Higher-order Equations Research Articles

Mixed problem for higher-order equations with fractional derivative and degeneration in both variables

UDC 517.9 We consider an initial-boundary-value problem for a higher-order equation with fractional Riemann – Liouville derivative in a rectangular domain degenerating in both variables. The solution to the problem is constructed in the explicit form by the method of separation of variables. Uniqueness is proved by the spectral method.

Vibration of nonlocal strain gradient functionally graded nonlinear nanobeams using a novel locally adaptive strong quadrature element method

The primary objective of this paper is to propose a novel method to derive Differential Quadrature Method matrices with several degrees of freedom at the boundaries that can be used to build Strong Quadrature Elements to solve fourth and higher-order equations of motion. The proposed method, referred to as Locally adaptive Strong Quadrature Element Method, is applied to higher-order equations of motion for nonlinear graded Timoshenko and Euler-Bernoulli nanobeams formulated using the Second Strain Gradient Theory or the Nonlocal Strain Gradient Theory. To limit the formulation complexity, the proposed approach is based on the regular formulation of the differential quadrature method combined with custom-built transfer matrices. Moreover, it does not require a different formulation for fourth and sixth-order equations and can be extended beyond sixth-order equations. Validation was carried out using examples from the literature as well as data obtained using the classical Locally adaptive Quadrature Element Method. Both linear and nonlinear frequencies were evaluated for a large number of configurations and boundary conditions. The proposed approach resulted in good accuracy and a convergence speed comparable to the conventional Locally adaptive Quadrature Element Method.

Algorithms analysis for solving geometrically nonlinear mechanics problems in the scheme of the semi-analytical finite element method

The effectiveness of using the semi-analytical finite element method (SAFEM) to research geometrically nonlinear construction mechanics problems for 3D bodies of revolution under the arbitrary loads based on a basic ring finite element is considered. An estimate of the rational application parameters of algorithms for taking into account the geometric nonlinearity of a defined above structures class, which has been obtained.
 In the process of numerically solving spatial problems of the theory of elasticity and plasticity with finite displacements, the choice of rational algorithms for solving systems of nonlinear equations is of fundamental importance. It is conditioned by the need of determining the coordinates of the discrete model in the actual configuration and changing the metric characteristics of the finite elements, which, in its turn, leads to the necessity for multiple solutions of systems of nonlinear equations of high order. Due to the introduction of additional hypotheses that do not reduce the accuracy of the approximation: the representation of deformations and stresses in physical terms and in accordance with the moment scheme of the finite element (MSFEM), it is possible, on the one hand, to avoid the time-consuming procedure of numerical integration over the cross-sectional area of the finite element, on the other hand- to maintain a high efficiency of spatial discretization. An important stage in the implementation of computer systems for solving spatial problems is the selection of optimal, from the point of the solution convergence speed and algorithms for solving equilibrium equations. The specificity of the algebraic equations of the SAFEM is conditioned by violation of the trigonometric function orthogonality in the space of the elasticity operator for bodies with variable stiffness and mass parameters along the guide. The clearly defined block structure of the stiffness matrix became the basis for using algorithms combining direct and iterative methods of solving. The reliability of the obtained results and the effectiveness of the approach are confirmed by the solution of a wide range of control examples under various boundary conditions and external loads.

A Priori Estimate of Solutions of One Boundary-Value Problem in a Strip for a Higher-Order Degenerate Elliptic Equation

In this paper, we prove coercive a priori estimates of solutions of a Dirichlet-type boundary-value problem in a strip for a certain higher-order degenerate elliptic equation containing weighted derivatives of a special form up to the order 2m and ordinary partial derivatives up to the order 2k−1 under the condition 2m > 2k −1. At the boundary of the strip, Dirichlet-type conditions are imposed. A coercive a priori estimate for solutions of the problem considered in special weighted Sobolev-type spaces is obtained.

Modified couple stress flexural–flexural quasi-static pull-in analysis of large deformable cantilever-based micro-gyroscopes

This paper aims to develop a size-dependent large deformable model to study the flexural–flexural quasi-static motion of cantilever-based micro-gyroscopes based on the modified couple stress theory. The micro-cantilever is assumed to undergo non-planar bending–bending–twisting motion. Based on this assumption, the displacement field associated with a point placed on the cross-section of the micro-beam is introduced. Employing the introduced displacement field, the components of the Green–Lagrange strain and curvature tensors are obtained. Neglecting the terms with the time derivatives, the nonlinear, coupled and higher-order equations governing on three-dimensional quasi-static motion of in-extensional cantilever-based micro-gyroscopes and their corresponding boundary conditions are then obtained. Normalizing the strain and kinetic energies as well as the works done by the external forces, the reduced equations governing the quasi-static motion of micro-gyroscopes are obtained through employing the Ritz method with linear mode-shapes of clamped-free beams as the approximating functions. Vanishing the Jacobian of the reduced equations, the governing stability equations of the system are then obtained. Afterward, employing the achieved reduced equations, both the stable and unstable branches of the micro-gyroscope equilibrium paths are extracted. To validate the accuracy of the present model, several verifications have been conducted. The performed verifications demonstrate that the current model can be considered as a promising tool in capturing the influence of small scales on micro-gyroscopes undergoing extremely large deformations.

Higher-order rogue waves solutions of the modified Gerdjikov–Ivanov equation with dispersion via generalized Darboux transformation

Higher-order rogue wave solutions have vital applications in nonlinear physics. The new modified Gerdjikov–Ivanov equation with dispersion is derived for the first time in this paper. Based on the extend Lax pair, the localized waves of the new model are obtained via generalized Darboux transformation, the higher-order rogue solutions are presented graphically and their dynamic properties are analyzed. The results obtained have spurred the study of higher-order dispersion equations in nonlinear optics.

A New Numerical Approach Using Chebyshev Third Kind Polynomial for Solving Integrodifferential Equations of Higher Order

There are several classifications of linear Integral Equations. Some of them include; Voltera Integral Equations, Fredholm Linear Integral Equations, Fredholm-Voltera Integrodifferential. In the past, solutions of higher-order Fredholm-Volterra Integrodifferential Equations [FVIE] have been presented. However, this work uses a computational techniques premised on the third kind Chebyshev polynomials method. The performance of the results for distinctive degrees of approximation (M) of the trial solution is cautiously studied and comparisons have been additionally made between the approximate/estimated and exact/definite solution at different intervals of the problems under consideration. Modelled Problems have been provided to illustrate the performance and relevance of the techniques. However, it turned out that as M increases, the outcomes received after every iteration get closer to the exact solution in all of the problems considered. The results of the experiments are therefore visible from the tables of errors and the graphical representation presented in this work.

On the global behavior of the rational difference equation \(y_{n+1}=\frac{\alpha_n+y_{n-r}}{\alpha_n+y_{n-k}}\)

In this article, we study the global behavior of the following higher-order nonautonomous rational difference equation 
 \[
 y_{n+1}=\frac{\alpha_n+y_{n-r}}{\alpha_n+y_{n-k}},\quad n=0,1,...,
 \]
 where \(\left\{\alpha_n\right\}_{n\geq0}\) is a bounded sequence of
 positive numbers, \(k,r\) are nonnegative integers such that \(r

Approximated Risk Probability of having Atherosclerosis, Cardiovascular Disease, and Stroke during the COVID-19 Period Using the Pre-COVID Period’S Dataset as Baseline of Calculations and Higher-Order Perturbation Equations from Quantum Mechanics Based on GH-Method: Math-Physical Medicine (No. 465)

This paper describes the author’s assessment on his risk probability percentage of having atherosclerosis, cardiovascular disease (CVD), and stroke. During the 7-quarters of the COVID-19 period from 2/5/2020 to 11/4/2021, he utilized his earlier 7-quarters data from 5/5/2018 to 2/4/2020 or the “pre-COVID period” and higher-order perturbation theory. The purpose of this study is to predict the present and future period’s CVD/stroke risks based on the previous period’s CVD/stroke risks via an effective approximation method of perturbation theory. In summary, this approximation method of perturbation equation from quantum mechanics offers high prediction accuracies on the present and future period’s (the “COVID period”) risks using the past pre-COVID period’s dataset and waveform as baseline calculations. The three orders of perturbation equations have provided the following high prediction accuracies in comparison against the COVID period’s calculated risks which includes the past 5-quarters of calculated risks and the future 2-quarters speculated risks of the COVID period.

Approximated Health Age Calculation Using Higher-Order Perturbation Equations from Quantum Mechanics and the 90-Days Moving Average Metabolism Index as its Primary Perturbation Factor Based on GH-Method: Math-Physical Medicine (No. 466)

The author became interested in geriatrics in early 2019, especially in regard to longevity. During the period of 2010-2013, he self-studied metabolism, endocrinology, chronic diseases, and food nutrition. In 2014, he applied topology concept and engineering finite element method to develop a mathematical model of metabolism. His collected big data of ~2 million on metabolism, chronic diseases, and lifestyle details allowed him to extend his research work into the area of longevity. In January 2020, he published his first geriatric paper, No. 223, regarding effective health age. Back in July 2010, he developed a simplified APP (application software program) on the iPhone for estimating a patients’ effective health age (Health Age) with or without chronic diseases to compare against the biological real age (Real Age). He then published his findings in paper, No. 292, where the data were utilized from four key medical conditions based on the health examination reports of 4 biomarkers weight, glucose, blood pressure, and lipids; along with the input by the patient of six lifestyle details involving diet, water intake, exercise, sleep, stress, and daily life routines. The APP can instantly calculate and show the metabolism index (MI) score and Health Age on the iPhone. Furthermore, he published two papers, No. 313 and No. 323, regarding the amplification factor used in the equation for effective health age.

Abundant exact solutions of higher-order dispersion variable coefficient KdV equation

Abstract In this article, various exact solutions of the fifth-order variable coefficient KdV equation with higher-order dispersion term are studied. Because of the complexity of the exact solution of the variable coefficient t, it has a certain influence on the tension waves at the fluid interface on the gravity surface. First, the bilinear KdV equation is derived by using the Hirota bilinear method, and four mixed solutions consisting of positive quartic function, quadratic function, exponential function, and hyperbolic function are constructed. Second, the linear superposition principle is used to obtain the resonance multisoliton solution, and two cases are taken as examples to illustrate the study of resonance multi soliton solution. In addition, 3D images and contour images are drawn by mathematical symbol calculation and appropriate parameters, and the process of tension fluctuation is vividly explained by physical phenomena. The results obtained greatly expand the exact solution of the KdV equation in the existing literature and enable us to understand nonlinear dynamical systems more deeply.

Solitary Wave Interaction with a Floating Pontoon Based on Boussinesq Model and CFD-Based Simulations

A mathematical model of solitary wave interaction with a pontoon-type rigid floating structure over a flat bottom is formulated based on Boussinesq-type equations under weakly nonlinear dispersive waves. Based on the higher-order Boussinesq equations, the solitary wave equation is derived, and a semi-analytical solution is obtained using the perturbation technique. On the other hand, brief descriptions of the application of wave2Foam and OceanWave3D on the aforementioned problem are presented. The analytical solitary wave profiles in the outer region are compared with Computational Fluid Dynamics (CFD) and OceanWave 3D model simulations in different cases. The comparison shows a good level of agreement between analytical, wave2Foam, and OceanWave3D. In addition, based on the wave2Foam and coupled OceanWave3D model, the horizontal, vertical wave forces, and the pressure distributions around the pontoon are analysed. Further, the effect of the Ursell number, pontoon length, and water depth on the solitary wave profiles are analysed based on the analytical solution. The paper validates each of the three models and performs intercomparison among them to assess their fidelity and computational burden.

Large time behavior of ODE type solutions to higher-order semilinear parabolic equations with small initial data

Let u be a solution to the Cauchy problem for semilinear parabolic equations(Pm){∂tu+(−Δ)mu=|u|αinRN×(0,∞),u(x,0)=λ+φ(x)>0inRN, where m=1,2,⋯,0<α<1,N≥1,λ>0,φ∈BC(RN)∩Lr(RN) with 1≤r<∞. In the case of m=1, by the comparison principle, one can easily show that the solution u to problem (Pm) behaves like a positive solution to ODE ζ′=ζα in (0,∞) as t→∞. In the case of m≥2, because of the lack of the comparison principle, it is not obvious that the solution u to problem (Pm) behaves like a solution to the ODE as t→∞. In this paper, by imposing a smallness condition‖φ‖L∞(RN)<c with a certain constant c dependent on m,α and λ, we prove that the solution u behaves like a solution to the ODE as t→∞. Furthermore, we obtain the precise description of the large time behavior of the solution u and reveal the relationship between the diffusion effect (Pm) has.

Dirichlet problem for higher-order equations in a ring domain

In this work, we study the solvability conditions and explicit solution of the Dirichlet problem for the inhomogeneous polyanalytic equation of arbitrary order in the ring domain. The higher-order problem is converted into the system of lower-degree ones.

Analytical Solitonic Solutions of Higher-Order Non-Linear Ito Equation

Analytical Solitonic Solutions of Higher-Order Non-Linear Ito Equation

On the control issues for higher-order nonlinear dispersive equations on the circle

The local and global control results for a general higher-order KdV-type operator posed on the unit circle are presented. Using spectral analysis, we are able to prove local results, that is, the equation is locally controllable and exponentially stable. To extend the local results to the global one we captured the smoothing properties of the Bourgain spaces, the so-called propagation of regularity, which are proved with a new perspective. These propagation, together with the Strichartz estimates, are the key to extending the local control properties to the global one, precisely, higher-order KdV-type equations are globally controllable and exponentially stabilizable in the Sobolev space Hs(T) for any s≥0. Our results recover previous results in the literature for the KdV and Kawahara equations and extend, for a general higher-order operator of KdV-type, the Strichartz estimates as well as the propagation results, which are the main novelties of this work.

Soliton solutions of weakly bound states for higher-order Ito equation

Soliton solutions of weakly bound states for higher-order Ito equation

Lump, soliton, and interaction solutions to a generalized two-mode higher-order nonlinear evolution equation in plasma physics

Lump, soliton, and interaction solutions to a generalized two-mode higher-order nonlinear evolution equation in plasma physics

A deep learning method for solving high-order nonlinear soliton equations

We propose an effective scheme of the deep learning method for high-order nonlinear soliton equations and explore the influence of activation functions on the calculation results for higher-order nonlinear soliton equations. The physics-informed neural networks approximate the solution of the equation under the conditions of differential operator, initial condition and boundary condition. We apply this method to high-order nonlinear soliton equations, and verify its efficiency by solving the fourth-order Boussinesq equation and the fifth-order Korteweg–de Vries equation. The results show that the deep learning method can be used to solve high-order nonlinear soliton equations and reveal the interaction between solitons.

Theoretical Approach to Synthetic Accelerations for the Solution of Neutron Transport Equation

A discrete ordinates neutron transport solution usually consists of solving alternately the transport and the diffusion equation. Here, the diffusion equation which is a P₁ approximation to the transport equation is solved only for an acceleration of transport solution to achieve the computational efficiency. The derivation of various low-order accelerating equations are generalized by using an operator form of acceleration in this paper. The source iteration, P₀, P₁, and P₂ acceleration schemes are derived and their spectral radii are calculated and compared by using Fourier convergent analysis. The result is that the P₁ acceleration outperforms the other schemes in accelerating transport solution. These results confirm that one cannot simply assume that the replacement of P₁ with a higher order equation would enhance the acceleration efficiency.