First-Order Ordinary Differential Research Articles

On Eigenfunction Expansions of First-Order Symmetric Systems and Ordinary Differential Operators of an Odd Order

We study general (not necessarily Hamiltonian) first-order symmetric systems \({J y'-B(t)y=\Delta(t) f(t)}\) on an interval \({\mathcal{I}=[a,b \rangle}\) with the regular endpoint a. It is assumed that the deficiency indices \({n_{\pm}(T_{\rm min})}\) of the minimal relation \({T_{\rm min}}\) satisfy \({n_{+}(T_{\rm min})\leq n_{-}(T_{\rm min})}\). We define special \({\lambda}\)-depending boundary conditions with Nevanlinna-type spectral parameter \({\tau=\tau(\lambda)}\) at the singular endpoint b. With boundary value problem involving such conditions, we associate an exit space self-adjoint extension \({\tilde{T}}\) of \({T_{\rm min}}\) and the \({N \times N}\)-matrix m-function \({m(\cdot)}\) of the size \({N={\rm dim ker} (iJ+I)}\). The role of \({m(\cdot)}\) is similar to that of the Weyl function for a Hamiltonian system. Using the m-function, we obtain the eigenfunction expansion with \({N \times N}\)-matrix spectral function \({\Sigma(\cdot)}\). Moreover, we represent \({\Sigma(\cdot)}\) immediately in terms of a boundary parameter \({\tau}\). We also characterize certain spectral properties of the extension \({\tilde{T}}\). Application of these results to ordinary differential operators of an odd order enables us to complete the results by Everitt and Krishna Kumar on the Titchmarsh–Weyl theory of such operators.

Web Geometry of a System of First-Order Autonomous Ordinary Differential Equations

The goal of this paper is to propose Cartan equivalence problem for a system of n first-order autonomous ordinary differential equations (ODEs) under a Web transformation. In fact, we obtain the necessary and sufficient conditions in the form of differential invariants of this system of n first-order ODEs in the Web geometry.

A Scalar Homotopy Method for Solving an Over/Under-Determined System of Non-Linear Algebraic Equations

Iterative algorithms for solving a system of nonlinear algebraic equa- tions (NAEs): Fi(xj) = 0, i, j= 1,. . . ,n date back to the seminal work of Issac New- ton. Nowadays a Newton-like algorithm is still the most popular one to solve the NAEs, due to the ease of its numerical implementation. However, this type of al- gorithm is sensitive to the initial guess of solution, and is expensive in terms of the computations of the Jacobian matrixFi=¶xj and its inverse at each iterative step. In addition, the Newton-like methods restrict one to construct an iteration proce- dure for n-variables by using n-equations, which is not a necessary condition for the existence of a solution for underdetermined or overdetermined system of equations. In this paper, a natural system of first-order nonlinear Ordinary Differential Equa- tions (ODEs) is derived from the given system of Nonlinear Algebraic Equations (NAEs), by introducing a scalar homotopy function gauging the total residual error of the system of equations. The iterative equations are obtained by numerically in- tegrating the resultant ODEs, which does not need the inverse ofFi=¶xj. The new method keeps the merit of homotopy method, such as the global convergence, but it does not involve the complicated computation of the inverse of the Jacobian ma- trix. Numerical examples given confirm that this Scalar Homotopy Method (SHM) is highly efficient to find the true solutions with residual errors being much smaller.

Upper and Lower Solutions for First-Order Discontinuous Ordinary Differential Equations

Upper and Lower Solutions for First-Order Discontinuous Ordinary Differential Equations

Eigenstructure bidirectional associative memory: an effective synthesis procedure

We propose a computationally efficient synthesis procedure for a class of bidirectional associative memories. Networks are described by a system of first-order ordinary difference equations which are defined on a closed hypercube of the state-space with solutions extended to the corner of the hypercube. The proposed algorithm possesses several advantages since it is possible: 1) to exert control over the number of spurious states; 2) to estimate the basins of attraction of the stable memories; and 3) under certain constraints, to effectively store a number of desired stable memories which by far exceed the order of the network. The applicability of the present results is demonstrated by means of a specific application in reconfigurable control of flexible structures.