A first-order system of equations on a compact star graph

Generalisation of the Lagrange multipliers for variational iterations applied to systems of differential equations

In this study, we develop the differential transform method in a new scheme to solve systems of first-order differential equations. The differential transform method is a procedure to obtain the coefficients of the Taylor series of the solution of differential and integral equations. So, one can obtain the Taylor series of the solution of an arbitrary order, and hence, the solution of the given equation can be obtained with required accuracy. Here, we first give some basic definitions and properties of the differential transform method, and then, we prove some theorems for solving the linear systems of first order. Then, these theorems of our system are converted to a system of linear algebraic equations whose unknowns are the coefficients of the Taylor series of the solution. Finally, we give some examples to show the accuracy and efficiency of the presented method.

A spectral theory is deduced for differential eigenvalue problems related to a formally selfadjoint differential equation Su=λ Tu, where u is complex-valued and λ is the eigenvalue parameter. The equation or rather the differential operators S and T are considered on an arbitrary open interval of the real axis. The lower order operator T is assumed to have a positive definite Dirichlet integral which serves as scalar product in spectral theorems determined by symmetric boundary conditions. The theory is given in terms of ordered pairs u/\(\dot u\)of functions. Thus symmetric boundary conditions are certain subrelations of {u/\(\dot u\): Su=T\(\dot u\)}. If for instance T is the identity operator the boundary conditions are equally well described as conditions on u only. As far as the spectral theorem is concerned the method of the paper is easily transferred to the case when S instead of T has a positive definite Dirichlet integral. In the here considered case with T positive a kernel representation of the resolvent is deduced and used to prove the regularity of the elements of eigenspaces belonging to finite intervals of the spectral axis. The theory was worked out independently of the investigations by F.W. Schafke, A. Schneider and H.-D. Niesser of systems of first order equations to which it seems related in different respects.

The defining equation is constructed for an algebra of infinitesimal Lie-symmetry-group operators of a first-order system of ordinary differential equations; some properties of the algebra are formulated; and the relation of the algebra with conservation laws is discussed. A similar approach is applied to a Hamilton system of general form.

In the paper estimates are established on the solution of systems of first-order differential equations subject to two-point conditions of the form which enable us, in particular, to obtain an estimate of the order of uniform convergence of the method of lines for solving periodic boundary-value problems for second-order nonlinear parabolic partial differential equations of form For problems with boundary conditions containing the derivatives where the functions satisfy, in a small neighborhood of the solutionu(t,x) being examined of Eq. (3), the inequalities , for which the approximation is only of the first order relative to the net steph, the uniform convergence of the approximate solutions to the exact one is established to the second order relative toh.

In this paper, we prove the existence and uniqueness for systems of first-order impulsive differential equations with periodic boundary conditions. To establish such results, sufficient conditions of limit forms are given.KeywordsPeriodic boundary value problemsImpulsive equationsFixed-point theory

11 - Snails, snakes, and first-order ordinary differential equations

A formal solution has been constructed for a countable system of first-order differential equations with two small parameters in the case where the principal matrix consists of Jordan blocks of equal dimension with distinct characteristic numbers. The asymptotic behavior of this solution has been investigated.The aim of the article is to determine the conditions under which a countable system of first-order linear differential equations with two small parameters, in the case where the principal matrix consists of Jordan blocks of equal dimension and distinct characteristic numbers, admits a solution; to construct a formal solution; and to prove its asymptotic behavior.

In Nambu mechanics an autonomous system of first-order ordinary differential equations du i /dt=F i(u) (i=1,…,n) is constructed with the help of (n−1) smooth functionsI i . These smooth functions are first integrals of this dynamical system. If the functionsI i are polynomials, then the system is algebraic completely integrable. We discuss the question whether the first integrals determine uniquely the autonomous system of first-order differential equations. Then we give a generalization of Nambu mechanics.

The present paper is the first attempt to study the transformation of spin-wave resonance spectra when symmetric boundary conditions are smoothly replaced by asymmetric. The transition is done by gradually reducing the thickness of one of the layers in a three-layer film. Spin deexcitation is caused by a dissipation mechanism. We find that in the transition region between symmetric and asymmetric boundary conditions the dispersion curve experiences a break, whose position depends on the degree of deexcitation (the thickness of the upper layer). The break is caused by the appearance of asymmetric transitional spin-wave modes, which cannot be excited under symmetric boundary conditions.

This paper focuses on studying the Lie symmetry and a conserved quantity of a system of first-order differential equations. The determining equations of the Lie symmetry for a system of first-order differential equations, from which a kind of conserved quantity is deduced, are presented. And their general conclusion is applied to a Hamilton system, a Birkhoff system and a generalized Hamilton system. Two examples are given to illustrate the application of the results.

We derive and compare first-order wave propagation systems for variable-tilt elastic and acoustic tilted transversely isotropic (TTI) media. Acoustic TTI systems are commonly used in reverse-time migration. Starting initially with homogeneous vertical transversely isotropic (VTI) media, and then extending to heterogeneous variable-tilt TI media, we derive a pseudoacoustic [Formula: see text] first-order system of differential equations by setting the shear-wave speeds to zero and simplifying the full-elastic system accordingly. This [Formula: see text] system conserves a complete energy, but only when the anelliptic anisotropy parameter [Formula: see text]. For [Formula: see text] (including isotropic media), the system allows linearly time-growing and spatially nonpropagating nonphysical solutions frequently taken for numerical noise. We modified this [Formula: see text] acoustic first-order system by changing the stress variables to obtain a system that stays stable for [Formula: see text]. This system for homogeneous VTI media is generalized to heterogeneous variable-tilt TI media by rotating the stress and strain variables in the full elastic system before setting the shear-wave speeds to zero; the system obtained can be greatly simplified by combining the rotational terms, resulting in only one rotation and extra lower-order terms compared to the [Formula: see text] first-order acoustic system for VTI media. This new system can be simplified further by neglecting the lower-order terms. Both systems (with and without lower-order terms) conserve the same complete energy. Finally, the corresponding [Formula: see text] full elastic system for variable-tilt acoustic TI media can be used for the purposes of benchmarking.

In this article, we examine two problems: a fractional Sturm-Liouville boundary value problem on a compact star graph and a fractional Sturm-Liouville transmission problem on a compact metric graph, where the orders α i {\alpha }_{i} of the fractional derivatives on the ith edge lie in ( 0 , 1 ) (0,1) . Our main objective is to introduce quantum graph Hamiltonians incorporating fractional-order derivatives. To this end, we construct a fractional Sturm-Liouville operator on a compact star graph. We impose boundary conditions that reduce to well-known Neumann-Kirchhoff conditions and separated conditions at the central vertex and pendant vertices, respectively, when α i → 1 {\alpha }_{i}\to 1 . We show that the corresponding operator is self-adjoint. Moreover, we investigate a discontinuous boundary value problem involving a fractional Sturm-Liouville operator on a compact metric graph containing a common edge between the central vertices of two star graphs. We construct a new Hilbert space to show that the operator corresponding to this fractional-order transmission problem is self-adjoint. Furthermore, we explain the relations between the self-adjointness of the corresponding operator in the new Hilbert space and in the classical L 2 {L}^{2} space.

Numerical solution of two-point boundary value problems

1 - INTRODUCTION