Higher-order Differential Equations Research Articles

Natural higher-derivatives generalization for the Klein–Gordon equation

We propose a natural family of higher-order partial differential equations generalizing the second-order Klein–Gordon equation. We characterize the associated model by means of a generalized action for a scalar field, containing higher-derivative terms. The limit obtained by considering arbitrarily higher-order powers of the d’Alembertian operator leading to a formal infinite-order partial differential equation is discussed. The general model is constructed using the exponential of the d’Alembertian differential operator. The canonical energy–momentum tensor densities and field propagators are explicitly computed. We consider both homogeneous and non-homogeneous situations. The classical solutions are obtained for all cases.

Application of Image Recognition Algorithm Based on Partial Differential Equation and Wavelet Transform in the Intelligent Linkage System of Government Urban Public Management

In the Internet age, social needs have become more diverse. The private sector has achieved great success in integrating resources to provide services to customers thanks to technological breakthroughs such as mobile Internet and big data, while the public sector has become even more dwarfed. The public sector urgently needs new theoretical support to integrate administrative resources more effectively, so as to provide the public with one-stop, holistic and seamless services as a whole. The “holistic governance” theory effectively responded to the abovementioned problems and gradually condensed the consensus of the academic and practical circles and became the most influential theory. The method of mathematical modeling is used to combine the improved wavelet domain Wiener filter and the comprehensive model of partial differential equations. This paper proposes an image recognition algorithm based on partial differential equations and wavelet transform. The algorithm uses weight coefficients to organically combine high-order partial differential equation models, retains the advantages of the second- and fourth-order partial differential recognition models, effectively improves the ability to maintain image edge detail information, and achieves better recognition results. This article starts with the description of the fragmentation of a street’s grassroots governance and sorts out the process of implementing overall governance with the “big linkage mechanism” as the main carrier. Due to the limitations of research ability and objective conditions, this article intercepts the “cross-section” of the big linkage mechanism in a certain street to demonstrate it and analyzes this “cross-section” more thoroughly. The experimental results of the data show that the image recognition algorithm based on partial differential equations and wavelet transform is effective and practical in the intelligent linkage system of the government urban public management.

The problem with non-separated multipoint-integral conditions for high-order differential equations and a new general solution

The problem with non-separated multipoint-integral conditions for high-order differential equations is considered. An interval is divided into m parts, the values of a solution at the beginning points of the subintervals are considered as additional parameters, and the high-order differential equations are reduced to the Cauchy problems on the subintervals for system of differential equations with parameters. Using the solutions to these problems, new general solutions to high-order differential equations are introduced and their properties are established. Based on the general solution, non-separated multipoint-integral conditions, and continuity conditions of a solution at the interior points of the partition, the linear system of algebraic equations with respect to parameters is composed. Algorithms of the parametrization method are constructed and their convergence is proved. Sufficient conditions for the unique solvability of considered problem are set. It is shown that the solvability of boundary value problems is equivalent to the solvability of systems composed. Methods for solving boundary value problems are proposed, which are based on the construction and solving these systems.

On a solvable system of p difference equations of higher order

On a solvable system of p difference equations of higher order

The hyper order and fixed points of solutions of a class of linear differential equations

In this paper, we precise the hyper order of solutions for a class of higher-order linear differential equations and investigate the exponents of convergence of the fixed points of solutions and their first derivatives for the second-order case. These results generalize those of Nan Li and Lianzhong Yang and of Chen and Shon.

On solutions of the Chazy equation

The Chazy system determines the necessary and sufficient conditions for the absence of movable critical points of solutions of the particular third order differential equation that was considered by Chazy in one of the first papers on the classification of higher-order ordinary differential equations with respect to the Painlevé property. The solution of the complete Chazy system in the case of constant poles has been already obtained. However, the question of integrating the Chazy equation remained open until now. In this paper, we prove that in the case of constant poles, under some additional conditions, this equation is integrated in elliptic functions.

Qualitative Behavior of a Nonlinear Generalized Recursive Sequence with Delay

Difference equations are of growing importance in engineering in view of their applications in discrete time-systems used in association with microprocessors. We will check out the global stability and boundedness for a nonlinear generalized high-order difference equation with delay.

Physical properties for bidirectional wave solutions to a generalized fifth-order equation with third-order time-dispersion term

Higher-order temporal-dispersion partial differential equations have its capability to visualize the evolution of steeper-waves for shorter wave-length better than the higher-order KdV does. In this work, we study the physical-structure propagations of generalized fifth-order nonlinear equation involving time-dispersion term. This model is recently proposed by Wazwaz (Xu and Wazwaz, 2020) and is considered to be the first type of integrable equations that involves a third-order time-dispersion term. We implement three functional-methods to seek solitary wave solutions to the proposed model. 2D-plots are provided to recognize the type of the obtained solutions. Finally, we propose some physical properties of the bidirectional waves that such model admits.

Uniqueness of solution for higher-order nonlinear fractional differential equations with multi-point and integral boundary conditions

Uniqueness of solution for higher-order nonlinear fractional differential equations with multi-point and integral boundary conditions

General theory of the higher-order quaternion linear difference equations via the complex adjoint matrix and the quaternion characteristic polynomial

Since the non-commutativity and particular structure of the quaternion algebra, the quternion difference equations (short for QDCEs) have a large difference from the classical theory of difference equations. In this paper, we establish a general theory of the higher-order linear QDCEs including the criteria of the linear independence (or dependence) of the discrete functions in the quaternion space, Liouville formulas, the structure theorems of the general solutions and the particular solutions with the quaternion power and trigonometric form, etc. By introducing the complex adjoint difference equations and the quaternion characteristic polynomial of the higher-order linear QDCEs, some basic results of the homogeneous and non-homogeneous difference equations are obtained. Through the analysis of the complex adjoint matrix and the quaternion eigenvalue, the general solutions of the higher-order linear QDCEs with variable and with constant coefficients are established. Several methods of obtaining the general solutions for the higher-order linear QDCEs are demonstrated and several examples are provided to illustrate the feasibility of our obtained results.

New analytic buckling solutions of non-Lévy-type cylindrical panels within the symplectic framework

In the analytic modeling of shell buckling, much attention has been paid to closed cylindrical shells or cylindrical panels with at least two opposite edges simply supported whose solutions are known to be Lévy-type solutions. However, there have been very few reports on analytic solutions of commonly used non-Lévy-type cylindrical panels, which is mainly attributed to the widely acknowledged difficulty in solving the governing higher-order partial differential equations under prescribed boundary conditions. To address this gap, the present study provides some new analytic buckling solutions of non-Lévy-type cylindrical panels within the Hamiltonian-system-based symplectic framework. The buckling of a cylindrical panel is first formulated in the Hamiltonian system to realize a new matrix-form governing equation. An original problem with non-Lévy-type boundary conditions is then treated as the superposition of two elaborated subproblems that are solved by the rigorous symplectic approach. The final solution is obtained according to the equivalence between the superposition of the subproblems and the original problem. For benchmark use, comprehensive buckling loads and buckling modes are presented for typical non-Lévy-type panels with different ratios of in-plane dimensions, different ratios of in-plane dimension to radius of curvature, and different ratios of thickness to in-plane dimension. The present study is the first successful extension of the symplectic superposition method to the buckling analysis of cylindrical panels, which may enable access to more new analytic solutions for similar issues.

Construction Of Three-Variable High-Order Defferential Equation Polynomial Solutions By Combinatorics Method

In this paper, we study how basic systems of polynomial solutions of a differential equation of high order with mixed derivatives of a function of three variables are constructed using combinatorial methods

Solving Higher-Order Fractional Differential Equations by the Fuzzy Generalized Conformable Derivatives

In this paper, the generalized concept of conformable fractional derivatives of order q ∈ n , n + 1 for fuzzy functions is introduced. We presented the definition and proved properties and theorems of these derivatives. The fuzzy conformable fractional differential equations and the properties of the fuzzy solution are investigated, developed, and proved. Some examples are provided for both the new solutions.

Variable coefficient higher-order nonlinear Schrödinger type equations and their solutions

In this communication we focus on certain higher-order partial differential equations of the nonlinear Schrödinger type with variable coefficients. By using a traveling wave reduction we obtain the general solution for the envelope in terms of the Jacobi elliptic sine function. The time dependence of the traveling wave coordinate and phase are explicitly determined as functions of the variable coefficients appearing in the equations. The equations investigated display cubic and fourth-order dispersion terms as well as higher nonlinearity and their constant coefficient versions have been the subject of a number of recent studies.

On the Solvability of One Boundary Value Problem for a Class of Higher-Order Nonlinear Partial Differential Equations

On the Solvability of One Boundary Value Problem for a Class of Higher-Order Nonlinear Partial Differential Equations

Exact Analytical Solutions of Generalized Fifth-Order KdV Equation by the Extended Complex Method

The recently introduced technique, namely, the extended complex method, is used to explore exact solutions for the generalized fifth-order KdV equation. Appropriately, the rational, periodic, and elliptic function solutions are obtained by this technique. The 3D graphs explain the different physical phenomena to the exact solutions of this equation. This idea specifies that the extended complex method can acquire exact solutions of several differential equations in engineering. These results reveal that the extended complex method can be directly and easily used to solve further higher-order nonlinear partial differential equations (NLPDEs). All computer simulations are constructed by maple packages.

Solving Non Linear Differential Equations By Using A G - Homotopy Analysis Method

This paper proposes an efficient hybrid analytical method named the Generalized Homotopy Analysis Method (GHAM) for solving nonlinear differential equations. The proposed hybrid method consists of topology based Homotopy Analysis Method (HAM) and Laplace typed integral transform (G-transform). Compared with HAM, GHAM does not require the effort to perform repeated integration and differentiation, when it is applied to solve higher order differential equations. Compared with G-transform, GHAM serves as an effective approach to tackle higher-order nonlinear differential equations. The effectiveness of GHAM is illustrated through the analysis of numerical examples, and the obtained results are graphically depicted. Also, GHAM is effectively utilized to analyze the density of the forest cover incorporating various parameters, which include the seed reproduction, seed deposition, seed establishment rates, old trees due to aging and the coefficients of mortality due to space variables.

Correctness conditions for high-order differential equations with unbounded coefficients

We give some sufficient conditions for the existence and uniqueness of the solution of a higher-order linear differential equation with unbounded coefficients in the Hilbert space. We obtain some estimates for the weighted norms of the solution and its derivatives. Using these estimates, we show the conditions for the compactness of some integral operators associated with the resolvent.

CALCULATION OF PARAMETERS OF BRANCHED MECHANICAL SYSTEMS WITH HARMONIC VIBRATIONS. Continuation

Parallel-series and series-parallel connections of mechanical power consumers are considered. According to the known parameters of systems and the disturbing harmonic effect, the velocities of the elements of mechanical systems and the forces applied to them are algebraically determined. For the considered branched mechanical schemes, the classical methods based on solving second-order differential equations become many times more complicated and require solving systems of equations that reduce to higher-order differential equations. The use of a symbolic (complex) description of mechanical processes and systems makes it possible to use instead simple and compact algebraic methods, the complexity of which is ten times less. Vector diagrams, not being a necessary component of the study of mechanical systems, are of significant methodological importance, since they show quantitative and phase relationships between the parameters of systems.

New benchmark free vibration solutions of non-Lévy-type thick rectangular plates based on third-order shear deformation theory

Higher-order shear deformation theories are generally adopted to study the dynamic behavior of substantially thick plates, of which Reddy’s third-order deformation theory (TSDT) is most extensively used in view of its conciseness and high accuracy. However, due to the intractability in solving the governing higher-order partial differential equations, analytic solutions of TSDT-based thick plate problems are limited to very few special cases, i.e., Navier-type or Lévy-type solutions. Since analytic solutions can not only serve as benchmarks for validation of various methods, but are also useful for structural designs, especially at the early design stage, developing new analytic methods that are applicable to more common non-Lévy-type thick plates is of much importance. In this paper, for the first time, a novel symplectic superposition method, which was proposed originally for thin/moderately thick plate problems, is extended to free vibration problems of thick rectangular plates based on the TSDT. Besides yielding Lévy-type solutions, new non-Lévy-type analytic free vibration solutions are obtained. Comprehensive benchmark natural frequencies and mode shapes are tabulated/plotted, all of which are well validated by the 3D theory-based solutions from the finite element method (FEM) and the literature, if any. The present method is implemented in the symplectic space under the Hamiltonian system framework, rather than in the Euclidean space under the Lagrangian system framework as implemented in conventional methods, providing a new route to more analytic solutions for complex plate and shell problems that have not been explored due to mathematical challenge.