Symmetric Differential Equations Research Articles

Processing Fractional Differential Equations Using ψ-Caputo Derivative

Recently, many scientists have studied a wide range of strategies for solving characteristic types of symmetric differential equations, including symmetric fractional differential equations (FDEs). In our manuscript, we obtained sufficient conditions to prove the existence and uniqueness of solutions (EUS) for FDEs in the sense ψ-Caputo fractional derivative (ψ-CFD) in the second-order 1<α<2. We know that ψ-CFD is a generalization of previously familiar fractional derivatives: Riemann-Liouville and Caputo. By applying the Banach fixed-point theorem (BFPT) and the Schauder fixed-point theorem (SFPT), we obtained the desired results, and to embody the theoretical results obtained, we provided two examples that illustrate the theoretical proofs.

Symmetric Difference Operator in Quantum Calculus

The main focus of this paper is to develop certain types of fundamental theorems using q, q(α), and h difference operators. For several higher order difference equations, we get two forms of solutions: one is closed form and another is summation form. However, most authors concentrate only on the summation part. This motivates us to develop closed-form solutions, and we succeed. The key benefit of this research is finding the closed-form solutions for getting better results when compared to the summation form. The symmetric difference operator is the combination of forward and backward difference symmetric operators. Using this concept, we employ the closed and summation form for q, q(α), and h difference symmetric operators on polynomials, polynomial factorials, logarithmic functions, and products of two functions that act as a solution for symmetric difference equations. The higher order fundamental theorems of q and q(α) are difficult to find when the order becomes high. Hence, by inducing the h difference symmetric operator in q and q(α) symmetric operators, we find the solution easily and quickly. Suitable examples are given to validate our findings. In addition, we plot the figures to examine the value stability of q and q(α) difference equations.

Geometric Inequalities via a Symmetric Differential Operator Defined by Quantum Calculus in the Open Unit Disk

The present investigation covenants with the concept of quantum calculus besides the convolution operation to impose a comprehensive symmetric q-differential operator defining new classes of analytic functions. We study the geometric representations with applications. The applications deliberated to indicate the certainty of resolutions of a category of symmetric differential equations type Briot-Bouquet.

Regular Fifth-Order Boundary Value Problems

The main purpose of this paper is to introduce a method to handle some boundary value problems generated by fifth-order formally symmetric differential equation and separated, real-coupled and complex-coupled boundary conditions. Moreover, the continuity properties of the eigenvalues of these problems on some data are studied and some Frechet derivatives of the eigenvalues are introduced.

Essential and continuous spectrum of symmetric difference equations

AbstractEssential and continuous spectrum of symmetric difference equations have been investigated. It has been shown that the deficiency indices and the existence of these components of the spectrum are determined by the growth conditions of the coefficients. In particular, the deficiency indices are superimposition of those clusters determined by the coefficient growth. Finally, we have proved the neccessary and sufficient conditions for the existence of essential spectrum of selfadjoint subspace extensions using subspace theory and asymtotic summation.

Global bifurcations close to symmetry

Heteroclinic cycles involving two saddle-foci, where the saddle-foci share both invariant manifolds, occur persistently in some symmetric differential equations on the 3-dimensional sphere. We analyse the dynamics around this type of cycle in the case when trajectories near the two equilibria turn in the same direction around a 1-dimensional connection — the saddle-foci have the same chirality. When part of the symmetry is broken, the 2-dimensional invariant manifolds intersect transversely creating a heteroclinic network of Bykov cycles. We show that the proximity of symmetry creates heteroclinic tangencies that coexist with hyperbolic dynamics. There are n-pulse heteroclinic tangencies — trajectories that follow the original cycle n times around before they arrive at the other node. Each n-pulse heteroclinic tangency is accumulated by a sequence of (n+1)-pulse ones. This coexists with the suspension of horseshoes defined on an infinite set of disjoint strips, where the first return map is hyperbolic. We also show how, as the system approaches full symmetry, the suspended horseshoes are destroyed, creating regions with infinitely many attracting periodic solutions.

Part-metric and its applications in discrete systems

This paper applies the part-metric method to study some types of higher-order symmetric difference equations with several different exponential parameters. These difference equations are proved to have unique equilibria and some useful inequalities regarding the difference equation functions are formulated. By use of the part-metric and a result given by Kruse and Nesemann (1999) [8], some sufficient conditions on the parameters are given to guarantee the global asymptotic stability of the equilibria. Furthermore, by the part-metric defined on matrices, this approach is also applicable to show the global asymptotic stability of some cyclic discrete dynamic systems. The results of this paper are considered a big improvement over many existing results found in the literature.

Dynamics near a periodically-perturbed robust heteroclinic cycle

Robust heteroclinic cycles (RHCs) arise naturally in collections of symmetric differential equations derived as dynamical models in many fields, including fluid mechanics, game theory and population dynamics. In this paper, we present a careful study of the complicated dynamics generated by small amplitude periodic perturbations of a stable robust heteroclinic cycle (RHC). We give a detailed derivation of the Poincaré map for trajectories near the RHC, asymptotically correct in the limit of small amplitude perturbations. This reduces the nonautonomous system in R3 to a 2D map.We identify three distinct dynamical regimes. The distinctions between these regimes depend subtly on different distinguished limits of the two small parameters in the problem. The first regime corresponds to the RHC being only weakly attracting: here we show that the system is equivalent to a damped nonlinear pendulum with a constant torque. In the second regime the periodically-perturbed RHC is more strongly attracting and the system dynamics corresponds to that of a (non-invertible or invertible) circle map. In the third regime, of yet stronger attraction, the dynamics of the return map is chaotic and no longer reducible to a one-dimensional map. This third regime has been noted previously; our analysis in this paper focusses on providing quantitative results in the first two regimes.

Global stability of some symmetric difference equations

Suppose r∈(0,1],m∈N and 1⩽k1<k2<⋯<k2m+1, and let S2m+1={1,2,…,2m+1}. We show that every positive solution to the difference equation yn=P2m+12m+1(yn-k1r,yn-k2r,…,yn-k2m+1r)P2m2m+1(yn-k1r,yn-k2r,…,yn-k2m+1r),n∈N0, where P2m+12m+1(x1,x2,…,x2m+1)=∑r=1rodd2m+1∑{t1,t2,…,tr}⊆S2m+1t1<t2<⋯<trxt1xt2⋯xtr and P2m2m+1(x1,x2,…,x2m+1)=1+∑r=2reven2m∑{t1,t2,…,tr}⊂S2m+1t1<t2<⋯<trxt1xt2⋯xtr, converges to one. This result confirms a quite recent conjecture posed by Liu and Yang (2010) in [10]. We also prove another result regarding a related equation.

Global Behavior of Two Families of Nonlinear Symmetric Difference Equations

We mainly investigate the global asymptotic stability and exponential convergence of positive solutions to two families of higher‐order difference equations, one of which was recently studied in Stević′s paper (2010). A new concise proof is given to a quite recent result by Stević and analogous parallel result of the other inverse equation, which extend related results of Aloqeili (2009), Berenhaut and Stević (2007), and Liao et al. (2009).

Inequalities among Eigenvalues of Second-Order Symmetric Equations on Time Scales

We consider coupled boundary value problems for second-order symmetric equations on time scales. Existence of eigenvalues of this boundary value problem is proved, numbers of their eigenvalues are calculated, and their relationships are obtained. These results not only unify the existing ones of coupled boundary value problems for second-order symmetric differential equations but also contain more complicated time scales.

A generalization of classical symmetric orthogonal functions using a symmetric generalization of Sturm–Liouville problems

In this research, usual Sturm–Liouville problems are extended for symmetric functions so that the corresponding solutions preserve the orthogonality property. Two basic examples as special samples of a generalized Sturm–Liouville problem are then introduced. The first example generalizes the associated Legendre functions having extensive applications in physics and engineering and the second example introduces a generic differential equation with various sub-cases having orthogonal solutions. For instance, this generic equation possesses a symmetric differential equation containing a basic solution of symmetric orthogonal polynomials.

Self-Adjoint Operators Generated from Non-Lagrangian Symmetric Differential Equations Having Orthogonal Polynomial Eigenfunctions

We discuss the self-adjoint spectral theory associated with a certain fourth-order non-Lagrangian symmetrizable ordinary differential equation l 4 [y] = Ay that has a sequence of orthogonal solutions. This example was first discovered by Jung, Kwon, and Lee. In their paper, they derive the remarkable formula for these polynomials {Q n (x)} ∞ n = 0 : Q n (x) = n ∫ x 1 PL n - 1 (t)dt, n ∈ N, where {PL n (x)} ∞ n = 0 are the left Legendre type polynomials. The left Legendre type polynomials and the spectral analysis of the associated symmetric fourth-order differential equation that they satisfy have been extensively studied previously by Krall, Loveland, Everitt, and Littlejohn. Despite the non-symmetrizability of the expression l 4 [.], we show that there exists a self-adjoint operator S in a certain Hilbert space H generated by l 4 [.] that has the polynomial sequence of ordered pairs { } ∞ n = 0 as a complete set of eigenfunctions in H. This operator S is related to the derivative of the self-adjoint operator T which has the left Legendre type polynomials {PL n (x)} ∞ n = 0 as eigenfunctions. We also develop a left-definite theory for l 4 [.]. This unexpected example casts further difficulties in the efforts to extend and generalize certain classification results in orthogonal polynomials and differential equations.

Global structure of homoclinic bifurcations: A combinatorial approach

A method for understanding the appearance of periodic orbits in families of symmetric differential equations is described, based on the changes in behaviour of the unstable manifold of a stationary point. The technique is illustrated using an example which exhibits chaotic behaviour and hysteresis (or bistability).

Iteration of some discretizations of the nonlinear Schrödinger equation

We consider several discretizations of the nonlinear Schrödinger equation which lead naturally to the study of some symmetric difference equations of the form φn+1+φn-1 = f(φn). We find a variety of interesting and exotic behaviour from simple closed orbits to intricate patterns of orbits and loops in the (φn+1,φn) phase-plane. Some analytical results for a special case are also presented.

A Titchmarsh-Weyl matrix function for symmetric differential equations of order 2n with an indefinite weight function

SynopsisA Titchmarsh-Weyl matrix function W(λ) is defined for the differential equation of order 2nwith po&gt;0, pk≧0, k = 1, 2, …, n on 005B;0, b), λєℂ and an indefinite weight function r. It is shown that this function W(λ) belongs to some class and that some operators associated with the above equation are definitizable in the Krein space . In the particular case n = 1, these results are contained in an earlier paper by the present author and H. Langer.

On the L2 nature of solutions of nth order symmetric differential equations and McLeod's conjecture

SynopsisWe investigate the integrable square properties of solutions of linear symmetric differential equations of arbitrarily large order 2m, whose coefficients involve a real multiple ɑr of certain positive real powers β of the independent variable x. Information on the L2 nature is obtained by variation of parameters from Meijer function solutions of an associated homogeneous equation of hypergeometric type. When the coefficients of the differential expressions are positive, it is possible, by a suitable choice of ɑr, β and m, to obtain between m and 2m —1 linearly independent solutions in L2(0, ∞). This proves a conjecture of J. B. McLeod that the deficiency index can take values between m and 2m —1 for such operators.

An approximation theory for generalized Fredholm quadratic forms and integral-differential equations

An approximation theory is given for a very general class of elliptic quadratic forms which includes the study of 2nth order (usually in integrated form), selfadjoint, integral-differential equations. These ideas follows in a broad sense from the quadratic form theory of Hestenes, applied to integral-differential equations by Lopez, and extended with applications for approximation problems by Gregory. The application of this theory to a variety of approximation problem areas in this setting is given. These include focal point and focal interval problems in the calculus of variations/optimal control theory, oscillation problems for differential equations, eigenvalue problems for compact operators, numerical approximation problems, and finally the intersection of these problem areas. In the final part of our paper our ideas are specifically applied to the construction and counting of negative vectors in two important areas of current applied mathematics: In the first case we derive comparison theorems for generalized oscillation problems of differential equations. The reader may also observe the essential ideas for oscillation of many nonsymmetric (indeed odd order) ordinary differential equation problems which will not be pursued here. In the second case our methods are applied to obtain the “Euler-Lagrange equations” for symmetric tridiagonal matrices. In this significant new result (which will allow us to reexamine both the theory and applications of symmetric banded matrices) we can construct in a meaningful way, negative vectors, oscillation vectors, eigenvectors, and extremal solutions of classical problems as well as faster more efficient algorithms for the numerical solution of differential equations. In conclusion it appears that many physical problems which involve symmetric differential equations are more meaningful presented as integral differential equations (effects of friction on physical processes, etc.). It is hoped that this paper will provide the general theory and present examples and methods to study integral differential equations.

Spherically symmetric differential equations, the rotation group, and tensor spherical functions

AbstractIn this paper a scheme is developed for handling tensor partial differential equations having spherical symmetry. The basic technique is that of Gelfand and Shapiro ((2), §8) by which tensor fields defined on a sphere give rise to scalar fields defined on the rotation group. These fields may be expanded as series of functions, where,mis fixed and the matricesTl(g) form a 21+ 1 dimensional irreducible representation of.Spherically symmetric operations, such as covariant differentiation of tensors and the contraction of tensors with other spherically symmetric tensor fields, are shown to act in a particularly simple way on the terms of the series mentioned above: terms with givenl, nare transformed into others with the same values ofl, n. That this must be so follows from Schur's Lemma and the fact that for eachmandlthe functionsform a basis for an invariant subspace of functions onof dimension 2l+ 1 in which an irreducible representation ofacts. Explicit formulae for the results of such operations are presented.The results are used to show the existence of scalar potentials for tensors of all ranks and the results for tensors of the second rank are shown to be closely related to those recently obtained by Backus(1).This work is intended for application in geophysics and other fields where spherical symmetry plays an important role. Since workers in these fields may not be familiar with quantum theory, some matter in sections 2–5 has been included in spite of the fact that it is well known in the quantum theory of angular momentum.

Symmetric differential equations

Symmetric differential equations