Fourth-order Equation Research Articles

Non-existence results for fourth order Hardy–Hénon equations in dimensions 2 and 3

We consider the fourth order Hardy–Hénon equationΔ2u=|x|σupin Rn with n≥2, σ>−4, and p>1. This is the fourth order analogy of the second order equation −Δu=|x|σup, known as the Hardy–Hénon equation, which was proposed by Hénon in 1973 as a model to study rotating stellar systems in astrophysics. Although there have been many works devoting to the study of the above fourth order equation, the assumption n≥4 is often assumed. In this work, we are interested in classical solutions to the equation in the case of low dimensions, namely 2≤n≤3. Here by a classical solution u we mean u belongs to the class C(Rn)∩C4(Rn∖{0}) if σ<0 and C4(Rn) if σ≥0. We show that if 2≤n≤3 then the equation admits no classical solution. In fact, we are able to provide a single treatment for 2≤n≤4.

The dimension reduction method of two-grid Crank–Nicolson mixed finite element solution coefficient vectors for nonlinear fourth-order reaction diffusion equation with temporal fractional derivative

Herein, we mainly resort to a proper orthogonal decomposition (POD) to study the dimension reduction of unknown solution coefficient vectors in the two-grid Crank–Nicolson mixed finite element (CNMFE) (TGCNMFE) method for the nonlinear fourth-order reaction diffusion equation with temporal fractional derivative and establish a new reduced-dimension extrapolated TGCNMFE (RDETGCNMFE) method. For this purpose, we firstly retrospect the TGCNMFE method for the nonlinear fourth-order reaction diffusion equation and provide the existence, unconditional stability, and error estimates of the TGCNMFE solutions. Thereafter, we use the POD method to reduce the dimensionality of the unknown TGCNMFE solution coefficient vectors of the nonlinear fourth-order reaction diffusion equation and develop the new RDETGCNMFE method, and use matrix analysis to analyze the existence, unconditional stability, and errors of the RDETGCNMFE solutions. Finally, we perform numerical experiments to verify our theoretical results and show the advantages of the RDETGCNMFE method. It is worth noting that the RDETGCNMFE method herein is completely distinguished from the existing ones. Hence, the work in this paper is brand-new.

Peristaltic transport of viscoelastic fluid in curved ducts with ciliated walls

The transport of complex rheological fluids in physiological ducts is often facilitated by the dynamic phenomenon of peristalsis. Additionally, peristaltic transport assisted by cilia plays a significant role in various natural processes such as respiration, circulation, locomotion, and reproduction. This study focuses on magnetically induced flow bounded by non-uniform curved walls, motivated by the importance of peristalsis and micro-organism motility. To characterize the complex rheology of the fluid liner, a viscoelastic model described by the constitutive equation of Jeffrey's fluid is employed. The flow problem is mathematically formulated using curvilinear coordinates. Subsequently, linear transformations and scaling factors are applied to convert the equations into dimensionless form, while considering biotic restrictions such as creeping transport and long wavelength to reduce dependent variables. By utilizing the stream function and cross-differentiation, a fourth-order equation is obtained and numerically approximated using the shooting method. The effects of various parameters on the flow are illustrated through graphs, and a physical interpretation of the graphical results is provided. It is observed that ciliated walls of the channel enhance the velocity and pumping, while trapping phenomena are more pronounced in a non-uniform channel compared to a uniform channel.

Hutchinson Trace Estimation for high-dimensional and high-order Physics-Informed Neural Networks

Physics-Informed Neural Networks (PINNs) have proven effective in solving partial differential equations (PDEs), especially when some data are available by seamlessly blending data and physics. However, extending PINNs to high-dimensional and even high-order PDEs encounters significant challenges due to the computational cost associated with automatic differentiation in the residual loss function calculation. Herein, we address the limitations of PINNs in handling high-dimensional and high-order PDEs by introducing the Hutchinson Trace Estimation (HTE) method. Starting with the second-order high-dimensional PDEs, which are ubiquitous in scientific computing, HTE is applied to transform the calculation of the entire Hessian matrix into a Hessian vector product (HVP). This approach not only alleviates the computational bottleneck via Taylor-mode automatic differentiation but also significantly reduces memory consumption from the Hessian matrix to an HVP’s scalar output. We further showcase HTE’s convergence to the original PINN loss and its unbiased behavior under specific conditions. Comparisons with the Stochastic Dimension Gradient Descent (SDGD) highlight the distinct advantages of HTE, particularly in scenarios with significant variability and variance among dimensions. We further extend the application of HTE to higher-order and higher-dimensional PDEs, specifically addressing the biharmonic equation. By employing tensor-vector products (TVP), HTE efficiently computes the colossal tensor associated with the fourth-order high-dimensional biharmonic equation, saving memory and enabling rapid computation. The effectiveness of HTE is illustrated through experimental setups, demonstrating comparable convergence rates with SDGD under memory and speed constraints. Additionally, HTE proves valuable in accelerating the Gradient-Enhanced PINN (gPINN) version as well as the Biharmonic equation. Overall, HTE opens up a new capability in scientific machine learning for tackling high-order and high-dimensional PDEs.

A novel collocation approach using Chebyshev wavelets for solving fourth-order Emden-Fowler type equations

This research paper presents a new approach using Chebyshev wavelets to solve fourth-order Emden-Fowler type equations. To address the singularity issue at x=0, we initially transform the problem into an equivalent Volterra integral equation. The paper establishes the existence and uniqueness of solutions for each integral equation. For numerical solutions, the proposed technique uses an approximation and collocation method. Notably, this technique eliminates the need to approximate higher derivatives, simplifying the computational process and improving accuracy by avoiding potential errors associated with derivative approximations. Furthermore, the paper includes several examples to demonstrate the effectiveness of the proposed approach. These examples showcase how the technique successfully solves fourth-order Emden-Fowler type equations and highlight its practical applicability.

Error Analysis of the Nonuniform Alikhanov Scheme for the Fourth-Order Fractional Diffusion-Wave Equation

This paper considers the numerical approximation to the fourth-order fractional diffusion-wave equation. Using a separation of variables, we can construct the exact solution for such a problem and then analyze its regularity. The obtained regularity result indicates that the solution behaves as a weak singularity at the initial time. Using the order reduction method, the fourth-order fractional diffusion-wave equation can be rewritten as a coupled system of low order, which is approximated by the nonuniform Alikhanov scheme in time and the finite difference method in space. Furthermore, the H2-norm stability result is obtained. With the help of this result and a priori bounds of the solution, an α-robust error estimate with optimal convergence order is derived. In order to further verify the accuracy of our theoretical analysis, some numerical results are provided.

Unconditionally optimal time two-mesh mixed finite element algorithm for a nonlinear fourth-order distributed-order time fractional diffusion equation

In this article, a fast second-order time two-mesh mixed finite element (TT-MMFE) algorithm is considered to numerically solve the nonlinear fourth-order distributed-order time fractional diffusion equation, where the generated formula for the fractional BDF2 and fractional trapezoidal rule through a free parameter θ (FBT-θ formula) combined with the composite trapezoid formula for the distributed-order derivative is used to discretize the time direction, and the MFE method is utilized to approximate the space direction, respectively. Further, to improve the computing efficiency of our numerical method, the TT-M algorithm is applied to deal with the nonlinear system. For the case of nonsmooth solution, the corrected TT-MMFE scheme by adding starting parts is designed. The existence and uniqueness of the numerical solution are proved in detail. Different from the traditional finite element analysis, the error between the exact solution and the finite element solution is split into temporal error and spatial error by introducing a so-called the error splitting technique, which helps us obtain the spatial error independent of the time step. Finally, numerical examples are given to validate that the TT-MMFE method can reduce computing time, and the corrected TT-MMFE scheme by adding starting parts can recover the convergence rate.

A fast computational technique to solve fourth-order parabolic equations: application to good Boussinesq, Euler-Bernoulli and Benjamin-Ono equations

In this article, we introduce a novel cubic spline method for the numerical solution of a class of fourth-order time-dependent parabolic partial differential equations. The method employs uniform mesh discretization, achieving a spatial accuracy of four and a temporal accuracy of two. Here, the key contribution is that this method can be directly applied to singular parabolic problems due to the consideration of half-step points in spatial direction. The model linear fourth-order partial differential equation is used to discuss the method's stability. The method we propose has multiple advantages, including high accuracy and seamless point-to-point interpolation capabilities. It is a versatile technique that can easily adapt to solve various nonlinear problems without the need to change or linearize the nonlinear terms to produce numerical solutions. Benchmark linear and nonlinear problems like single and double-soliton of the good Boussinesq equation, the Euler-Bernoulli beam model and the Benjamin-Ono equation, are solved.

Effect of axial and radial components of the magnetic field on the electrostatic resistive instabilities in Hall thruster plasma

A two-fluid model is used to investigate the influence of the axial component of the magnetic field on the growth rate of electrostatic resistive instabilities with cross field electron transport in a Hall thruster. The axial component of the magnetic field plays an important role in instabilities. It provides additional confinement to electrons and ions near the channel axis. Also, it helps to protect the walls from the direct impacts of particles, thereby reducing erosion and extending the operational lifetime of the system. A fourth-order dispersion equation is derived using plasma perturbed densities into Poisson's equation to observe the various effects on the growing waves in plasma. It is observed that the growth rate and the real frequency increase with axial and radial components of the magnetic field, respectively. The order of the real frequency of the wave is found to be 106/s. For the fixed value of the azimuthal wavenumber (ky=500/m), the amplitude of the growth rate of the instability dropped to almost 40% if the axial component of the magnetic field is considered. Similarly, the amplitude of the real frequency increases by almost 74% (at ky=500/m) by incorporating the contribution of the axial component of the magnetic field. In addition, it is also observed that the amplitude of the growth rate increases with low values of radial and axial components of the magnetic field, but it decreases at the higher value of the magnetic field due to the resonance of electron cyclotron frequency with plasma frequency.

A fast compact finite difference scheme for the fourth-order diffusion-wave equation

In this paper, the H 2 N 2 method and compact finite difference scheme are proposed for the fourth-order time-fractional diffusion-wave equations. In order to improve the efficiency of calculation, a fast scheme is constructed with utilizing the sum-of-exponentials to approximate the kernel t 1 − γ . Based on the discrete energy method, the Cholesky decomposition method and the reduced-order method, we prove the stability and convergence. When K 1 < 3 2 , the convergence order is O ( τ 3 − γ + h 4 + ϵ ) , where K 1 is diffusion coefficient, γ is the order of fractional derivative, τ is the parameters for the time meshes, h is the parameters for the space meshes and ε is tolerance error. Numerical results further verify the theoretical analysis. It is find that the CPU time is extremely little in our scheme.

Determination of the rational shift value the center of the magnetic system electromagnetic bearing

The article sets the task of determining the rational value of the displacement of the center of the magnetic system of an electromagnetic bearing relative to the axis of rotation, at which, with equal currents in opposite electromagnets, the weight force falling on one axis of the bearing will be completely compensated. To achieve the set, the equations of motion of the rotor in the field of electromagnets are considered. It is shown that the rational value of the displacement of the center of the magnetic system is determined from a fourth-order algebraic equation. The analytical solution of Descartes-Euler of this equation is applied. Analytical expressions are found that allow to determine the rational value of the displacement of the center of the magnetic system relative to the axis of rotation according to the known parameters of the electromagnetic bearing. A study was made of the stability of a three-circuit control system of an electromagnetic bearing when the center of the magnetic system is shifted by a rational value relative to the axis of rotation. It is proved that the displacement of the center of the magnetic system by the calculated value does not affect the stability of the control system of the electromagnetic bearing with the settings of the regulators selected for the central position of the rotor.

On the stability of rotation in an environment with resistance of a free system of two elastically connected rigid bodies

The equations of rotation in a medium with resistance of a free system of two rigid bodies connected by an elastic spherical joint or Hooke's joint are obtained. Under the assumption that the center of mass of solid bodies is located on the third axis of inertia, equations of perturbed motion are obtained. These equations are represented as eight ordinary differential equations with periodic coefficients. For two Lagrange gyroscopes, a fourth-order characteristic equation is obtained. Based on the Liénard-Schipart criterion, written in the form of innors, conditions for the asymptotic stability of uniform rotations of Lagrange gyroscopes are obtained. These conditions are presented in the form of three inequalities. Analytical studies of stability conditions were carried out. It has been proven that the first inequality is always satisfied. From the third inequality it follows that if the gyroscopes have equal axial moments of inertia and rotate with the same angular velocities in different directions or there is no elasticity in the hinge, then the characteristic equation has a multiple root. In this case, the issue of sustainability requires additional research. The conditions for stability in terms of kinetic moments are written out and it is shown that the leading inequality coefficients are positive, from which it follows that stability will always be possible for sufficiently large values of one or two kinetic moments. Similar conclusions were obtained with the Hooke's hinge, and it was also shown that with identical gyroscopes the characteristic equation splits into two equations. In the absence of elasticity, a multiple zero root appears in the hinge and stability requires additional study.

Reconstruction of the time-dependent source in thermal grooving by surface diffusion

In hot polycrystalline materials, when a vertical flat grain boundary meets a horizontal surface the grain boundary forms a groove in the surface. Mathematically modelling features of such thermal grooving mechanism is therefore very important in characterizing polycrystalline materials composed of tiny grains intersecting an external free surface. With this aim in mind, we formulate and investigate a novel inverse problem of reconstructing the unknown time-dependent source term entering the fourth-order parabolic equation of thermal grooving by surface diffusion from a given integral observation. We formulate and prove in Theorems 2.3–2.7 that this linear inverse problem is well-posed. However, in practice, the ideal regularity of data under which the inverse source problem is stable is never satisfied due to the inherent non-smoothness of the measurement. Consequently, this leads to the inverse problem with raw data becoming ill-posed. In order to obtain accurate and stable solutions, we develop and compare two numerical methods, namely, a time-discrete method and an optimization method. We obtain error estimates and convergence rates for the time-discrete method. For the optimization method, an objective functional, which is proved to be Fréchet differentiable, is introduced and the conjugate gradient method (CGM), regularized by the discrepancy principle, is developed to compute the minimizer yielding the source term. The results of two numerical tests illustrate the performance of the two methods for both exact and noisy measured data.

An approximate secular equation of Rayleigh-like waves in coated elastic half-space containing voids

Propagation of Rayleigh-like surface waves is studied in an isotropic elastic solid half-space coated with a thin isotropic elastic solid layer. The half-space and the thin coated layer are in welded contact with each other and contain a uniform distribution of small void pores. Effective boundary condition method is employed to derive an approximate secular equation of second-, third-, and fourth-orders in terms of dimensionless wavenumber. The corresponding secular equations are solved numerically to obtain the speed of propagating Rayleigh-like waves for a particular model. The computed results are presented graphically and compared with those obtained from exact secular equation. The fourth-order approximate secular equation is found to have high accuracy as it provides solutions that are in close vicinity of those obtained from the exact secular equation in the considered model. The presence of voids in the model is found to influence the speed of Rayleigh-like waves theoretically and verified numerically. By ignoring the presence of voids in the model, the secular equation is found to be in complete agreement to the earlier known results in the literature for the corresponding model.

Novel dynamical behaviors of interaction solutions of the new (3+1)-dimensional integrable fourth-order nonlinear equation

The breather, rogue and interaction waves of the (3+1)-dimensional integrable fourth-order nonlinear equation are reviewed. The breather and rouge waves are attained with the aid of the extended homoclinic test. Thereafter, the interaction solutions between a lump wave and a 1-kink or 2-kink soliton are researched. Additionally, four kinds of interaction solutions between lump, kink, and periodic waves through a ” rational-cosh-cos” test function are established. Furthermore, the dynamic attributions of the attained solutions are presented utilizing the graphical analysis.

Unconditional analysis of the linearized second-order time-stepping scheme combined with a mixed element method for a nonlinear time fractional fourth-order wave equation

In this article, we present a numerical algorithm for solving a two-dimensional nonlinear fourth-order time fractional wave model, where a Galerkin mixed finite element method yielded by introducing two auxiliary variables v=ut and σ=Δu−f(u) is used in the spatial direction and a second-order θ scheme with the weighted shifted Grünwald difference (WSGD) formula is applied in the time direction. Utilizing the space-time splitting technique without limiting the relationship between spatial mesh size and temporal step size, we derive the unconditional optimal error estimate in L2-norm and the stability result. Further, for verifying the feasibility and effectiveness of our algorithm with or without starting parts, we implement numerical calculations by choosing nonsmooth solutions.

The “Legendre scalarization” of nonlinear gravity theories

We discuss the proposal of a new method to transform an [Formula: see text] metric gravity theory into a general relativistic theory including an auxiliary scalar field, recently introduced by Cotsakis et al. We argue that (i) the fact that the fourth-order equations of [Formula: see text] metric gravity can be recast (via a Legendre transformation) into Einstein equations without any conformal rescaling has been thoroughly clarified in the previous literature, and (ii) the newly proposed method produces a set of equations that are not equivalent to the original theory. In the conclusion, a comment is added on another aspect of the Legendre transformation which seems to be generally overlooked.

Global existence and blow-up of weak solutions for a fourth-order parabolic equation with gradient nonlinearity

Global existence and blow-up of weak solutions for a fourth-order parabolic equation with gradient nonlinearity

Vibration of Non-Uniform Bernoulli-Euler Beam Under Moving Distributed Masses Resting on Pasternak Elastic Foundation Subjected to Variable Magnitude

This paper investigates the dynamic response of a clamped-clamped non-uniform Bernoulli-Euler beam resting on a Pasternak elastic foundation to variable magnitude moving distributed masses. The predicament is dictated by a partial differential equation of fourth order, which features coefficients that are both variable and singular. The primary aim is to derive an analytical solution for this category of a dynamic problem. To achieve this, we employ the method of Galerkin with a series representation of the Heaviside function to reduce the equation to second-order ordinary differential equations with variable coefficients. We simplify these transformed equations using (i) the Laplace transformation technique in conjunction with convolution theory for solving moving force problems, and (ii) finite element analysis in conjunction with the Newmark method for solving analytically unsolvable moving mass problems due to their harmonic nature. We first solve the moving force problem using the finite element method and compare it against analytical solutions as validation for its accuracy in solving analytically unsolvable moving mass problems. The numerical solution obtained from the finite element method is shown to be comparable favorably against analytical solutions of our moving force problem. Lastly, we calculate displacement response curves for both moving distributed force and mass models at various time t for our dynamical problem presentation purposes.

On solvability of some inverse problems for a nonlocal fourth-order parabolic equation with multiple involution

&lt;abstract&gt;&lt;p&gt;In this paper, the solvability of some inverse problems for a nonlocal analogue of a fourth-order parabolic equation was studied. For this purpose, a nonlocal analogue of the biharmonic operator was introduced. When defining this operator, transformations of the involution type were used. In a parallelepiped, the eigenfunctions and eigenvalues of the Dirichlet type problem for a nonlocal biharmonic operator were studied. The eigenfunctions and eigenvalues for this problem were constructed explicitly and the completeness of the system of eigenfunctions was proved. Two types of inverse problems on finding a solution to the equation and its righthand side were studied. In the two problems, both of the righthand terms depending on the spatial variable and the temporal variable were obtained by using the Fourier variable separation method or reducing it to an integral equation. The theorems for the existence and uniqueness of the solution were proved.&lt;/p&gt;&lt;/abstract&gt;