задача по поиску неизвестной правой части, которая зависит от одной переменной. Собственные функции и присоединенные функции соответствующей спектральной задачи, и их биортогональные функции полны и образуют базис Рисса в пространстве L2​(0,1). Установлены критерии единственности и существования решения рассматриваемой нелокальной обратной задачи для уравнения четвертого порядка. Единственность решения нелокальной обратной задачи вытекает из полноты системы биортогональных функций. Решение задачи построено в виде суммы ряда по собственным и присоединенным функциям, соответствующей спектральной задачи. Установлены достаточные условия на граничные функции, которые гарантируют теоремы существования и устойчивости решения рассматриваемой задачи. В замкнутой области показана абсолютная и равномерная сходимость найденного решения обратной задачи в виде ряда, а также рядов, полученных почленным дифференцированием по t и x соответственно два и четыре раза, в зависимости от гладкости функции заданными начальными условиями. При этом возникают малые знаменатели, затрудняющие сходимость этих рядов. Доказано, что в зависимости от размера области, множество ненулевых решений выражения в знаменателе не пусто. А также, показано, что если этот знаменатель равен нулю, то данная задача будет иметь нетривиальное решение при однородных условиях. Доказана также, устойчивость решения обратной задачи по нормам пространств L2​,W2n​ и C(Ω±), относительно изменения входных данных. In this paper, we consider a nonlocal inverse problem of finding an unknown right-hand side with one variable for a fourth-order partial differential equation in a rectangular domain. The eigenfunctions and functions of the corresponding spectral problem and its biorthogonal functions are complete and form a Riesz basis in the space L2​(0,1). Criteria for the uniqueness and existence of a solution to the considered nonlocal inverse problem for a fourth-order equation are established. The uniqueness of the solution of the nonlocal inverse problem follows from the completeness of the system of biorthogonal functions. The solution of the problem is constructed as the sum of a series in terms of eigenfunctions and associated functions of the corresponding spectral problem. Sufficient conditions are established for the boundary functions that guarantee existence and stability theorems for the solution of the problem under consideration. In a closed domain, absolute and uniform convergence of the found solution of the inverse problem is shown in the form of a series, as well as series obtained by term-by-term differentiation with respect to t and x two and four times, respectively, depending on the smoothness of the function given the initial conditions. In this case, small denominators arise, which hinder the convergence of these series. It is proved that, depending on the size of the domain, the set of non-zero solutions of the expression in the denominator is not empty. And also, it is shown that if this denominator is equal to zero, then this problem will have a non-trivial solution under homogeneous conditions. It is also proved that the solution of the inverse problem is stable in the norms of the spaces L2​,W2n​ and C(Ω±) with respect to changes in the input data.

Analysis of wave energy device dynamics and power optimization

As an important renewable resource, wave energy has considerable development value and application prospects. How to design a wave energy conversion device to maximize the output power is an urgent problem in its practical application. In this paper, we mainly study the principle of wave energy device capacity, the motion and power model of the constructed wave energy device, and finally design a specific scheme to achieve the maximum output power of wave energy through kinetic analysis, the fourth-order ordinary differential equation solved by the Longacurta method, and the Simpson integral method. At the end of the paper, the developed model is objectively evaluated and the potential value of the developed model in practical wave energy device applications is illustrated.

A meshless generalized finite difference scheme for the stream function formulation of the Naiver-Stokes equations

In this study, a meshless numerical scheme, which is the combination of the generalized finite difference method, the fictitious-nodes technique, and the two-step Newton-Raphson method, was proposed to solve the stream function formulation of the Navier-Stokes equations. To reduce the numbers of unknowns and partial differential equations for the two-dimensional flow field of a viscous incompressible fluid, the stream function formulation, which is a fourth-order nonlinear partial differential equation with only one unknown variable, can be derived by introducing the definition of vorticity and stream function. By applying the generalized finite difference method, the derivatives in the stream function formulation can be simply expressed as a linear combination of functional data and weighting values at several nearest nodes. To deal with the two boundary conditions for the fourth-order partial differential equations, the fictitious-nodes technique was utilized to set up the fictitious nodes outside the computational domain, and cooperate with the generalized finite difference method to form a system of nonlinear algebraic equations. The aforementioned numerical procedure can yield a square Jacobian matrix, so the Newton-Raphson method can be adopted to efficiently acquire the numerical solutions. Several numerical examples were tested to verify the practicability of the proposed meshless numerical scheme.

Dephasing Dynamics in a Non-Equilibrium Fluctuating Environment.

We performed a theoretical study of the dephasing dynamics of a quantum two-state system under the influences of a non-equilibrium fluctuating environment. The effect of the environmental non-equilibrium fluctuations on the quantum system is described by a generalized random telegraph noise (RTN) process, of which the statistical properties are both non-stationary and non-Markovian. Due to the time-homogeneous property in the master equations for the multi-time probability distribution, the decoherence factor induced by the generalized RTN with a modulatable-type memory kernel can be exactly derived by means of a closed fourth-order differential equation with respect to time. In some special limit cases, the decoherence factor recovers to the expression of the previous ones. We analyzed in detail the environmental effect of memory modulation in the dynamical dephasing in four types of dynamics regimes. The results showed that the dynamical dephasing of the quantum system and the conversion between the Markovian and non-Markovian characters in the dephasing dynamics under the influence of the generalized RTN can be effectively modulated via the environmental memory kernel.

IVOMFuse: An image fusion method based on infrared-to-visible object mapping

Due to the significant difference in the structure of infrared and visible images, most of the existing algorithms use globally consistent strategy, which may weaken the contrast of the target region. To solve this problem, we propose an infrared and visible image fusion method based on infrared-to-visible object mapping (IVOMFuse) First of all, we use the probability induced intuitionistic fuzzy C-means algorithm (PIIFCM) to process the infrared image, and extract the target region of the infrared image according to the image structure features. However, because of the poor contrast of the target of the visible image, it can not be segmented directly. For this reason, we map the infrared target region to the visible image, and the target region with the same coordinates of the two images can be obtained very quickly. Secondly, due to the significant differences in image structure, we set a hybrid fusion strategies for different regions. We use the fusion strategy based on the Expectation-Maximization (EM) algorithm to iterate and optimize the probability model for the target region. The fourth-order partial differential equation (FPDE) is used for the background region to decompose the source image into high-frequency texture and low-frequency approximation images respectively. Then, the principal component analysis (PCA) and the average fusion strategy are used to obtain the fused image. Finally, we evaluated our algorithm with fourteen other state-of-the-art image fusion methods on three commonly used datasets, namely TNO, CVC14, and RoadScene. Among the ten evaluation metrics, our algorithm achieved the best six times.