Order Homogeneous Differential Equations Research Articles

Atmospheric Ekman flows of the Northern Hemisphere with latitude and height in ellipsoidal coordinates

In this paper, we study the atmospheric Ekman flows of the Northern Hemisphere related to latitude and height in ellipsoidal coordinates. Firstly, by using a suitable transformation, we convert the atmospheric Ekman equation into a second order elliptic partial differential equation. Further, we prove it is equivalent to a second order homogeneous ordinary differential equation with variable coefficients. Finally, we construct the explicit solution and give the laws of motion of the atmospheric Ekman flows.

On Extended Transformation Formulae Associated to the k-Hypergeometric Function

Recently the second order homogeneous k-hypergeometric differential equation and the definition of the function as its solution at the origin have been presented. Motivated by the current research work in the study of k-hypergeometric functions, our aim is to establish certain extended transformation formulae related to the k-hypergeometric function with the Euler's identity for Gauss hypergeometric function as a special case.

Some coefficient estimates on real kernel 𝛼-harmonic mappings

We call the solution of a kind of second order homogeneous partial differential equation as real kernel α − \alpha - harmonic mappings. For this class of mappings, we explore its Heinz type inequality. Furthermore, for a subclass of real kernel α − \alpha - harmonic mappings with real coefficients, we estimate their coefficients. At last, we study the extremal function of Schwartz type lemma for the class of real kernel α − \alpha - harmonic mappings.

Numerical Solution of Fourth Order Homogeneous Parabolic Partial Differential Equations (PDEs) using Non-Polynomial Cubic Spline Method (NPCSM)

Non-polynomial cubic spline functions are already being used in the field of engineering, computer sciences, and natural sciences to solve ordinary differential equations (ODEs) and partial differential equations (PDEs). However, many of the above-mentioned problems do not have an exact, stable, or convergent exact solution. There are different approximations and methods that can be applied to solve these problems. This study implemented the purposed method on homogeneous parabolic PDEs having different dimensions. The results obtained were compared with the exact solution and results of other existing methods in tabular and graphical form. Mathematica was used to find the mathematical and graphical results.EMATICA.
 Keywords: Adomian decomposition method (ADM), non-polynomial cubic spline method (NPCSM), continuous approximation, finite difference approximations, fourth order homogeneous parabolic partial differential equations (PDEs)
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Extraction of Some New Solutions of Linear Differential Equations by Dual Space

In this study, the solutions of second order linear homogeneous ordinary differential equations being variable and constant coefficients have been obtained by using power series method which is used frequently in dual space. The solutions of second order homogeneous ordinary differential equations being constant and variable coefficients in dual space by using power series have given and some applications have implemented.

On Liouvillian Solutions of Third Order Homogeneous Linear Differential Equations

In this article we will consider third order homogeneous differential equations:  L(y)=y'''+a_1y'+a_0y(a_0,a_1 ∈k) whose Galois group G（L） is imprimitive. This case is characterised by the fact that the third symmetric power equation L ^ⓢ3(y)=0 has an exponential solution whose square is rational (Singer & Ulmer 1993). If L(y)=0 has a Liouvillian solution z whose logarithmic derivative u=z'/z  is algebraic over a differential field (k,') ,we will give an algorithm to find the relation between a_0, a_1 , the semi-invariant S=Y_1Y_2Y_3 which is unique up to multiplication by a constant, the coefficients C_0, C_1 of the minimal polynomial P(u) of u  and their derivatives. The aim of this work is to diminutize the number of constants C_m  stated in the algorithm of Singer & Ulmer (Singer & Ulmer 1993 Algorithm p. 31) whose determination is not easy to do, and we will achieve this by using Groebner Basis.

Ternary SU(3)-group symmetry and its possible applications in hadron-quark substructure. Towards a new spinor-fermion structure

The questions on the existence of the three color quark symmetry and three quark-lepton generations could have the origin associated with the new exotic symmetries outside the Cartan-Killing-Lie algebras/groups. Our long-term search for these symmetries has been began with our Calabi-Yau space classification on the basis of the n-ary algebra for the reflexive projective numbers and led us to the expansion of the binary n = 2 complex and hyper complex numbers in the framework of the n-ary complex and hyper-complex numbers with n = 3, 4, … where we constructed new Abelian and non-Abelian symmetries. We have studied then norm-division properties of the Abelian nary complex numbers and have built the infinite chain of the Abelian groups U(n–1) = [U(1) × … × U(1)](n–1). We have developed the n-ary holomorphic (polymorphic) analysis on the n-ary complex space NC{n}, which led us to the generalization of the quadratic Laplace equations for the harmonic functions. The generalized Laplace equations for the n-ary harmonic functions give us the n-th order homogeneous differential equations which are invariant with respect to the Abelian n-ary groups U(n–1) and with some new spatial properties. Further consideration of the non-Abelian n-ary hyper-complex numbers opens the infinite series of the non-Abelian TnSU(n)-Lie groups(n=3,4,…) and its corresponding tnsu(n) algebras. One of the exceptional features of these symmetry groups is the appearance of some new n-dimensional spinors that could lead to an extension of the concept of the SU(2)-spin, to the appearance of n-dimensional quantum structures -exotic “n-spinor” matter(n = 3, 4, … - maarcrions). It is natural to assume that these new exotic “quantum spinor states” could be candidates for the pra-matter of the quark-charge leptons or/and for the dark matter. We will be also interested in the detection of the exotic quantum ’n-spinor” matter in the neutrino and hadron experiments.

Application of the two-parameter single shooting method for natural vibration analysis of Euler-Bernoulli beams

The paper presents a new method of calculating the natural frequencies and modes of Euler-Bernoulli beams. The formulated algorithm is based on the fourth order homogeneous ordinary differential equation of motion that allows different support conditions to be considered. The idea of the method is to replace a two-point boundary value problem with an initial value problem. Distinct from the classic approach, where only one unknown parameter is considered, the method enables problems with two parameters to be solved. The proposed two-parameter version of the single shooting method is verified on the example of a non-prismatic simply supported beam and a prismatic propped cantilever beam resting on Winkler foundation and loaded by axial force. The obtained results confirm the correctness and high effectiveness of the presented method.

Bonded contact and general crack problems in generalized materials and their relationships

The term of generalized material is introduced here as the material, whose state is described by the second order homogeneous differential equations with constant coefficients. The generalized point sources are described by homogeneous expressions, containing the first order derivatives with constant coefficients. The Green’s function for a half-space, made of generalized material, subjected to the action of generalized point sources, is derived in the form of a single integral over a circle. Some of the components of the surface Green’s function are presented in finite form, no computation of any integral is needed. The bonded contact problem is described as mixed–mixed boundary value problem for a half-space, with normal and tangential displacements prescribed inside domain S and vanishing tractions outside this domain on the plane $$x_{3}=0$$ . A set of governing integral equations for bonded contact problem was derived, with some of the kernels defined in finite form. The general crack problem is defined as that of a flat crack of arbitrary shape S in the plane $$x_{3}=0$$ , with normal tractions $$\sigma _{33}$$ applied symmetrically to the crack faces and tangential tractions $$\sigma _{31}$$ and $$\sigma _{23}$$ applied anti-symmetrically to the crack faces. A set of 3 governing integral equations is derived, with some of the kernels presented in the finite form. The relationship between the integrands in Fourier transform of the kernels of the sets of the integral equations of both problems is established: they are related in such a way, as if they were the coefficients in the schematic sets of algebraic equations. This relationship can also be presented as a general relationship between matrices of arbitrary rank n, with their components being determinants of certain matrices of rank q, with $$q\ge n$$ .

Pedagogy of second order homogeneous differential equations: A holistic approach using a Matlab workbook

ABSTRACTA Matlab workbook is developed to facilitate a better understanding of the multiple approaches available to solve homogeneous second order differential equations with constant coefficients. With minimal input consisting of the coefficients and initial conditions, the workbook displays all aspects of the differential equation such the phase plane plot, state of the equilibrium, solutions based on roots of the characteristic equation, Laplace transforms and Runge‐Kutta methods. The workbook format is such that it displays results as if the instructor is solving the differential equation on a blackboard with definitions, explanations, solution of unknown constants and comparison of solutions from multiple approaches. Several examples are provided which demonstrate the potential of the workbook in enhancing the pedagogy by letting instructors and students observe all aspects, thereby providing a holistic approach. The workbook (with examples and the Matlab script as a .p file) is available to all from the author's university website. © 2015 Wiley Periodicals, Inc. Comput. Appl. Eng. Educ. 24:114–121, 2016; View this article online at wileyonlinelibrary.com/journal/cae; DOI 10.1002/cae.21677

First order homogeneous ordinary differential equation with initial value as triangular intuitionistic fuzzy number

In this paper first order homogeneous ordinary differential equation is described in intuitionistic fuzzy environment. Here initial condition of the said differential equation is considered as triangular intuitionistic fuzzy number. It is illustrated by numerical examples. Finally two elementary applications on oil production and weight loss problems are described in intuitionistic fuzzy environment.

A Simplified Linear Approach for Elastic Collision Analysis between the Fixed Middle Pier and the 2-Span Continuous Girder

Based on a collision analysis between the fixed middle girder and the 2-span continuous girder in longitudinal direction, motion equations of the girder and the pier are established by mode decomposition method. Displacements’ relationship at the collision interface is used to establish the equation about the collision force, and Duhamel integrals in the equation are eliminated so as to get a 6 order homogeneous ordinary differential equation about the collision force. Then an approach to solve the equation is put forward. The analysis results show that the time history curve of collision force is a half sine waveform, and the initial velocity of girder towards pier top is proportional to the peak collision force but has nothing to do with the duration of collision.

The Helmholtz conditions for systems of second order homogeneous differential equations

Variationality of systems of second order ordinary differential equations is studied within the class of positive homogeneous systems. The concept of a higher order positive homogeneous function, related to Finsler geometry, is represented by the well-known Zermelo conditions, and applied to the theory of variational equations. In particular, it is shown that every system of m+1 second order variational and positive homogeneous differential equations is linearly dependent and admits subsystems of m differential equations which are variational in sense of parameter-invariant variational problems, and vice versa. An example of a positive homogeneous variational system of second order differential equations is given.

Distortional solutions for loaded semi-discretized thin-walled beams

For thin-walled beams, the classic theory for flexural and torsional analysis of open and closed cross-sections can be generalized by including distortional displacements. In a companion paper it is shown that using a novel semi-discretization process, it is possible to determine specific distortional displacement fields which decouple the reduced order differential equations. In this process the cross section is discretized into finite cross-section elements, and the natural distortional modes as well as the related axial variations are found as solutions to the established coupled fourth order homogeneous differential equations of GBT. In this paper the non-homogeneous distortional differential equations of GBT are formulated using this novel semi-discretization process. Transforming these non-homogeneous distortional differential equations into the natural eigenmode space by using the distortional modal matrix found for the homogeneous system, we get the uncoupled set of differential equations including the distributed loads. This uncoupling is very important in GBT, since the shear stiffness contribution from St. Venant torsional shear stress as well as “Bredt's shear flow” cannot be neglected nor approximated by the combination of axial stiffness and transverse stiffness, especially for closed cross sections. The full analytical solutions of these linear non-homogeneous differential equations are given, including four illustrative examples, which illustrate the strength of this novel approach to GBT. This new approach is a considerable theoretical achievement, since it without approximation gives the full analytical solution for a given discretization of the cross section including distributed loading. The boundary conditions considered in the examples of this paper are restricted to built in ends, which are needed for future displacement formulation of an exact first-order distortional beam element.

Solution of a Novel Quasi-Exactly Solvable Potential via Asymptotic Iteration Method

Since the wave function contains all the necessary information to describe a quantum system fully, it is of high importance of obtaining exact or approximate solutions of Schrodinger equation in quantum mechanics. It is known that there are not so many potentials that can be solved exactly. Therefore, many techniques are suggested (and in use) to find the approximate solutions of the potentials that are not exactly solvable. A recent technique, called the Asymptotic Iteration Method (AIM),1),2) has been used to obtain eigenvalues of second order homogeneous differential equations. In the case of Schrodinger equation, it has been found that the method reproduces the energy spectrum for most exactly solvable potentials,2)–6) and it produces very good results for non-exactly solvable potentials.7)–10) It is well known in quantum mechanics that the potentials having bound states can be classified as solvable, non solvable and quasi exactly solvable (QES). The potentials in the last category are characterized by the fact that part of their spectrum can be found analytically while the remaining part is unknown. In a sense, they constitute the intermediate step between exactly solvable equations (associated with potentials such as the harmonic oscillator one, the Coulomb one, · · · ) whose all analytic solutions can be obtained, and the analytically unsolvable ones requiring a numerical treatment.11) Some QES potentials can be given in terms of a parameter J and then one can obtain exactly the first J -th energy values and wavefunctions, for positive integer (and half-integer) values of J. Thus, one can find the solution of the QES system in the framework of Lie-algebraic formalism.12)–16) In their study, Cho et al.17) have found a novel QES model with symmetric inverted potentials which are unbounded from below. They have obtained a discrete spectrum of bound states for these inverted potentials and have shown the solvable part of the spectrum is the QES states they have discovered. Using the AIM, we determine the energy eigenvalues and the eigenfunctions of

Identifying a differential equation for lip motion

A second order homogeneous differential equation is fitted to lip trajectories during speech at fast, normal, and slow speaking rates. The objective is to characterize lip motion in terms of biomechanical parameters such as stiffness, damping, and applied external forces. The results show that, at fast speaking rates, the lip behaves like a simple mass-spring oscillator moving at its natural frequency, and with low variability across repetitions. At slow speaking rates, on the other hand, the lip is under a stronger external control to conform to the required trajectory, which introduces a larger variability accross repetitions. The case of normal rate falls between the fast and slow rate cases.

A duality principle for homogeneous vectorfields with applications

We introduce a duality principle for homogeneous vectorfields. As an application of this duality principle, stability and boundedness results for negative order homogeneous differential equations are obtained, starting from known results for positive order homogeneous differential equations.

The general solutions of difference vector-equations with constant coefficients

This paper deals with general solutions of difference vector-equations with constant coeffici­ents. We find the simple formulae between solution of the first order homogeneous differential equations and solution of difference equation. This allows easily get all solutions of difference equations if we know the differential solution.

Generalized Partial Differential Equation and Fermat's Last Theorem

The equivalence of Fermat's Last Theorem and the non-existence of solutions of a generalized n th order homogeneous hyperbolic partial differential equation in three dimensions and periodic boundary conditions defined in a cubic lattice is demonstrated for all positive integer, n > 2. For the case n = 2, choosing one variable as time, solutions are identified as either propagating or standing waves. Solutions are found to exist in the corresponding problem in two dimensions.

Finite element collocation approximation of the characteristic exponent in BVPs with periodic coefficients

We consider the application of the finite element collocation method, with Hermite cubic elements, for the determination of the characteristic exponent in Floquet's theory for the solution of the general second order homogeneous differential equation with periodic coefficients. The freedom in choosing the collocation points and the high order of approximation were the main reasons for the employment of the collocation method. The determination of the Gaussian points as optimal interior collocation points gave us the additional advantage of an optimum method. The computational comparison with the classical matrix method reveals the superiority of the collocation method.