Applications of Borel’s methods

In this paper the order and the hyper-order of the solutions of higher-order homogeneous linear differential equations is investigated.

Using power series methods, Harris and Sibuya [3,4] recently showed that if k k is an ordinary differential field of characteristic zero and y ≠ 0 y \ne 0 is an element of a differential extension of k k such that y y and 1 / y 1/y satisfy linear differential equations with coefficients in k k , then y ′ / y y\prime /y is algebraic over k k . Using differential galois theory, we generalize this and characterize those polynomial relations among solutions of linear differential equations that force these solutions to have algebraic logarithmic derivatives. We also show that if f f is an algebraic function of genus ≥ 1 \geq 1 and if y y and f ( y ) f(y) or y y and e ∫ y {e^{\int y}} satisfy linear differential equations, then y y is an algebraic function.

In this paper, we investigate the order and the hyper-order of growth of solutions of the linear differential equation where n≥2 is an integer, Aj (z) (≢0) (j = 1,2) are entire functions with max {σ A(j) : (j = 1,2} < 1, Q (z) = qmzm + ... + q1z + q0 is a nonoonstant polynomial and a1, a2 are complex numbers. Under some conditions, we prove that every solution f (z) ≢ 0 of the above equation is of infinite order and hyper-order 1.

We discuss some changes of the traditional scheme for finding rational solutions of linear difference equations (homogeneous or inhomogeneous) with rational coefficients. These changes in some cases allow one to detect the absence of rational solutions in an early stage of computation and to stop the work. The corresponding strategy may be useful for the algorithms based on finding rational solutions of equations of considered form, e.g., for some symbolic summation algorithms.

The limit superior of the nth root of |x(n)|, where x is any nonzero solution of a scalar homogeneous linear difference equation of finite order, is identified in an elementary manner with the modulus of a root of the associated characteristic polynomial. Similar results are obtained for sequences of determinants of certain finite Hankel matrices of the same order generated by such solutions.

Differential equations of the form f'' + A(z)f' + B(z)f = 0 (*) are considered, where A(z) and B(z) not equiv 0 are entire functions. The Lindelöf function is used to show that for any rho in (1/2, infty ), there exists an equation of the form (*) which possesses a solution f with a Nevanlinna deficient value at 0 satisfying rho =rho (f)ge rho (A)ge rho (B), where rho (h) denotes the order of an entire function h. It is known that such an example cannot exist when rho le 1/2. For smaller growth functions, a geometrical modification of an example of Anderson and Clunie is used to show that for any rho in (2, infty ), there exists an equation of the form (*) which possesses a solution f with a Valiron deficient value at 0 satisfying rho =rho _{log }(f)ge rho _{log }(A)ge rho _{log }(B), where rho _{log }(h) denotes the logarithmic order of an entire function h. This result is essentially sharp. In both proofs, the separation of the zeros of the indicated solution plays a key role. Observations on the deficient values of solutions of linear differential equations are also given, which include a discussion of Wittich’s theorem on Nevanlinna deficient values, a modified Wittich theorem for Valiron deficient values, consequences of Gol’dberg’s theorem, and examples to illustrate possibilities that can occur.

The growth of solutions of linear differential equations with coefficients of iterated order in the unit disc

This paper use multi-splittings waveform relaxation (MSWR) method to determine the periodic solutions of linear differential equations. Based on the spectral radius of the derived operator by decoupled process, we obtain some convergent conditions for the MSWR method. The MSWR method is an acceleration technique of the periodic waveform relaxation. Numerical example is provided to further confirm the theoretical analysis and also show that the multi-splitting technique can effectively accelerate the convergent performance of the iterative process.

An algorithm of multisummation of formal power series, solutions of linear ODE equations

We extend and apply the Galois theory of linear differential equations equipped with the action of an endomorphism. The Galois groups in this Galois theory are difference algebraic groups, and we use structure theorems for these groups to characterize the possible difference algebraic relations among solutions of linear differential equations. This yields tools to show that certain special functions are difference transcendent. One of our main results is a characterization of discrete integrability of linear differential equations with almost simple usual Galois group, based on a structure theorem for the Zariski dense difference algebraic subgroups of almost simple algebraic groups, which is a schematic version, in characteristic zero, of a result due to Z. Chatzidakis, E. Hrushovski, and Y. Peterzil.

A definite theory exists for solving linear systems of ordinary differential equations with constant coefficients. “This theory is essentially a branch of linear algebra, and allows us to solve all autonomous linear equations”, ([4], p. 95). Furthermore, “In view of the fact that the solution of these equations does not, in principle, present any great difficulties, they are often considered to be of no great interest for theory, and in textbooks they are usually relegated to the position of simple exercises appended to the general theory of linear equations”, ([81], p. 41). Modern textbooks [8, 21, 26, 44, 45, 46, 50, 90] give a detailed and rigorous treatment of the standard methods available for linear problems, supplemented with numerous mechanical, electrical, and engineering applications. The voluminous literature accumulated on the general properties of the solutions of linear differential equations must therefore no doubt leave the impression that no further contribution could be made to the standard normal linear dynamic system with real and constant coefficients. This impression, however, will fade rather quickly as we go on. Paradoxically, some fundamental aspects of this simple dynamic system are still unnoticed.

This study shows how to obtain least-squares solutions to initial value problems (IVPs), boundary value problems (BVPs), and multi-value problems (MVPs) for nonhomogeneous linear differential equations (DEs) with nonconstant coefficients of any order. However, without loss of generality, the approach has been applied to second-order DEs. The proposed method has two steps. The first step consists of writing a constrained expression, that has the DE constraints embedded. These kind of expressions are given in terms of a new unknown function, g ( t ) , and they satisfy the constraints, no matter what g ( t ) is. The second step consists of expressing g ( t ) as a linear combination of m independent known basis functions. Specifically, orthogonal polynomials are adopted for the basis functions. This choice requires rewriting the DE and the constraints in terms of a new independent variable, x ∈ [ − 1 , + 1 ] . The procedure leads to a set of linear equations in terms of the unknown coefficients of the basis functions that are then computed by least-squares. Numerical examples are provided to quantify the solutions’ accuracy for IVPs, BVPs and MVPs. In all the examples provided, the least-squares solution is obtained with machine error accuracy.

Certain classes of linear differential equations are investigated for which the distribution of zeros of their solutions determines their asymptotic behaviour. These results generalize those already obtained for the second order linear differential equations to equations of arbitrary order.

Piecewise-linear approximation of solution of linear differential equations by Walsh functions

We establish a relationship between the growth rates of algebroid solutions and coefficients of nth-order linear differential equations.