Second-order Nonlinear Equations Research Articles

On the Theory of Entropy Solutions of Nonlinear Degenerate Parabolic Equations

We consider a second-order nonlinear degenerate parabolic equation in the case when the flux vector and the nonstrictly increasing diffusion function are merely continuous. In the case of zero diffusion, this equation degenerates into a first order quasilinear equation (conservation law). It is known that in the general case under consideration an entropy solution (in the sense of Kruzhkov-Carrillo) of the Cauchy problem can be non-unique. Therefore, it is important to study special entropy solutions of the Cauchy problem and to find additional conditions on the input data of the problem that are sufficient for uniqueness. In this paper, we obtain some new results in this direction. Namely, the existence of the largest and the smallest entropy solutions of the Cauchy problem is proved. With the help of this result, the uniqueness of the entropy solution with periodic initial data is established. More generally, the comparison principle is proved for entropy suband super-solutions, in the case when at least one of the initial functions is periodic. The obtained results are generalization of the results known for conservation laws to the parabolic case.

Existence of positive solutions for second-order nonlinear neutral dynamic equations on time scales

AbstractIn this article we study the existence of positive solutions for second-order nonlinear neutral dynamic equations on time scales. The main tool employed here is Schauder’s fixed point theorem. The results obtained here extend the work of Culakova, Hanustiakova and Olach [12]. Two examples are also given to illustrate this work.

Instantaneous blow-up of solutions to the Cauchy problem for the fractional Khokhlov-Zabolotskaya equation

Abstract In this paper, we consider the Cauchy problem for a second-order nonlinear equation with mixed fractional derivatives related to the fractional Khokhlov-Zabolotskaya equation. We prove the nonexistence of a classical local in time solution. The obtained instantaneous blow-up result is proved via the nonlinear capacity method.

Improved Oscillation Results for Functional Nonlinear Dynamic Equations of Second Order

In this paper, the functional dynamic equation of second order is studied on an arbitrary time scale under milder restrictions without the assumed conditions in the recent literature. The Nehari, Hille, and Ohriska type oscillation criteria of the equation are investigated. The presented results confirm that the study of the equation in this formula is superior to other previous studies. Some examples are addressed to demonstrate the finding.

A Novel Alignment Method for SINS with Large Misalignment Angles Based on EKF2 and AFIS.

In order to achieve the fine alignment of strapdown inertial navigation (SINS) under large misalignment angles, a novel filtering alignment method is proposed based on the second-order extended Kalman filter (EKF2) and adaptive fuzzy inference system (AFIS). Firstly, the quaternion is employed to represent the attitude errors of SINS. A second-order nonlinear state equation is made based on the nonlinear velocity error model and attitude error model, and the linear measurement equation is based on the velocity outputs from SINS. Then, the filtering scheme is designed based on EKF2 and AFIS. The error estimation and fine alignment can be achieved by using the proposed filtering scheme. The results of Monte Carlo Simulation show that the errors of pitch, roll and yaw misalignment angles quickly decrease to about 14″, 15″ and 7.62′ respectively in 350 s under the condition of any misalignment angles with pitch error from −40° to 40°, roll error from −40° to 40°, and yaw error from −50° to 50°. Even when the initial misalignment angles are all very large such as (80°, 120°, 170°), the proposed nonlinear alignment method still can converge normally by utilizing the adaptive fuzzy inference system (AFIS) to adjust the covariance matrix Pk/k−1. Finally, the turntable experiment was performed, and the effectiveness and superiority of the proposed method were further verified by compared with other nonlinear methods.

Solving second-order nonlinear evolution partial differential equations using deep learning**The project is supported by the National Natural Science Foundation of China (No. 11675054), Shanghai Collaborative Innovation Center of Trustworthy Software for Internet of Things (Grant No. ZF1213) and Science and Technology Commission of Shanghai Municipality (No. 18dz2271000).

Solving nonlinear evolution partial differential equations has been a longstanding computational challenge. In this paper, we present a universal paradigm of learning the system and extracting patterns from data generated from experiments. Specifically, this framework approximates the latent solution with a deep neural network, which is trained with the constraint of underlying physical laws usually expressed by some equations. In particular, we test the effectiveness of the approach for the Burgers’ equation used as an example of second-order nonlinear evolution equations under different initial and boundary conditions. The results also indicate that for soliton solutions, the model training costs significantly less time than other initial conditions.

Generalized sedeonic equations of hydrodynamics

We present a generalization of the equations of hydrodynamics based on the noncommutative algebra of space-time sedeons. It is shown that for vortex-less flow the system of Euler and continuity equations is represented as a single nonlinear sedeonic second-order wave equation for scalar and vector potentials, which is naturally generalized on viscous and vortex flows. As a result we obtained the closed system of four equations describing the diffusion damping of translational and vortex motions. The main peculiarities of the obtained equations are illustrated on the basis of the plane wave solutions describing the propagation of sound waves.

Oscillatory behavior for nonlinear homogeneous neutral difference equations of second order with coefficient changing sign

In this article, we obtain sufficient conditions so that all solutions of the neutral difference equation $$ \Delta^{2}\big(y_n-p_n L(y_{n-s})\big) + q_nG(y_{n-k})=0, $$ and all unbounded solutions of the neutral difference equation $$ \Delta^{2}\big(y_n-p_n L(y_{n-s})\big) + q_nG(y_{n-k}) -u_nH(y_{\alpha(n)})=0 $$ are oscillatory, where \(\Delta y_n = y_{n+1}-y_n\), \(\Delta^2 y_n =\Delta(\Delta y_n)\). Different types of super linear and sub linear conditions are imposed on \(G\) to prevent the solution approaching zero or \(\pm \infty\). For more information see https://ejde.math.txstate.edu/Volumes/2020/87/abstr.html

Correspondence between multiwavelet shrinkage and nonlinear diffusion

Wavelet/frame shrinkage and nonlinear diffusion filtering are two popular methods for signal and image denoising. The relationship between these two methods has been studied recently. This relationship leads to new types of diffusion equations and helps to design the wavelet/frame-inspired diffusivity functions, and on the other hand it helps to design the diffusion-inspired shrinkage functions with a better performance in signal and image denoising.Multiwavelets have important properties such as orthogonality, short support, and symmetry, etc. that scalar orthogonal wavelets cannot possess simultaneously. There is rich literature on the theoretical study, construction and applications of multiwavelets. In particular, it has been shown that multiwavelets perform better than the scalar wavelets in signal and image denoising. Recently multiwavelet denoising has been applied in different applications including rolling bearing fault detection and study of load spectrum of computer numerical control lathe. Therefore it is worth to further study multiwavelet denoising.In this paper we investigate the correspondence between multiwavelet denoising and nonlinear diffusion. We show that the multiwavelet shrinkages of the commonly used CL(2) and DGHM multiwavelets are associated with a second-order nonlinear diffusion equation. We also derive high-order nonlinear diffusion equations associated with general multiwavelet shrinkages. The experimental results carried out in this paper show that the diffusion-inspired multiwavelet shrinkage performs better than the traditional multiwavelet hard- and soft-thresholding shrinkages.

Electrical properties of an n-order network with Y circuits

This paper introduces an n-order network model with Y circuits to investigate the electrical properties, and it is an entirely new question that has seldom been addressed in the academic world. We carry out the research by means of building an equivalent model. Firstly, we use Kirchhoff’s law to build a second-order nonlinear recurrence equation model. Secondly, by using variable substitution we transform the nonlinear equation into a linear difference equation. Thirdly, we name the process recurrence transformation of n-order network, called N-RT for short. Based on the linear difference equation, an analytical solution is offered to the problem of the equivalent resistance of the n-order network with Y circuits. This research is going to find specific application in many cases. What’s important, this paper further investigates the equivalent complex impedance of LC network and the electrical properties of complex impedance, and illustrates the oscillation and resonance characteristics of LC network with 3D graphs.

Analytical Solutions to the Singular Problem for a System of Nonlinear Parabolic Equations of the Reaction-Diffusion Type

The paper deals with a system of two nonlinear second-order parabolic equations. Similar systems, also known as reaction-diffusion systems, describe different chemical processes. In particular, two unknown functions can represent concentrations of effectors (the activator and the inhibitor respectively), which participate in the reaction. Diffusion waves propagating over zero background with finite velocity form an essential class of solutions of these systems. The existence of such solutions is possible because the parabolic type of equations degenerates if unknown functions are equal to zero. We study the analytic solvability of a boundary value problem with the degeneration for the reaction-diffusion system. The diffusion wave front is known. We prove the theorem of existence of the analytic solution in the general case. We construct a solution in the form of power series and suggest recurrent formulas for coefficients. Since, generally speaking, the solution is not unique, we consider some cases not covered by the proved theorem and present the example similar to the classic example of S.V. Kovalevskaya.

Numerical solution of second-order two-dimensional hyperbolic equation by bi-cubic B-spline collocation method

A method based on B-splines has been introduced for the solution of second-order nonlinear hyperbolic equation in 2-dimensions subject to appropriate initial and Dirichlet boundary conditions. We first convert the second-order equation into a system of first-order partial differential equations. Then, collocation of bi-cubic B-splines is used to discretize spatial variables and their derivatives to further obtain first-order ordinary differential equations which have block tri-diagonal structure. Computation technique is discussed to handle the thus obtained block tri-diagonal matrices, which are then solved by two-step, second-order strong-stability-preserving Runge--Kutta method (SSP RK-22). The efficiency and accuracy of the proposed method are demonstrated by its application to a few test problems and by comparing the results with analytic solutions and with the results obtained by using other numerical methods available in the literature.

Second-Order Elliptic Equation with Singularities

On the compact Riemannian manifold of dimension n≥5, we study the existence and regularity of nontrivial solutions for nonlinear second-order elliptic equation with singularities. At the end, we give a geometric application of the above singular equation.

Existence of positive strongly decaying solutions of second-order nonlinear q-difference equations

This paper investigates the existence of positive strongly decaying solutions of second-order nonlinear q-difference equation with q-regularly varying coefficients a and b. By means of a fixed point theorem in partially ordered Banach space, necessary and sufficient conditions for the existence of solutions satisfying and as are established. Moreover, with the help of q-regular varying theory, precise asymptotic behaviour of such solutions is described.

Modelling the phytoremediation of formaldehyde from indoor air by Chamaedorea Elegans using artificial intelligence, genetic algorithm and response surface methodology

Abstract Recently, indoor air quality has become a great concern for human health. Volatile organic compounds (VOCs), are the most common pollutants in the indoor air. Some VOCs such as formaldehyde are toxic and carcinogenic. A lot of researches have proved that potted plants through phytoremediation can help to reduce these pollutants, but modelling on these systems is not developed adequately and without creating the appropriate model, there will be no ability to predict the performance of phytoremediation. In this study, Formaldehyde removal from indoor air by Chamaedorea Elegans was modelled through Artificial Neural Networks (ANN) and Response Surface Methodology (RSM), and optimized by genetic algorithm (GA). Here, the formaldehyde concentration, leaf surface area, light intensity and relative humidity were considered as input variables. The experimental design was conducted using full factorial design method. The RSM results showed that the second-order nonlinear equation was in high accordance with the experiments. A Multilayer Perceptron ANN was developed and network performance was evaluated with several training algorithms, in which, the Levenberg Marquardt (LM) algorithm showed the best performance. Finally, the optimum value of the variables was determined by GA. The results indicated that the ANN model exhibits a higher degree of delicacy and accuracy compared to the RSM.

Synchronizing Stability Analysis and Region of Attraction Estimation of Grid-Feeding VSCs Using Sum-of-Squares Programming

Grid-synchronising Stability (GSS) is an emerging issue that exists in grid-feeding voltage-source converters (VSCs). Its occurrence is primarily related to the nonlinear dynamics of a type of vastly applied synchronization units - Phase-locked Loops (PLL). Dynamic characterisation and modelling for the GSS analysis can be achieved by using a simplified system model, which is a second-order and autonomous nonlinear equation; however, with the presence of an indefinite damping term. As revealed and demonstrated in this work, this indefinite damping effect can result in an inaccurate Region-of-Attraction (ROA) estimation of the traditional Equal Area Principle (EAP)-based method. To overcome this issue and achieve a valid ROA estimation, this paper adopts the sum-of-squares (SOS) programming technique, which is a numeric optimisation method with SOS relaxations. The development and implementation of the SOS program for ROA estimation are presented. Numerical case studies and time-domain verifications demonstrate that this method is valid for GSS analysis, and an almost precise estimation is achieved in the first quadrant. This evidence makes the SOS method a promising tool for GSS analysis because the GSS problem is most concerned with the stability within the first swing cycle, i.e., in the first quadrant.

Group resonant interactions between surface and internal gravity waves in a two-layer system

Nonlinear interactions between surface and internal gravity waves in a two-layer system are studied using explicit second-order nonlinear evolution equations in Hamiltonian form. Motivated by the detailed experiment of Lewis, Lake & Ko (J. Fluid Mech., vol. 63, 1974, pp. 773–800), our focus is on surface wave modulation by the group resonance mechanism that corresponds to near-resonant triad interactions between a long internal wave and short surface waves. Our numerical solutions show good agreement with laboratory measurements of the local wave amplitude and slope, and confirm that the surface modulation becomes significant when the group velocity of the surface waves matches the phase speed of the internal wave, as the linear modulation theory predicts. It is shown, however, that, after the envelope amplitude is increased sufficiently, the surface and internal waves start to exchange energy through near-resonant triad interactions, which is found to be crucial to accurately describe the long-term surface wave modulation by an internal wave. The reduced amplitude equations are also adopted to validate this observation. For oceanic applications, numerical solutions are presented for a density ratio close to one and it is found that significant energy exchanges occur through primary and successive resonant triad interactions.

Integrability Properties of Functions with a Given Behavior of Distribution Functions and Some Applications

We establish that if the distribution function of a measurable function v defined on a bounded domain Ω in ℝn (n ≥ 2) satisfies, for sufficiently large k, the estimate meas {|v| > k} ≤ k−αϕ(k)/ψ(k), where α > 0, ϕ: [1,+∞) → ℝ is a nonnegative nonincreasing measurable function such that the integral of the function s → ϕ(s)/s over [1,+∞) is finite, and ψ: [0,+∞) → ℝ is a positive continuous function with some additional properties, then |v|αψ(|v|) ∈ L1(Ω). In so doing, the function ψ can be either bounded or unbounded. We give corollaries of the corresponding theorems for some specific ratios of the functions ϕ and ψ. In particular, we consider the case where the distribution function of a measurable function v satisfies, for sufficiently large k, the estimate meas {|v| > k} ≤ Ck−α(ln k)−β with C, α > 0 and β ≥ 0. In this case, we strengthen our previous result for β > 1 and, on the whole, we show how the integrability properties of the function v differ depending on which interval, [0, 1] or (1,+∞), contains β. We also consider the case where the distribution function of a measurable function v satisfies, for sufficiently large k, the estimate meas {|v| > k} ≤ Ck−α(ln ln k)−β with C, α > 0 and β ≥ 0. We give examples showing the accuracy of the obtained results in the corresponding scales of classes close to Lα(Ω). Finally, we give applications of these results to entropy and weak solutions of the Dirichlet problem for second-order nonlinear elliptic equations with right-hand side in some classes close to L1(Ω) and defined by the logarithmic function or its double composition.

Oscillatory behavior of solutions of dynamic equations of higher order on time scales

We study the n -th-order nonlinear dynamic equations x [ n ] ( t ) + p ( t ) ϕ α n − 1 [ ( x [ n − 2 ] ( t ) ) Δ σ ] + q ( t ) ϕ γ ( x ( g ( t ) ) ) = 0 on an unbounded time scale 𝕋 , where n ≥ 2 and for i = 1 , … , n − 1 x [ i ] ( t ) : = r i ( t ) ϕ α i [ ( x [ i − 1 ] ( t ) ) Δ ] , with r n = α n = 1 and x [ 0 ] = x ; here the constants α i and the functions r i , i = 1 , … , n − 1 , are positive and p , q are nonnegative functions. Criteria are established for the oscillation of solutions for both even- and odd-order cases. The results improve several known results in the literature on second-order, third-order, and higher-order linear and nonlinear dynamic equations. In particular our results can be applied when g is not (delta) differentiable and the forward jump operator σ and g do not commute.

On the singular second-order elliptic equation

Let (M,g) be a smooth and compact Riemannian manifold with dimensions n≥3. This paper deals with the existence of nontrivial solutions for a nonlinear second-order elliptic equation with a singular term and critical Sobolev growth. We provide a geometric application of this singular equation.