On algebraic identities between generalized hypergeometric functions

Operational matrices to solve nonlinear Riccati differential equations of arbitrary order

Algorithms and methods in differential algebra

Existence and comparison theorems for eigenvalues of (k, n -k)-focal point and (k, n -Ac)-conjugate point problems are proved for a class of nth order linear differential equations for arbitrary n.

The purpose of this paper is to establish comparison criteria, by which the oscillatory and asymptotic behavior of linear retarded differential equations of arbitrary order is inherited from the oscillation of an associated second order linear ordinary differential equation. These criteria are new even in the case of ordinary differential equations.

The simulation problem for high order linear differential systems

A matrix bilinear form expression of Green's formula is derived and is used to develop the Riccati equation for systems described by a one-dimensional scalar linear parabolic partial differential equation of arbitrary order with mixed boundary condition having distributed and boundary control. An example is presented.

In this paper, we study the problem on the asymptotic behavior of solutions of the equation l n y = λy of arbitrary order (even or odd) with complex-valued coefficients in a neighborhood of the point +∞. The result obtained allows one to find defect numbers of the corresponding closed minimal symmetric differential operator.

Chapter 8: Arithmetic methods applied to solutions of linear differential equations of arbitrary order

Forced oscillations for differential equations of arbitrary order

This is the first of two papers on the propagation of high-frequency electromagnetic waves through a magnetized plasma in curved space-time. We first show that the nonlinear system of equations governing the plasma and the electromagnetic field in a given, external gravitational field has locally a unique solution for any initial data set obeying the appropriate constraints, and that this system is linearization stable at any of its solu­tions. Next we prove that the linearized perturbations of a ‘background’ solution are characterized by a third-order (not strictly) hyperbolic, constraint-free system of three partial differential equations for three unknown functions of the four space-time coordinates. We generalize the algorithm for obtaining oscillatory asymptotic solutions of linear systems of partial differential equations of arbitrary order, depending polynomially on a small parameter such that it applies to the previously established perturbation equation when the latter is rewritten in terms of dimensionless variables and a small scale ratio. For hyperbolic systems we then state a sufficient condition in order that asymptotic solutions of finite order, constructed as usual by means of a Hamiltonian system of ordinary differential equations for the characteristic strips and a system of transport equations determining the propagation of the amplitudes along the rays, indeed approximate solutions of the system. The pro­cedure is a special case of a two-scale method, suitable for describing the propagation of locally approximately plane, monochromatic waves through a dispersive, inhomogeneous medium. In the second part we shall apply the general method to the perturbation equation referred to above.

We estimate the growth of the spherical derivative for meromorphic solutions of algebraic differential equations of arbitrary order under certain conditions on the growth of the coefficients.

The article discusses the solution of the problem of the synthesis of the parameters of the control laws of nonlinear continuous and pulse automatic control systems with the polynomial approximation of nonlinearities. As a mathematical apparatus, the inversion of the direct variational method of analysis - the generalized Galerkin method - is used to solve the problem. This makes it possible to completely algebraize the solution of the problem posed for the ACS class under study, whose dynamics are described by nonlinear differential equations of arbitrary order.

appears as the last result of L.E.J. Brouwer’s paper [5] developing a theory of degree for continuous mappings between two oriented manifolds of the same finite dimension. Brouwer did not make any comment on the result, and never returned to it in any subsequent paper. Despite of its topological importance and its potential for application, this theorem, whose history can be found in [17], did not catch much attention until 1922, when G.D. Birkhoff and O.D. Kellogg [4] gave, in order to obtain new existence theorems for semilinear ordinary differential equations of arbitrary order on [a, b] with general linear multipoint boundary conditions, the first extension of Brouwer fixed point theorem to the spaces C([a, b]) and L(a, b) :

The interval G = ( - 1 , 1 ) {G=(-1,1)} turns into a Lie group under the group operation x ∘ y := ( x + y ) ⁢ ( 1 + x ⁢ y ) - 1 {x\circ y:=(x+y)(1+xy)^{-1}} , x , y ∈ G {x,y\in G} . This enables us to define of the invariant measure d ⁢ G ⁢ ( x ) := ( 1 - x 2 ) - 1 ⁢ d ⁢ x {dG(x):=(1-x^{2})^{-1}dx} and the Fourier transformation ℱ G {{\mathcal{F}}{}_{G}} on the interval G and, as a consequence, we can consider Fourier convolution operators W G , a 0 := ℱ a G - 1 ℱ G {W^{0}_{G,a}:={\mathcal{F}}{}_{G}^{-1}a{\mathcal{F}}{}_{G}} on G. This class of convolutions includes the celebrated Prandtl, Tricomi and Lavrentjev–Bitsadze equations and, also, differential equations of arbitrary order with the generic differential operator 𝔇 G ⁢ u ⁢ ( x ) = ( 1 - x 2 ) ⁢ u ′ ⁢ ( x ) {\mathfrak{D}_{G}u(x)=(1-x^{2})u^{\prime}(x)} , x ∈ G {x\kern-1.0pt\in\kern-1.0ptG} . Equations are solved in the scale of generic Bessel potential ℍ p s ⁢ ( G , d ⁢ G ⁢ ( x ) ) {\mathbb{H}^{s}_{p}(G,dG(x))} , 1 ⩽ p ⩽ ∞ {1\kern-1.0pt\leqslant\kern-1.0ptp\kern-1.0pt\leqslant\kern-1.0pt\infty} , and Hölder–Zygmund ℤ ν ⁢ ( G ) {\mathbb{Z}^{\nu}(G)} , 0 < μ , ν < ∞ {0<\mu,\nu<\infty} , spaces, adapted to the group G. The boundedness of convolution operators (the problem of multipliers) is discussed. The symbol a ⁢ ( ξ ) {a(\xi)} , ξ ∈ ℝ {\xi\in\mathbb{R}} , of a convolution equation W G , a 0 ⁢ u = f {W^{0}_{G,a}u=f} defines solvability as follows: the equation is uniquely solvable if and only if the symbol a is elliptic. The solution is written explicitly with the help of the inverse symbol. Also, we shortly touch upon the multidimensional analogue – the Lie group G n {G^{n}} .

Solution for fractional generalized Zakharov equations with Mittag-Leffler function