Higher-order Equations Research Articles

On the Conditions for the Solvability of Boundary-Value Problems for Higher-Order Equations with Discontinuous Coefficients

В работе для уравнения высокого четного порядка с разрывными коэффициентами изучена задача типа Дирихле. Дан критерий единственности решения. Решение построено в виде ряда Фурье по собственным функциям одномерной задачи. При обосновании сходимости ряда возникает проблема малых знаменателей. Получены достаточные условия отделимости знаменателя от нуля. Показана, что на разрешимость задачи влияет не только размерность прямоугольника, но и порядки задаваемых производных на нижней границе прямоугольника. Библиография: 23 названия.

Applications of OHAM and MOHAM for Fractional Seventh-Order SKI Equations

In this article, a comparative study between optimal homotopy asymptotic method and multistage optimal homotopy asymptotic method is presented. These methods will be applied to obtain an approximate solution to the seventh-order Sawada-Kotera Ito equation. The results of optimal homotopy asymptotic method are compared with those of multistage optimal homotopy asymptotic method as well as with the exact solutions. The multistage optimal homotopy asymptotic method relies on optimal homotopy asymptotic method to obtain an analytic approximate solution. It actually applies optimal homotopy asymptotic method in each subinterval, and we show that it achieves better results than optimal homotopy asymptotic method over a large interval; this is one of the advantages of this method that can be used for long intervals and leads to more accurate results. As far as the authors are aware that multistage optimal homotopy asymptotic method has not been yet used to solve fractional partial differential equations of high order, we have shown that this method can be used to solve these problems. The convergence of the method is also addressed. The fractional derivatives are described in the Caputo sense.

Differential inequality and non-oscillation of fourth order differential equation

The oscillatory theory of fourth order differential equations has not yet been developed well enough. The results are known only for the case when the coefficients of differential equations are power functions. This fact can be explained by the absence of simple effective methods for studying such higher order equations. In this paper, the authors investigate the oscillatory properties of a class of fourth order differential equations by the variational method. The presented variational method allows to consider any arbitrary functions as coefficients, and our main results depend on their boundary behavior in neighborhoods of zero and infinity. Moreover, this variational method is based on the validity of a certain weighted differential inequality of Hardy type, which is of independent interest. The authors of the article also find two-sided estimates of the least constant for this inequality, which are especially important for their applications to the main results on the oscillatory properties of these differential equations.

Local existence for class of nonlinear higher-order wave equation with logarithmic source term

Local existence for class of nonlinear higher-order wave equation with logarithmic source term

Proposing Algorithm Using YOLOV4 and VGG-16 for Smart-Education

In this paper, we propose an algorithm to identify and solve systems of high-order equations. We rely on traditional solution methods to build algorithms to solve automated equations based on deep learning. The proposal method includes two main steps. In the first step, we use YOLOV4 (Kumar et al. 2020; Canu, 2020) to recognize equations and letters associated with the VGG-16 network (Simonyan and Zisserman, 2015) to classify them. We then used the SymPy model to solve the equations in the second step. Data are images of systems of equations that are typed and designed by ourselves or handwritten from other sources. Besides, we also built a web-based application that helps users select an image from their devices. The results show that the proposed algorithm is set out with 95% accuracy for smart-education applications.

Gravitational waves in Hořava-Lifshitz anisotropic gravity

We show that in the anisotropic Ho\v{r}ava-Lifshitz gravity there is a well-defined wave zone where the physical degrees of freedom propagate according to a non-relativistic linear evolution equation of high order in spatial derivatives, which reduces to the wave equation at low energy. This is so, provided the coupling parameters satisfy some restrictions which we study in detail. They are imposed to obtain a finite ADM gravitational energy, which depends manifestly on the terms which break the Lorentz symmetry of the formulation. The analysis we perform is beyond the linearized approach and includes all high order terms of the Hamiltonian potential.

On the Solvability of the Generalized Neumann Problem for a Higher-Order Elliptic Equation in an Infinite Domain

We consider the generalized Neumann problem for a 2lth-order elliptic equation with constant real higher-order coefficients in an infinite domain containing the exterior of some circle and bounded by a sufficiently smooth contour. It consists in specifying of the (kj-1){(k_j - 1)}th-order normal derivatives where 1≤k1...kl≤2l{1\le k_1 < ... < k_l \le 2l}; for kj=j{k_j = j} it turns into the Dirichlet problem, and for kj=j+1{k_j = j +1} into the Neumann problem. Under certain assumptions about the coefficients of the equation at infinity, a necessary and sufficient condition for the Fredholm property of this problem is obtained and a formula for its index in Holder spaces is given.

Averaging of Higher-Order Parabolic Equations with Periodic Coefficients

In L2(Rd;Cn), we consider a wide class of matrix elliptic operators A of order 2p (where p2) with periodic rapidly oscillating coefficients (depending on x/). Here 0 is a small parameter. We study the behavior of the operator exponent e-A for 0 and small . We show that the operatore-A converges as 0 in the operator norm in L2(Rd;Cn) to the exponent e-A0 of the effective operator A0. Also we obtain an approximation of the operator exponent e-A in the norm of operators acting from L2(Rd;Cn) to the Sobolev space Hp(Rd; Cn). We derive estimates of errors of these approximations depending on two parameters: and . For a fixed 0 the errors have the exact order O(). We use the results to study the behavior of a solution of the Cauchy problem for the parabolic equation u(x,)= -(A u)(x,)+F(x,) in Rd.

A new form of general soliton solutions and multiple zeros solutions for a higher-order Kaup–Newell equation

Due to the fact that the higher-order Kaup–Newell (KN) system has more complex and diverse solutions than the classical second-order flow KN system, the research on it has attracted much attention. In this paper, we consider a higher-order KN equation with third-order dispersion and fifth-order nonlinearity. Based on the theory of the inverse scattering, the matrix Riemann–Hilbert problem is established. Through the dressing method, the solution matrix with simple zeros without reflection is constructed. In particular, a new form of solution is given which is more direct and simpler than previous methods. In addition, through the determinant solution matrix, the vivid diagrams and dynamic analysis of single-soliton solution and two-soliton solution are given in detail. Finally, by using the technique of limit, we construct the general solution matrix in the case of multiple zeros, and the examples of solutions for the cases of double zeros, triple zeros, single–double zeros, and double–double zeros are especially shown.

Homogenization of the Higher-Order Hyperbolic Equations with Periodic Coefficients

Homogenization of the Higher-Order Hyperbolic Equations with Periodic Coefficients

BOUNDARY VALUE PROBLEM WITH A NONLOCAL BOUNDARY CONDITION OF INTEGRAL FORM FOR A MULTIDIMENSIONAL EQUATION OF IV ORDER

The aim of this paper is to study the solvability of solution of non-local problem with integral condition in spatial variables for high-order linear equation in the classe of regular solutions (which have all the squaredderivatives generalized by S.L. Sobolev that are included in the corresponding equation). It is indicated that at first similar problems were studied for high-order equations either in the one-dimensional case, or under certain conditions of smallness by the value of T. A list of new works for the multidimensional case is also given. In this paper, we present new results on the solvability of non-local problem with integral spatial variables for high-order equation a) in the multidimensional case with respect to spatial variables; b) in the absence of smallness conditions by the value T; however, this condition exists for the kernel K(x; y; t). The research method is based on obtaining a priori estimates of the solution of the problem, which implies its existence and uniqueness in a given space.

From a generalized discrete NLS equation in discrete alpha helical proteins to the fourth-order NLS equation

In this paper, we consider a generalized integrable discrete nonlinear Schrödinger (NLS) equation, which can describe the dynamics of discrete alpha helical proteins with higher-order excitations and lead to the higher-order NLS equation in the continuum limit. The Darboux transformation (DT) and the soliton solutions of this generalized discrete NLS equation are implemented. It is shown that the integrable properties of the generalized discrete NLS equation, including the discrete Lax pair, the DT and the discrete soliton solutions, give rise to their continuous counterparts as the discrete space step tends to zero.

Rogue wave solutions for the generalized fifth-order nonlinear Schrödinger equation on the periodic background

In this paper, we construct the rogue wave solutions on the background of the Jacobi elliptic functions for a generalized fifth-order nonlinear Schrödinger (NLS) equation. Using the Jacobi elliptic function expansion method, we reduce this higher-order nonlinear equation to a lower-order ordinary differential equation. Through the approach of the nonlinearization of spectral problem and then the Darboux transformation method, two kinds of rogue periodic waves which are on dn and cn Jacobi elliptic functions background are obtained. Furthermore, we represent the nonlinear dynamics of the rogue periodic wave solutions of this higher-order equation.

A symmetry-preserving difference scheme and analytical solutions of a generalized higher-order beam equation

Under investigation in this work is a generalized higher-order beam equation, which is an important physical model and describes the vibrations of a rod. By considering Lie symmetry analysis, and using the power series method, we derive the geometric vector fields, symmetry reductions, group invariant solutions and power series solutions of the equation, respectively. The convergence analysis of the power series solutions are also provided with detailed proof. Furthermore, by virtue of the multipliers, the local conservation laws of the equation are derived as well. Furthermore, an effective and direct approach is proposed to study the symmetry-preserving discretization for the equation via its potential system. Finally, the invariant difference models of the generalized beam equation are successfully constructed. Our results show that it is very useful to construct the difference models of the potential system instead of the original equation.

Trigonometric Ratios Using Algebraic Methods

The main aim of this article is to start with an expository introduction to the trigonometric ratios and then proceed to the latest results in the field. Historically, the exact ratios were obtained using geometric constructions. The geometric methods have their own limitations arising from certain theorems. In view of the certain limitations of the geometric methods, we shall focus on the powerful techniques of equations in deriving the exact trigonometric ratios using surds. The cubic and higher-order equations naturally arise while deriving the exact trigonometric ratios. These equations are best expressed using the expansions of the cosines and sine of multiple angles using the Chebyshev polynomials of the first and second kind respectively. So, we briefly present the essential properties of the Chebyshev polynomials. The equations lead to the question of reduced polynomials. This question of the reduced polynomials is addressed using the Euler's totient function. So, we describe the techniques from theory of equations and reduced polynomials. The trigonometric ratios of certain rational angles (when measured in degrees) give rise to rational trigonometric ratios. We shall discuss these along with the related theorems. This is a frontline area of research connecting trigonometry and number theory. Results from number theory and theory of equations are presented wherever required.

Thermal and solutal performance of Cu/CuO nanoparticles on a non-linear radially stretching surface with heat source/sink and varying chemical reaction effects

Current simulations, heat and mass transport of a variable hydromagnetic flow of Copper and Copper oxide-water nanoparticles suspended in nanofluid on a non-linear radially stretching surface in the presence of various impacts have been analyzed. The mathematical model scrutinizes the impacts of thermal radiation, viscous dissipation, and heat source/sink in thermal transport equation and variable chemical reaction in mass transport equation Additionally, velocity slip is imposed in boundary conditions. Further, the impacts of two shapes nanoparticle named column and lamina on the over nonlinearly radially stretched surface. The higher-order equations modeling the problem are mutated into nonlinear ordinary differential equations via similarity transformations approach, while the solutions to the modified equations are sought using Keller Box technique.

Exploring pro-environmental behavior in Azerbaijan: an extended value-belief-norm approach

PurposeThis paper aims to explore pro-environmental behavior (PEB) in Azerbaijan. Therefore, the authors used value-belief-norm (VBN) theory, extended by the construct of social norms (SN), as a basis.Design/methodology/approachData were collected by establishing a link within various social media platforms. The final sample consisted of 407 respondents. The authors analyzed four dimensions of PEB using higher-order structural equations. The authors also examined both direct and (serial) indirect effects for cross-cultural validation of the extended VBN theory.FindingsThe authors were able to confirm the VBN theory in its entirety. However, SN, which are influential in collectivistic and Sunni-majority states, do not contribute significantly to explaining PEB in predominantly Shiite Azerbaijan.Research limitations/implicationsThe authors could not establish a direct effect of SN on PEB within this study. However, the authors observed an indirect “values-beliefs-norms-behavior” effect. The different (partly abbreviated) effect channels of the four tested value antecedents provide interesting insights for marketing research.Practical implicationsBased on the results, it seems crucial to make Muslim consumers aware of the negative outcomes of their consumption behavior and to emphasize individual responsibility. However, SN may not need to be addressed depending on cultural and/or religious values.Originality/valueThe authors examined PEB in Azerbaijan by testing the serial mediation effects in the VBN model. Further, the authors tested the influence of SN within the framework of the original VBN theory, contributing to a better understanding of the possibility of integrating components of the theory of planned behavior.

P-Order Secant Method for Rapidly Solving the Ray Inverse Problem of Underwater Acoustic Positioning

The computational efficiency of underwater acoustic positioning based on the ray tracing is mainly limited to a great amount of calculation of ray inverse problem. We propose two kinds of p-order secant methods to improve the efficiency of traditional method, and the proposed methods can be regarded as a generalization of the traditional secant method from two points to p points for rapidly solving the inverse problem. In the proposed methods, the calculation information in previous iterations is utilized to fit a polynomial model to speed up the algorithm convergence. In the first-kind method, the inverse problem is calculated by solving a polynomial equation approximating the function mapping from the emission angle to the radial distance of the ray. In the second-kind method, the inverse problem is however directly solved by approximating the function mapping from the radial distance to the emission angle. As the first-kind method needs to solve a p-order polynomial equation, the practicability of this method is limited to the complexity of solving the high-order equation, while the second-kind method can directly approximate the solution of the inverse problem, which is more practical and flexible. The proposed methods have been verified in deep-sea trial. It shows that, the proposed methods can precisely produce the solution of the acoustic ray inverse problem within one iteration, and the computational efficiency of proposed method is about 6 times faster than that of the traditional method.

Growth and fixed points of solutions and their arbitrary-order derivatives of higher-order linear differential equations in the unit disc

In this paper, we investigate the growth and fixed points of solutions of higher-order linear differential equations in the unit disc. We extend the coefficient conditions to a type of one-constant-control coefficient comparison and obtain the same estimates of iterated order of solutions. We also obtain better estimates by providing a precise value of iterated order of solution instead of a range of that in the case of coefficient characteristic function comparison. Moreover, we utilize iteration to investigate and estimate the fixed points of solutions’ arbitrary-order derivatives with higher-order equations f^{(k)}+A_{k-1}(z)f^{(k-1)}+{cdots }+A_{1}(z)f'+A_{0}(z)f=0 and provide a concise method to judge if the items generated by the iteration do not vanish identically and ensure the iteration proceeds. Our results are an improvement over those by B. Belaïdi, T. B. Cao, G. W. Zhang and A. Chen.

Hybrid solutions for the (2+1)-dimensional variable-coefficient Caudrey-Dodd-Gibbon-Kotera-Sawada equation in fluid mechanics

In fluid mechanics, the higher-dimensional and higher-order equations are constructed to describe the propagations of nonlinear waves. In this paper, we investigate the (2+1)-dimensional variable-coefficient Caudrey-Dodd-Gibbon-Kotera-Sawada equation in fluid mechanics. The Nth-order Pfaffian solutions are constructed and proved via the modified Pfaffian technique. The higher-order soliton, first- and second-order breather solutions are constructed based on the Nth-order Pfaffian solutions. We graphically demonstrate that the amplitudes and velocities of the solitons are affected by some variable coefficients. Hybrid solutions composed of breathers, lumps and solitons are illustrated graphically. It can be found that when certain parameters are chosen, the breathers, lumps and solitons included in the hybrid solutions possess the same properties as those of the breather and lump solutions.