The simulation problem for high order linear differential systems

Operational matrices to solve nonlinear Riccati differential equations of arbitrary order

Addition theorems for solutions to linear homogeneous constant coefficient ordinary differential equations

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.

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.

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.

Calibration of low-fidelity ship motion programs through regressions of high-fidelity forces

Polynomial Control Toolbox: A New Aid for Teaching Linear Systems in MATLAB

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.

In this paper, we obtain new algebraic identities between hypergeometric functions satisfying linear differential equations of arbitrary order as well as between the fundamental matrices of hypergeometric equations.

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

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