Expansion problems associated with a system of integral equations

3 - LINEAR SYSTEMS

We consider boundary value problems with periodic boundary conditions for first-order linear systems of impulsive ordinary differential equations with unknown right-hand sides and jumps of solutions at the impulse points entering into the statement of these problems which are assumed to be subjected to some quadratic restrictions. From indirect noisy observations of their solutions on a finite system of intervals, we obtain the optimal, in certain sense, estimates of images of their right-hand sides under linear continuous operators. Under the condition that the unknown correlation functions of noises in observations belong to some special sets, it is established that such estimates and estimation errors are expressed explicitly via solutions of special periodic boundary value problems for linear impulsive systems of ordinary differential equations.

For each associative, commutative, and unitary algebra over the field of real or complex numbers and an integrable nonlinear ordinary differential equation we can to construct integrable systems of ordinary differential equations and integrable systems of partial differential equations. In this paper we consider in some sense the inverse problem. Determine the conditions under which a given system of ordinary differential equations can be represented as a differential equation in some associative, commutative and unitary algebra. It is also shown that associativity is not a necessary condition.

A half-space containing a surface-breaking crack of uniform depth is subjected to three-dimensional dynamic loading. The elastodynamic stress-analysis problem has been decomposed into two problems, which are symmetric and antisymmetric, respectively, relative to the plane of the crack. The formulation of each problem has been reduced to a system of singular integral equations of the first kind. The symmetric problem is governed by a single integral equation for the opening-mode dislocation density. A pair of coupled integral equations for the two sliding-mode dislocation densities govern the antisymmetric problem. The systems of integral equations are solved numerically. The stress-intensity factors are obtained directly from the dislocation densities. The formulation is valid for arbitrary 3-D loading of the half-space. As an example, an applied stress field corresponding to an incident Rayleigh surface wave has been considered. The dependence of the stress-intensity factors on the frequency, and on the angle of incidence, is displayed in a set of figures.

This paper introduces the concept of first-order systems of linear fuzzy differential equations for \(\mathcal {S}\)-linearly correlated fuzzy processes. These fuzzy processes have range embedded in Banach spaces of fuzzy numbers, and the fuzzy initial value problems studied are given in terms of the Fréchet derivative of these fuzzy functions. An equivalence between the first-order systems of linear fuzzy differential equations and a family of classical first-order systems of linear differential equations is established. Also, conditions on the existence and uniqueness of the solutions are presented. Lastly, an application on the multiple mass-spring system is provided.KeywordsBanach spacesStrongly linearly independence\(\mathcal {S}-\)linearly correlated fuzzy processesSystems of fuzzy differential equationsMass-spring multiple system

It is well known that a model of any electrical circuit can be translated into a system of ordinary differential equations. Usually, the simulation is performed in the time-domain using classical discrete approaches convenient for both linear and non-linear systems. An other method consists in the quantization on the variable magnitude which is based on discrete events. The discrete event system specification formalism is adapted to simulate any electrical circuit described with a system of ordinary differential equations using the system state quantification. In this paper, a solution to make more efficient the magnitude quantification for a system of linear ordinary differential equations is given. The proposed technique is used to simulate a simple three phase coupled power systems and it is compared to other solutions given by the explicit integration using formal language and the time-domain discretization using fixed-step fifth order Runge-Kutta integration method.

In the case of n functions of a single variable, the vanishing of the wronskian is the most familiar criterion for their linear dependence. The wronskian also plays an important part in connection with the theory of a single ordinary homogeneous linear differential equation of the nth order, even in questions which do not concern the linear dependence of solutions of the equation. It is the purpose of the present paper to generalize the fundamental facts connected with these topics, to the case of functions of several variables. The characteristic property of an ordinary homogeneous linear differential equation is that any solution is expressible linearly, with constant coefficients, in terms of a fundamental set of solutions. The natural generalization to several independent variables is afforded by the completely integrable system of homogeneous linear partial differential equations, in one dependent variable, any solution of which is likewise linearly dependent on a finite number of solutions of the system. It is such systems which are discussed in the latter part of this paper; the subject of the linear dependence of functions of several variables is developed in the first part, chiefly with a view to its subsequent application to completely integrable systems. In our discussion of linear dependence, we consider throughout n functions Yl, Y2, Y n,, of p independent variables u1, u2, * up . Functions and variables may be either real or complex; if any of the variables be complex, we suppose the functions to be analytic in those variables. All of the main theorems, however, are stated and proved under the supposition that the independent variables are real; the modifications usually simplifications and omissions-are easily supplied for cases in which one or more of the independent variables is complex. The n functions, then, of the real variables

A mixed mode crack problem in functionally graded materials is formulated to a system of Cauchy singular Fredholm integral equations, then the system is solved by the singular integral equation method (SIEM). This specific crack problem has already been solved by N. Konda and F. Erdogan (Konda & Erdogan 1994). However, many mathematical details have been left out. In this paper we provide a detailed derivation, both analytical and numerical, on the formulation as well as the solution to the system of singular Fredholm integral equations. The research results include crack displacement profiles and stress intensity factors for both mode I and mode II, and the outcomes are consistent with the paper by Konda & Erdogan (Konda & Erdogan 1994).

The paper deals with a numerical algorithm to reduce the overall system of integral equations describing the heat transfer process at any geometrically complex area (both twodimensional and three-dimensional), to the iterative solution of a system of independent onedimensional integral equations. This approach has been called "string method" and has been used to solve a number of applications, including the problem of the detonation wave front for the calculation of heat loads in pulse detonation engines. In this approach "the strings" are a set of limited segments parallel to the coordinate axes, into which the whole solving area is divided (similar to the way the strings are arranged in a tennis racket). Unlike other grid methods where often for finding solutions, the values of the desired function in the region located around a specific central point here in each iteration step is determined by the solution throughout the length of the one-dimensional "string", which connects the two end points and set them values and determine the temperature distribution along all the strings in the first step of an iterative procedure.

The Monte Carlo method is applied to solve Cauchy problems for a system of linear and nonlinear ordinary differential equations. The Monte Carlo method is relevant for the solution of large systems of equations and in the case of small smoothness of initial functions. In this case, the system is reduced to an equivalent system of integral equations of the Volterra type. For linear systems, this transformation allows removing constraints connected with a convergence of a majorizing process. Examples of estimates of solution functionals are provided, and a behavior of their variances are discussed. In the general case, a solution interval is divided into finite subintervals, on which the nonlinear function is approximated by a polynomial. The obtained integral equation is solved by using branched Markov chains with absorption. Algorithm parallelization problems arising in this case are discussed in this paper. A one-dimensional cubic equation is considered as an example. A choice of transition densities of branching is discussed. A method of generations is described in detail. Numerical results are compared with a solution obtained by the Runge–Kutta method.

THE purpose of this paper is to describe a short method for solving arbitrary systems of linear algebraic equations, and evaluating determinants, the quantities involved being either real or complex. For proofs and for treatment of complex systems whose symmetrical coefficients are conjugate, see appendix. The cases considered are: 1. Arbitrary systems with real coefficients, which occur in obtaining stresses in structures, in solving systems of linear differential equations with constant coefficients (transient problems), etc. 2. Symmetrical systems with real coefficients, which occur with d-c networks, undamped vibration, deflections in structures, least square processes, Ritz's method, etc. 3. Symmetrical systems with complex coefficients, which occur with a-c networks, and forced vibration with dissipation. 4. Arbitrary systems with complex coefficients, which occur in certain vibration problems involving gyroscopic action. 5. Systems involving two sets of variables, which occur when the currents in a network are to be found for a variety of impressed voltages, also in the approximate solution of integral equations arising in electric field problems.

In a recent paper (L. Greengard and V. Rokhlin, On the Numerical Solution of Two‐Point Boundary Value Problems. in Communications on Pure and Applied Mathematics, Volume XLIV, 1991, pages 419‐452). L. Greengard and V. Rokhlin introduce a numerical technique for the rapid solution of integral equations resulting from linear two‐point boundary value problems for second‐order ordinary differential equations. In this paper, we extend the method to systems of ordinary differential equations. After reducing the system of differential equations to a system of second kind integral equations, we discretize the latter via a high‐order Nyström scheme. A somewhat involved analytical apparatus is then constructed which allows for the solution of the discrete system using O(N.p2.n3) operations with N the number of nodes on the interval, p the desired order of convergence, and n the number of equations in the system. Thus, the advantages of the integral equation formulation (small condition number, insensitivity to boundary layers, insensitivity to endpoint singularities, etc.) are retained, while achieving a computational efficiency previously available only to finite difference or finite element methods.We in addition present a Newton method for solving boundary value problems for nonlinear first‐order systems in which each Newton iterate is the solution of a second kind integral equation; the analytical and numerical advantages of integral equations are thus obtained for nonlinear boundary value problems. © 1994 John Wiley & Sons. Inc.

The purpose of this paper is to derive a means of scaling the dependent variables of a system of linear differential equations so that the normalized system accurately reflects the dependence of the system on initial data. The theory is developed for the special case of a system of first order hyperbolic partial differential equations in two independent variables. Several other systems of differential equations, to which the same normalization could be applied, are also indicated. The scaling is equivalent to row and column scaling commonly used in linear algebra. For the case of positive matrices (that is, for the case where the diagonalizing matrix and its inverse have no zero elements), the theory for the “1” and “$\infty $ ” norms has been thoroughly developed by Bauer [1]. The assumption of positive matrices is too stringent for some applications. Thus, the theory is generalized to show that for irreducible matrices a best scaling exists (for both the “1” and “ $\infty $” norms), while for simply reducible matrices a best scaling exists for at least one of these two norms. Several examples are given.

Stability of solutions of systems of Volterra integral equations

In this paper, we define sufficient conditions for the solvability in quadratures of systems of linear ordinary differential equations of higher orders with variable coefficients. The method of variation of constants for equations of higher orders and for systems of equations of first order is generalized to the case of such systems. Results are obtained by reducing the systems under consideration to matrix equations of higher orders studied by methods of the theory of Lie algebras.