Fourth-order Equation Research Articles

Solving a Class of High-Order Elliptic PDEs Using Deep Neural Networks Based on Its Coupled Scheme

Deep learning—in particular, deep neural networks (DNNs)—as a mesh-free and self-adapting method has demonstrated its great potential in the field of scientific computation. In this work, inspired by the Deep Ritz method proposed by Weinan E et al. to solve a class of variational problems that generally stem from partial differential equations, we present a coupled deep neural network (CDNN) to solve the fourth-order biharmonic equation by splitting it into two well-posed Poisson’s problems, and then design a hybrid loss function for this method that can make efficiently the optimization of DNN easier and reduce the computer resources. In addition, a new activation function based on Fourier theory is introduced for our CDNN method. This activation function can reduce significantly the approximation error of the DNN. Finally, some numerical experiments are carried out to demonstrate the feasibility and efficiency of the CDNN method for the biharmonic equation in various cases.

Reheating and particle creation in unimodular f(R, T) gravity

We study cosmological inflation and reheating in the unimodular f(R, T) gravity. During the reheating era, which takes place just after the end of inflation, the energy density of inflaton is converted to radiation energy through, for instance, rapid oscillation about the minimum of the potential. We quantify our investigation by calculating the reheating temperature. This quantity is written in terms of the spectral index and the power spectrum, which provides a suitable framework to constrain the parameter space of the model. We discuss the massless particle creation for a spatially flat, homogeneous and isotropic universe in the context of unimodular f(R, T) gravity. We obtain the number of created particles per unit volume of space. To avoid the complexity of solving the fourth order equations, we analyze the reheating in the Einstein frame by considering some specific illustrative examples and obtain the corresponding analytical solutions in addition to some numerical estimations.

Exact Solutions for Vector Phase-Matching Conditions in Nonlinear Uniaxial Crystals

The transcendental equations of vector phase matching are transformed into a fourth-order polynomial equation that admits an analytical solution. The real roots of this equation provide the optical axis orientations that are useful for efficient down-conversion in nonlinear uniaxial crystals. The production of entangled photon pairs is discussed in both collinear and non-collinear configurations of the spontaneous parametric down-conversion (SPDC) process. Degenerate and non-degenerate cases are also distinguished. As a practical example, SPDC processes of type-I and type-II are studied for beta-barium borate (BBO) crystals. The predictions are in very good agreement with experimental measurements already reported in the literature and include theoretical results of other authors as particular cases. Some properties that seem to be exclusive to BBO crystals are reported; the experimental verification of the latter would allow a better characterization of these crystals.

Compact difference scheme for time-fractional nonlinear fourth-order diffusion equation with time delay

We construct a compact difference scheme for two-dimensional time-fractional nonlinear fourth-order diffusion equation with time delay. By choosing the second-order spatial derivative of the original unknown as an additional variable, the fourth-order problem is transformed into a second-order system. Then the linearized compact difference scheme is formulated by employing the L2−1σ formula and the compact difference operator to approximate temporal Caputo derivative and spatial second-order derivatives, respectively. The unique solvability of the proposed scheme is proved. Then the scheme is proved to be stable and convergent with the rate of second order in time and fourth order in space. The efficiency of the scheme is supported by some numerical experiments.

Direct dynamic tomographic reconstruction without explicit blood input function

With dynamic positron emission tomography (PET), we study the metabolic processes by monitoring the uptake of a radioactive tracer. The goal is to reconstruct the time-activity curves (TACs) of the voxels, from which the relevant kinetic parameters can be obtained. These curves are assumed to have the algebraic form of a pre-determined kinetic model. Plausible algebraic forms have a limited number of free parameters and thus can be used even for low-statistic measurements. The kinetic model typically involves the amount of radiotracer in the blood, which should be determined either by direct measurement or by extracting it from the PET data using image-derived or model-based methods. However, the direct measurement of the blood concentration is complicated, and the results of the image-derived and the model-based methods are also not reliable because of the partial volume effect and the unknown fraction of blood in the voxels. Moreover, in direct dynamic tomography, the kinetic model is fit from the very beginning of the reconstruction, thus the blood input function is needed even before the image-derived or model-based approaches can provide it. In this paper, we propose a method that is based on compartmental modeling and the Feng blood input function model defined by a fourth-order exponential equation with a pair of repeated eigenvalues, but does not require the blood input function and the fraction of blood parameters explicitly. Thus, the model can be used in cases when we wish to have the robustness and advantages of compartmental modeling, but no blood input function measurement is available. The test results show that the added error in the TACs of not knowing the blood input is reduced in comparison with the spline-based TAC representation. The method can be applied in reference tissue analysis and in image-derived input function approaches.

A wavelet collocation method based on Gegenbauer scaling function for solving fourth-order time-fractional integro-differential equations with a weakly singular kernel

A high resolution wavelet collocation method based on Gegenbauer polynomials is proposed for the solution of fourth-order time-fractional integro-differential equations (FTFIDE) with a weakly singular kernel. A Riemann-Liouville fractional integral operator for the Gegenbauer scaling function (RLFIO-G) is constructed using the definition of Riemann-Liouville (R-L) operator with the aid of Gegenbauer scaling function. The application of Gegenbauer scaling function and RLFIO-G to FTFIDE gives a system of linear algebraic equations, which can be quickly solved for unknown coefficients. With the aid of these coefficients, we get the approximate solution. We have also established the convergence analysis of the proposed method. We have tested the presented method on some numerical examples to demonstrate the accuracy. Moreover, we have compared the developed method with the existing method to conclude the superiority of the proposed method.

An efficient computational approach for solving two-dimensional extended Fisher–Kolmogorov equation

The extended Fisher–Kolmogorov (EFK) system is a strong nonlinear fourth-order reaction diffusion evolution equation. So far, the numerical methods for solving this problem are only a few. This paper deals with the finite difference solution to the two-dimensional EFK equation. The scheme is proved to be uniquely solvable and second-order convergent in both time and space in maximum norm. Finally, some numerical examples have been illustrated to verify the efficiency and simplicity of the proposed technique.

Stackelberg-Nash exact controllability for the Kuramoto-Sivashinsky equation with boundary and distributed controls

This paper deals with a multi-objective control problem for the Kuramoto-Sivashinsky equation by following a Stackelberg-Nash strategy. We have a distributed control called Leader, and two boundary controls called Followers, each of them has to act over the equation to influence the behavior of the state in a particular way, by reaching or approaching to many targets at once. To be more precise, the Leader wants to drive the solution to a prescribed target at a final time, and the followers have to minimize some given cost functionals, adapting themselves to what the Leader wants. The main difficulty here is that, since the Followers are in the boundary, the problem turns to be equivalent to prove a partial null controllability result for a system of nonlinear fourth-order equations with boundary coupling terms.

Q-Pearson pair and moments in q-deformed ensembles

The generalisation of continuous orthogonal polynomial ensembles from random matrix theory to the q-lattice setting is considered. We take up the task of initiating a systematic study of the corresponding moments of the density from two complementary viewpoints. The first requires knowledge of the ensemble average with respect to a general Schur polynomial, from which the spectral moments follow as a corollary. In the case of little q-Laguerre weight, a particular \(_3 \phi _2\) basic hypergeometric polynomial is used to express density moments. The second approach is to study the q-Laplace transform of the un-normalised measure. Using integrability properties associated with the q-Pearson equation for the q-classical weights, a fourth-order q-difference equation is obtained, generalising a result of Ledoux in the continuous classical cases.

Improved blowup time estimates for fourth-order damped wave equations with strain term and arbitrary positive initial energy

We propose a new differential inequality that improve the upper bound of the blowup time estimate for nonlinear fourth-order damped wave equations with strain term and arbitrary positive initial energy. We also give two new initial conditions to expand the range of the initial data leading to the finite time blowup of solutions. We obtain a sharp result of finite time blowup for the special case of the new differential inequality.We illustrate our results with some simulations.

Active exterior cloaking of an inclusion with evanescent multipole devices for flexural waves in thin plates.

We present an active exterior cloak for flexural waves propagating in a Kirchhoff plate of infinite extent. The evanescent multipole devices are characterized by Macdonald functions of the required order, which, assuming time-harmonic vibrations, are solutions of the fourth-order biharmonic equation. It is shown that in the region of interfering waves, which emanate from the devices, a field is recreated which cancels the incident wave to yield a region of ‘stillness’. An inclusion is then positioned in this region for further investigation, with additional attention given to the boundary condition.This article is part of the theme issue ‘Wave generation and transmission in multi-scale complex media and structured metamaterials (part 2)’.

Nontrivial Solutions of a Class of Fourth-Order Elliptic Problems with Potentials

This paper deals with a fourth-order elliptic equation with Dirichlet boundary conditions. Using a variant form of the mountain pass theorem, we prove the existence of nontrivial solutions to this problem. Furthermore, we discuss the fundamental properties of the representation of the solution by considering two cases. Our results not only make previous results more general but also show new insights into fourth-order elliptic problems.

An energy-conserving finite element method for nonlinear fourth-order wave equations

In this paper, an energy-conserving finite element method is developed and intensively analyzed for a class of nonlinear fourth-order wave equations in a general sense for the first time, where the two-level, Crank-Nicolson type of temporal discretization scheme is designed to cooperate with the Lagrange finite element approximation in space in order to achieve the conservation of discrete energy at each time step. The energy conservation is crucial in engineering field to preserve the total energy as constant for dynamic vibration problems of beams and thin plates that can be modeled by the presented wave equations. In addition, the optimal spatial convergence properties in both L2- and H1-norm and the second-order temporal approximation rate are obtained for finite element solutions of u (deflection), Δu (bending moment) and/or ut (deflection speed) at the same time. Numerical experiments are carried out to validate all attained theoretical results.

Fast compact finite difference schemes on graded meshes for fourth-order multi-term fractional sub-diffusion equations with the first Dirichlet boundary conditions

In this paper, fast compact finite difference schemes are derived for fourth-order multi-term fractional sub-diffusion equations with initial singularity under the first Dirichlet boundary conditions. In contrast to the direct scheme, the fast algorithm adopted to approximate the Caputo derivative reduces the computation costs effectively. Sharp error estimate of the proposed scheme for linear models is rigorously presented by the energy method. Furthermore, to handle more intricate nonlinear models, a crucial Grönwall inequality is then deduced to analyse the stability and convergence of the linearized scheme. It is worth pointing out that the Grönwall inequality is also helpful in numerical analysis of multi-step schemes for other problems, such as, integro-differential equations with multiple fractional derivatives. Ultimately, numerical examples are provided to verify the efficiency of the established difference schemes.

Global well-posedness of fourth-order Petrovsky equation with weak and strong damping terms

In this paper, a semilinear fourth-order Petrovsky equation with nonlinear weak damping term and linear strong damping term is considered in the frame of the potential well. We prove the existence and uniqueness of the global strong solution with sub-critical initial energy. In addition, we obtain the global existence, asymptotic behaviour and finite time blowup for the weak (non-steady state) solution with a critical initial energy level.

EXPONENTIAL GROWTH OF SOLUTIONS FOR A VARIABLE-EXPONENT FOURTH-ORDER VISCOELASTIC EQUATION WITH NONLINEAR BOUNDARY FEEDBACK

In this paper we study a variable-exponent fourth-order viscoelastic equation of the form$$|u_{t}|^{\rho(x)}u_{tt}+\Delta[(a+b|\Delta u|^{m(x)-2})\Delta u]-\int_{0}^{t}g(t-s)\Delta^{2}u(s)ds=|u|^{p(x)-2}u,$$in a bounded domain of $R^{n}$. Under suitable conditions on variable exponents and initial data, we prove that the solutions will grow up as an exponential function with positive initial energy level. Our result improves and extends many earlier results in the literature such as the on by Mahdi and Hakem (Ser. Math. Inform. 2020, https://doi.org/10.22190/FUMI2003647M).

Specializations of partial differential equations for Feynman integrals

Starting from the Mellin–Barnes integral representation of a Feynman integral depending on a set of kinematic variables zi, we derive a system of partial differential equations w.r.t. new variables xj, which parameterize the differentiable constraints zi=yi(xj). In our algorithm, the powers of propagators can be considered as arbitrary parameters. Our algorithm can also be used for the reduction of multiple hypergeometric sums to sums of lower dimension, finding special values and reduction equations of hypergeometric functions in a singular locus of continuous variables, or finding systems of partial differential equations for master integrals with arbitrary powers of propagators. As an illustration, we produce a differential equation of fourth order in one variable for the one-loop two-point Feynman diagram with two different masses and arbitrary propagator powers.

Stability, Boundedness, and Square Integrability of Solutions of Neutral Fourth-Order Differential Equations

The purpose of this paper is to establish a new result, which guarantees the asymptotic stability and boundedness of the zero solution and the square integrability of solutions and their derivatives to neutral type nonlinear differential equations of fourth-order. We illustrate our results by an example at the end of the paper.

Convergence analysis of the Galerkin finite element method for the fourth-order Rosenau equation

In this paper, the Galerkin finite element method (FEM) for solving the fourth-order Rosenau equation is proposed with the bicubic Hermite element. The existence and uniqueness of the approximate solution is demonstrated through the Browder fixed point theorem and the convergent result of order O(h2) in H2-norm is derived for the semi-discrete scheme. The linearized fully-discrete scheme is constructed and its error estimation of order O(h2+τ2) in H2-norm is deduced. Finally, some numerical results are provided to confirm our theoretical analysis. Here and later h and τ denote the mesh size and the time step, respectively.

Oscillation results via comparison theorems for fourth-order delay three-terms difference equations

In this paper we establish sufficient conditions for the oscillation of all solutions of equation Δ4x(n)+p(n)Δx(n+1)+q(n)x(n-τ)=0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\varDelta ^4x(n)+p(n)\\varDelta x(n+1)+q(n)x(n-\ au )=0 \\end{aligned}$$\\end{document}via comparison with some first order delay difference equations whose oscillatory characters are known. The presented criterion is easily verifiable. Examples are also given to illustrate the main result.