Meromorphic Solutions of Algebraic Difference Equations

A function f meromorphic on the complex plane c is called an α-Yosida function if \( {{f}^{\# }}(z) = O(|z{{|}^{\alpha }})\,as|z| \to \infty ,\alpha \geqslant - 1\;where\,{{f}^{\# }}(z) = |f'(z)|(/1 + |f(z){{|}^{2}}) \) We give a sufficient and necessary condition for a meromorphic function f to be not an α-Yosida function by proving a relationship between α-Yosida functions and normal families. We also give some integral characterizations of α-Yosida functions. Applying our results we know that the meromorphic solutions of some higher order algebraic differential equations are α-Yosida functions. Moreover, some estimates for the order of growth of these meromorphic solutions by studying α-Yosida functions are given.Keywordsα-Yosida functionsmeromorphic solutionsalgebraic differential equations

We find all non-rational meromorphic solutions of the equation $ww"-(w')^2=\alpha(z)w+\beta(z)w'+\gamma(z)$, where $\alpha$, $\beta$ and $\gamma$ are rational functions of $z$. In so doing we answer a question of Hayman by showing that all such solutions have finite order. Apart from special choices of the coefficient functions, the general solution is not meromorphic and contains movable branch points. For some choices for the coefficient functions the equation admits a one-parameter family of non-rational meromorphic solutions. Nevanlinna theory is used to show that all such solutions have been found and allows us to avoid issues that can arise from the fact that resonances can occur at arbitrarily high orders. We actually solve the more general problem of finding all meromorphic solutions that are admissible in the sense of Nevanlinna theory, where the coefficients $\alpha$, $\beta$ and $\gamma$ are meromorphic functions.

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In this paper, we mainly consider the existence of meromorphic solutions of nonlinear q-difference equation of typewhere the right-hand side is irreducible, P(z, f (z)) and Q(z, f (z)) are polynomials in f with rational coefficients, and q is a nonzero complex constant.We obtain that such equation has no transcendental meromorphic solution when |q| = 1 and m = degAnd we investigate the growth of transcendental meromorphic solutions of nonlinear q-difference equation and find lower bounds for their characteristic functions for transcendental meromorphic solutions of such equation for the case |q| = 1.

In this article, we investigate the growth and value distribution of finite-order transcendental meromorphic solutions for some non-linear difference equations. For instance, we show that demanding the existence of a transcendental meromorphic solution of a large class of difference equations containing difference Painleve I reduces the class either to a smaller one which still contains the difference Painleve I, or to a class of first-order equations. In addition, for certain classes of equations, we prove the existence of rational solutions and give their forms.

The modeling of a system composed by a gas phase and organic aerosol particles, and its numerical resolution are studied. The gas-aerosol system is modeled by ordinary differential equations coupled with a mixed-constrained optimization problem. This coupling induces discontinuities when inequality constraints are activated or deactivated. Two approaches for the solution of the optimization-constrained differential equations are presented. The first approach is a time splitting scheme together with a fixed-point method that alternates between the differential and optimization parts. The ordinary differential equations are approximated by the Crank-Nicolson scheme and a primal-dual interior-point method combined with a warm-start strategy is used to solve the minimization problem. The second approach considers the set of equations as a system of differential algebraic equations after replacing the minimization problem by its first order optimality conditions. An implicit 5th-order Runge-Kutta method (RADAU5) is then used. Both approaches are completed by numerical techniques for the detection and computation of the events (activation and deactivation of inequality constraints) when the system evolves in time. The computation of the events is based on continuation techniques and geometric arguments. Moreover the first approach completes the computation with extrapolation polynomials and sensitivity analysis, whereas the second approach uses dense output formulas. Numerical results for gas-aerosol system made of several chemical species are proposed for both approaches. These examples show the efficiency and accuracy of each method. They also indicate that the second approach is more efficient than the first one. Furthermore theoretical examples show that the method for the computation of the activation is of second order for the first approach and exact for the second one.

In this article, we treat complex difference equation of the form where A j (z)(j = 0, 1, … , n, n + 1) are meromorphic functions. We give answers to the growth estimates of the meromorphic solutions, and firstly consider the deficiency and fixed points of the meromorphic solutions of these equations. Some examples are listed to show that the result about the properties of fixed points is the best possible in a certain sense.

Let h be a meromorphic function with few poles and zeros. By Nevanlinna's value distribution theory we prove some new properties on the polynomials in h with the coefficients being small functions of h. We prove that if f is a meromorphic function and if <TEX>$f^m$</TEX> is identically a polynomial in h with the constant term not vanish identically, then f is a polynomial in h. As an application, we are able to find the entire solutions of the differential equation of the type <TEX>$$f^n+P(f)=be^{sz}+Q(e^z)$$</TEX>, where P(f) is a differential polynomial in f of degree at most n-1, and Q(<TEX>$e^z$</TEX>) is a polynomial in <TEX>$e^z$</TEX> of degree k <TEX>$\leqslant$</TEX> max {n-1, s(n-1)/n} with small functions of <TEX>$e^z$</TEX> as its coefficients.

Multibody dynamics comprises methodologies for the design and analysis of mechanical systems, with Kalman filters being the principal state estimation methods. The Kalman filters are generally formulated for unconstrained systems, the dynamics of which are described by ordinary differential equations in state-space models. However, multibody systems are constrained systems, and the commonly used method to describe their dynamics involves using differential algebraic equations (DAEs) comprising differential and algebraic equations. The differential equations also include the Lagrange multipliers. Hence, incorporating multibody systems, described by DAEs into the scheme of Kalman filters, cannot be achieved straightforwardly, which facilitates different strategies being addressed. This study develops a novel method for converting DAEs into a state-space model. A transition model of the time derivatives of Lagrange multipliers and a Lagrange multiplier constraint vector are devised and used in the state and output equations, respectively. The continuous- and discrete-time extended Kalman filters (CEKF and DEKF) are constructed using the proposed state-space model, and state estimations are simulated on the benchmark planar four- and five-bar linkages. Further, a demonstration of system observability is conducted, and sensitivity to the initial state estimates is studied. These tests demonstrate that the proposed state-space model achieves observable systems and that both the CEKF and DEKF, constructed using the proposed state-space model, can estimate the states with a wide range of initial conditions.

We are concerned with difference Riccati equations in the complex plane. We study the continuous limit of difference Riccati equations to differential Riccati equations, and compare asymptotic expansions of both of solutions. Some examples are constructed. We also consider ramification relations and value distributions of transcendental meromorphic solutions to difference Riccati equations.

In this paper, we treat the problem of determining the rate of growth of meromorphic functions on the plane, which are solutions of nth order algebraic differential equations whose coefficients are arbitrary entire functions (i.e., equations of the form, Ω(z,y,dy/dz, • • •, dny/dzn) — 0, where Ω is a polynomial in y, dy/dz, • • •, dny/dzn whose coefficients are arbitrary entire functions of z.)

A method is proposed to transform a system of differential algebraic equations (D.A.E.) to a system of ordinary differential equations (O.D.E.) which can be solved relatively easily by standard numerical techniques.Two examples, including a model of an absorption tower, are given to illustrate the utility of the method. The example problems reveal that this easily implemented technique offers significant savings in CPU time compared to the numerical solution of the untransformed D.A.E.s particularly when the algebraic equations are nonlinear.Furthermore, it appears to be faster and/or more reliable than other numerical schemes which have been recently developed for equations of this type.

Eremenko and Sodin proved that meromorphic solution f (z) of the Schröder equation f (sz) = R (f (z)), |s| > 1, has no Valiron deficiency other than exceptional values of R(z). We consider transcendental meromorphic solutions of non-autonomous equation f (sz) =R (z, f (z)), |s| > 1. It is shown that there exists an equation of this form possessing a transcendental meromorphic solution, which has a Valiron deficiency other than a Nevanlinna deficiency. We also give some generalizations of the Eremenko and Sodin theorem for algebraic functions as targets.

Let f(z) be a meromorphic function; we shall investigate the asymptotic behaviour of the ratio T(r, f(z + α))/T(r, f(z)) and T(r, f(αz))/T(r, f(z)), and discuss the growth of the meromorphic solutions of some functional equations.

In this paper, we investigate the growth of meromorphic solutions of higher order linear differential equation \({f^{( k) }+A_{k-1}( z)e^{P_{k-1}( z) }f^{( k-1)}+\cdots+A_{1}( z) e^{P_{1}( z) }f'+A_{0}( z)e^{P_{0}( z) }f=0 ( k\geq 2)}\) , where \({P_{j}( z) ( j=0,1,\cdots,k-1)}\) are nonconstant polynomials such that deg \({P_{j}=n ( j=0,1,\cdots,k-1)}\) and \({A_{j}( z) (\not\equiv 0) ( j=0,1,\cdots,k-1)}\) are meromorphic functions with order \({\rho ( A_{j}) < n ( j=0,1,\cdots,k-1) }\).