Iterative Functional 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.

Some sufficient conditions which allow to extend every local solution of a simultaneous system of equations in a single variable of the form \[ \varphi(x) = h (x, \varphi[f_1(x)],\ldots,\varphi[f_m(x)]),\] \[\varphi(x) = H (x, \varphi[F_1(x)],\ldots,\varphi[F_m(x)]),\] to a global one are presented. Extensions of solutions of functional equations, both in single and in several variables, play important role (cf. for instance [M. Kuczma, Functional equations in a single variable, Monografie Mat. 46, Polish Scientific Publishers, Warsaw, 1968, M. Kuczma, B. Choczewski, R. Ger, Iterative functional equations, Encyclopedia of Mathematics and Its Applications v. 32, Cambridge, 1990, J. Matkowski, Iteration groups, commuting functions and simultaneous systems of linear functional equations, Opuscula Math. 28 (2008) 4, 531-541]).

Fractal and Hausdorff dimensions for systems of iterative functional equations

In 1968–1969, A. V. Jategaonkar published his famous constructions of left but not right noetherian rings that provided counterexamples to several important conjectures of that era. These examples, and others like them, seemed to indicate that, in general, the task of completely understanding the structure of indecomposable injective modules over one-sided noetherian rings was hopeless. In this paper I show how to deduce by natural methods, directly from the known description of these rings and their properties, explicit computational descriptions of the indecomposable injective left modules over Jategaonkar's rings. I use these explicit descriptions to answer some simple structural questions about the indecomposables. The structure of indecomposable injective modules over a commutative noetherian ring has been well understood since the work of Matlis[10] in the 1950's. There is a natural correspondence between the indecomposable injectives and the localizations of the ring, and the structure of ...

We study linear systems of ordinary differential equations of higher order with an identically singular matrix at the higher derivative of the required vector-valued function in the domain of definition. We give definitions of the index and singular points for these systems, formulate the conditions of solvability, and deduce a formula for the general solution. The algorithms for finding the index and singular points are presented.

Equations including unknown functions are called function al equations([1]). Among them, functionalequations containing only one independent variable are cal led iterative functional equations([9]). On theother hand, functional equations, like Cauchy's equation( [10]), containing two or more independent vari-ables for unknown functions of a single variable are called f unctional equations of several variables([2]).Among iterative functional equations, the problem of itera tive roots is important. If the n -th iterateof a function f is equal to g, i.e.f

This paper is concerned with an iterative functional equation from the problem of invariant curves. By reducing the equation with the Schröder transformation to another functional equation without iteration of the unknown function f. For technical reasons, in previous work the constant λ given in the Schröder transformation is required to fulfil that λ is off the unit circle or lies on the circle with the Diophantine condition. In this paper, we obtain analytic solutions in the case of λ at resonance, i.e., at a root of the unity and the case of λ near resonance under the Brjuno condition.KeywordsIterative functional equationConvergenceBrjuno conditionAnalytic solutionMajorant series

In previous chapters, we have learned to solve a wide range of differential equations. Primarily, our focus has been on linear differential equations: first-order linear equations, higher-order linear equations with constant coefficients, and systems of linear equations with constant coefficients. Indeed, we have learned through a variety of techniques that under the proviso that a differential equation or system is linear, we can almost always find a solution.

A new technique for solving the large system of linear algebraic equations associated with implicit differencing of multidimensio nal partial differential equations is presented. The coefficient matrix of the equations is factored, and then approximations to certain terms in the matrix are obtained from series expansions. The resulting system of equations is solved easily. The method is developed and demonstrated using a simple representative two-dimensional equation. Very good results are obtained when one direction is dominant. MPLICIT finite-difference schemes for the solution of multidimensional partial differential equations are usually stable and therefore applicable to a large class of problems. However, they are difficult to implement and may require an excessive amount of computer storage and time. The long computing time arises from the need to solve the large system of linear algebraic equations that result from the differencing. The computing time can be reduced significantly by approximating the coefficient matrix of the linear equations with a matrix that produces a system of equations that are relatively easy to solve. Among such methods are the alternating direction method (ADI)l used by Beam and Warming2 and Stone's strongly implicit method,3 which has been tested by Linetal.4 In this paper, a new technique for solving the large system of linear algebraic equations associated with implict differencing of multidimensional partial differential equations is presented. This method, called the pseudo-elimination method (PE), is shown to be faster than Stone's method for certain problems. The method is directly applicable to linear and linearized nonlinear systems of parabolic or elliptic partial differential equations. In order to discuss the method, a simple linear partial differential equation will be used; however, it should be kept in mind that the PE method is applicable to much more complicated problems. The question of whether the method will work when applied to difficult problems is not addressed. The scope of this paper is limited to presenting the method and illustrating, via a simple problem, that the method has some merit and deserves further study.

One proves theorems regarding the estimates of the solutions of systems of linear algebraic equations and inequalities. On the basis of these theorems one suggests a method for estimating the norm of the inverse matrix of a system of difference equations which approximates a boundary-value problem for an integrodifferential equation. The method allows us to eliminate the restrictions which are usually imposed on the coefficients of the integrodifferential equation in order to ensure the diagonal dominance in the system of the difference equations. One considers an application to nonlinear problems.

Identifying integrable coupled nonlinear ordinary differential equations (ODEs) of dissipative type and deducing their general solutions are some of the challenging tasks in nonlinear dynamics. In this paper we undertake these problems and unearth two classes of integrable coupled nonlinear ODEs of arbitrary order. To achieve these goals we introduce suitable nonlocal transformations in certain linear ODEs and generate the coupled nonlinear ODEs. In particular, we show that the problem of solving these classes of coupled nonlinear ODEs of any order effectively reduces to solving a single first order nonlinear ODE. We then describe a procedure to derive explicit general solutions for the identified integrable coupled ODEs, when the above mentioned first order nonlinear ODE reduces to a Bernoulli equation. The equations which we generate and solve include the two coupled versions of modified Emden equations (in second order), coupled versions of Chazy equations (in third order), and their variants, higher dimensional coupled Ricatti and Abel’s chains, as well as a new integrable chain and higher order equations.

The application of mathematics to natural phenomena often brings up the question : when a linear equation is changed slightly, how much of a change results in the solution? Here vector is used to include, for example, vectors in Hubert space and elements of (linear) function spaces; also it is assumedC) that the equation is uniquely solvable. We formulate and answer the question in §2 for a linear equation in a Banach space, then specialize to deduce perturbation limits for linear algebraic systems, infinite systems, and integral equations. Finally we obtain error limitations for certain approximate methods of solving infinite linear systems (method of segments) and integral equations (method of Goursat-Schmidt). It is hoped that these limitations may be useful to applied mathematicians. The methods and results of this paper unify and extend investigations concerning algebraic systems by F. R. Moulton [15](2), Etherington [6] and the author [13, 14]; and on integral equations of Fredholm type and second kind by Tricomi [21]. However, they do not cover perturbation questions associated with characteristic values and characteristic functions, which have been studied by Lord Rayleigh [16, p. 115], Courant [4, p. 296] and Mrs. Adams [l]. §1 is expository, containing as much about abstract spaces as is needed for §2. 1. Vector spaces. It will be useful to collect here some facts about normed linear spaces ispaces L), Banach spaces ispaces B), and linear transformations. A space L is linear: if xEL and a is any complex number, the product ax is defined and axEL; if also y EL, the sum x+y is defined and x+yEL. With each element x of L is associated a non-negative real number ||*||, its norm; ||*|| >0 unless x = @, the zero-element of L; |J@|| =0. The norm has properties of an absolute value: ||a*|| = \a -11*11, ||*+y|| a||*||+||y||(We have described a complex space L; in a real space, number a must be real.) A normed linear space is a Banach space 5 [2, p. 53] if it is furthermore complete: if {*„} is an infinite sequence of elements of 5, and if II*»—*»||—*0 as m, m—»oo, there exists a xo of 5 such that ||*„—*o||—»0 as n—+ oo (strong completeness). Examples of such spaces are listed in Banach's book [2, pp. 10-12, examples 3-10], and several occur in the remainder of this paper.

An approach to solutions of systems of linear partial differential equations with applications

Introduction. This study investigates the possibility of increasing the accuracy of numerically solving boundary value problems using the modified Bubnov-Galerkin method with a linear ordinary differential equation, where the coefficients and the right-hand side are continuous functions. The order of the differential equation n must be less than the number of coordinate functions m.Materials and Methods. A modified Petrov-Galerkin method was used to numerically solve the boundary value problem. It employs a system of linearly independent power-type basis functions on the interval [−1,1], each normalized by the unit Chebyshev norm. The system of linear algebraic equations includes only the linearly independent boundary conditions of the original problem.Results. For the first time, an integral quadrature formula with a 22nd order error was developed for a uniform grid. This formula is used to calculate the matrix elements and coefficients in the right-hand side of the system of linear algebraic equations, taking into account the scalar product of two functions based on the new quadrature formula. The study proves a theorem on the existence and uniqueness of a solution for boundary value problems with general non-separated conditions, provided that n linearly independent particular solutions of a homogeneous differential equation of order n are known.Discussion and Conclusion. The hydrodynamic problem in a viscous strong boundary layer with a third-order equation was precisely solved. The analytical solution was compared with its numerical counterpart, and the uniform norm of their difference did not exceed 5·10‒15. The formulas derived using the generalized Bubnov-Galerkin method may be useful for solving boundary value problems with linear ordinary differential equations of the third and higher orders.

A system of linear interval equations is called solvable if each system of linear equations contained therein is solvable. In the main result of this paper it is proved that solvability of a general rectangular system of linear interval equations can be characterized in terms of nonnegative solvability of a finite number of systems of linear equations which, however, is exponential in matrix size; the problem is proved to be NP-hard. It is shown that three earlier published results are consequences of the main theorem, which is compared with its counterpart valid for linear interval inequalities that turn out to be much less difficult to solve.