Differential Equations Of Arbitrary Order Research Articles

Integrating Factors and λ–Symmetries

We investigate the relationship between integrating factors and λ–symmetries for ordinary differential equations of arbitrary order. Some results on the existence of λ–symmetries are used to prove an independent existence theorem for integrating factors. A new method to calculate integrating factors and the associated first integrals is derived from the method to compute λ–symmetries and the associated reduction algorithm. Several examples illustrate how the method works in practice and how the computations that appear in other methods may be simplified.

Application of least square method to arbitrary-order problems with separated boundary conditions

In this paper, differential equations of arbitrary order with separated boundary conditions are converted into an optimal control problem. Then a convergent approximate solution is constructed such that the exact boundary conditions are satisfied. In fact it will be shown that for every ε > 0 , there exists an approximate solution v ε of B-spline functions such that the corresponding least square error is less than ε > 0 , and also v ε satisfies the exact boundary conditions. Some examples are given and the optimal errors are obtained for the sake of comparison.

On the Applications of a New Technique to Solve Linear Differential Equations, with and without Source

A general method for solving linear differential equations of arbitrary order, is used to arrive at new representations for the solutions of the known differential equa- tions, both without and with a source term. A new quasi-solvable potential has also been constructed taking recourse to the above method. Linear differential equations play a crucial role in various branches of science and mathematics. Second order differential equations routinely manifest in the study of quantum mechanics, in connection with Schrodinger equation. There are various techniques available to solve a given differential equation, e.g., power series method, Laplace transforms, etc. Not many general methods applicable to differential equations of arbitrary order exist in the literature (see (1) and references therein). We make use of a general method for solving linear differential equations of arbitrary order to construct new representations for the solutions of the known second order linear differential equations, both without and with a source term. This method has found applications in solving linear single and multi variable differential equations. The solutions of linear differential equation with a source and development of a new Quasi-Exactly Soluble (QES) system are the new results of this paper.

On the fractional-order logistic equation

The topic of fractional calculus (derivatives and integrals of arbitrary orders) is enjoying growing interest not only among mathematicians, but also among physicists and engineers (see [E.M. El-Mesiry, A.M.A. El-Sayed, H.A.A. El-Saka, Numerical methods for multi-term fractional (arbitrary) orders differential equations, Appl. Math. Comput. 160 (3) (2005) 683–699; A.M.A. El-Sayed, Fractional differential–difference equations, J. Fract. Calc. 10 (1996) 101–106; A.M.A. El-Sayed, Nonlinear functional differential equations of arbitrary orders, Nonlinear Anal. 33 (2) (1998) 181–186; A.M.A. El-Sayed, F.M. Gaafar, Fractional order differential equations with memory and fractional-order relaxation–oscillation model, (PU.M.A) Pure Math. Appl. 12 (2001); A.M.A. El-Sayed, E.M. El-Mesiry, H.A.A. El-Saka, Numerical solution for multi-term fractional (arbitrary) orders differential equations, Comput. Appl. Math. 23 (1) (2004) 33–54; A.M.A. El-Sayed, F.M. Gaafar, H.H. Hashem, On the maximal and minimal solutions of arbitrary orders nonlinear functional integral and differential equations, Math. Sci. Res. J. 8 (11) (2004) 336–348; R. Gorenflo, F. Mainardi, Fractional calculus: Integral and differential equations of fractional order, in: A. Carpinteri, F. Mainardi (Eds.), Fractals and Fractional Calculus in Continuum Mechanics, Springer, Wien, 1997, pp. 223–276; D. Matignon, Stability results for fractional differential equations with applications to control processing, in: Computational Engineering in System Application, vol. 2, Lille, France, 1996, p. 963; I. Podlubny, A.M.A. El-Sayed, On Two Definitions of Fractional Calculus, Solvak Academy of science-institute of experimental phys, ISBN: 80-7099-252-2, 1996. UEF-03-96; I. Podlubny, Fractional Differential Equations, Academic Press, 1999] for example). In this work we are concerned with the fractional-order logistic equation. We study here the stability, existence, uniqueness and numerical solution of the fractional-order logistic equation.

Invariant cones and stability of linear dynamical systems

We present a method for the investigation of the stability and positivity of systems of linear differential equations of arbitrary order. Conditions for the invariance of classes of cones of circular and ellipsoidal types are established. We propose algebraic conditions for the exponential stability of linear positive systems based on the notion of maximal eigenpairs of a matrix polynomial.

On polynomial solutions of the Heun equation

By making use of a recently developed method to solve linear differential equations of arbitrary order, we find a wide class of polynomial solutions to the Heun equation. We construct the series solution to the Heun equation before identifying the polynomial solutions. The Heun equation extended by the addition of a term, $- \s/x$, is also amenable for polynomial solutions.

Numerical modelling method for wave propagation in a linear viscoelastic medium with singular memory

SUMMARY A numerical modelling method for wave propagation in a linear viscoelastic medium with singular memory is developed in this paper. For a demonstration of the method, the Cole‐ Cole model of viscoelastic relaxation is adopted here. A formulation of the Cole‐Cole model based on internal variables satisfying fractional relaxation equations is applied. In order to avoid integrating and storing of the entire history of the variables, a new method for solving fractional differential equations of arbitrary order based on a set of secondary internal variables is developed. Using the new method, the velocity‐stress equations and the fractional relaxation equations are reduced to a system of first-order ordinary differential equations for the velocities, stresses, primary internal variables as well as the secondary internal variables. The horizontal spatial derivatives involved in the governing equations are calculated by the Fourier pseudospectral (PS) method, while the vertical ones are calculated by the Chebychev PS method. The physical boundary conditions and the non-reflecting conditions for the Chebychev PS method are also discussed. The global solution of the first-order system of ordinary differential equations is advanced in time by the Euler predictor‐corrector methods. For the demonstration of our method, some numerical results are presented.

Method for Computing Scattering Matrices for General Dissipative and Selfadjoint Elliptic Problems in Domains with Cylindrical Ends

In a domain with finitely many cylindrical ends at infinity, we consider dissipative and formally selfadjoint elliptic problems for systems of differential equations of arbitrary order. As is known, one can regard cylindrical ends as waveguides and introduce families of incoming and outgoing waves. The amplitudes of such waves can grow at infinity at power or even exponential rate. The scattering matrices account finitely many waves. We suggest and justify a numerical method for finding such scattering matrices. Bibliography: 18 titles.

Movable Singularities of a Class of Nonlinear Ordinary Differential Equations of Arbitrary Order

In this paper we consider nonlinear ordinary differential equations y(n)=F(y′, y, x) of arbitrary order n≥ 3, where F is algebraic in y, y′ and locally analytic in x. We prove that for n > 3 these equations always admit movable branch points. In the case n= 3 these equations admit movable branch points unless they are of the known class y′′′=a(x)(y′)2+ (b2(x)y2+b1(x)y+b0(x))y′+ (c4(x)y4+c3(x)y3+c2(x)y2+c1(x)y+c0(x)), where a, bj, cj are locally analytic in x.

A general construction of linear differential equations with solutions of prescribed properties

Effective constructions of ordinary linear differential equations of arbitrary orders are presented that give equations with solutions of prescribed qualitative properties, like solutions in the classes L p , converging to zero or bounded solutions, etc. Connections with transformations of equations and distribution of the zeros of solutions are considered as well. Results generalize those obtained for the second-order linear differential equations.

A note on space approximation of parabolic evolution equations

Our point of interest is the numerical approximation of the abstract Cauchy problem u ̇ (t)=Au(t), u(0)=u 0∈X, where A generates an analytic semigroup in X, X being a separable Banach space. Such equations include e.g. parabolic partial differential equations of arbitrary order defined on a bounded domain. We apply semigroup methods and interpolation theory to get the order of convergence for spatially approximation in arbitrary separable Banach spaces.

The simulation problem for high order linear differential systems

In this paper we consider linear constant coefficient ordinary differential equations of arbitrary order of the form R( d dt )w=M( d dt )f with R and M given, but otherwise arbitrary, polynomial matrices. To these equations we associate a behavior B defined as the set of all vector-valued distributions w, f that solve the equations themselves. We think of f as a given distribution, while w is to be found in such a way that the equations defining the behavior are satisfied. Alongside R( d dt )w=M( d dt )f we also consider initial conditions obtained by imposing the value at time t=0 of a linear combination of the variables w and their derivatives, yielding S( d dt )w(0)=Ta with S a polynomial matrix, T a fixed real matrix, and a a real vector. The three main issues we address are: (i) Solvability of the behavior, meaning the problem of finding conditions that assure that corresponding to a (or any) given vector distribution f, a trajectory w exists such that ( w, f) belong to B . This question is formalized and answered in Section 4. (ii) The index of the behavior, which means investigating how the smoothness of the given distribution f is related to smoothness of the solution w. Section 5 is dedicated to this issue. (iii) Compatibility of initial conditions, in which case the initial conditions S( d dt )w(0)=Ta are considered alongside the behavior B . We first check that S( d dt )w(0) is well defined. If it is, we provide conditions under which there exists a (unique) distribution w such that w, f belong to B and such that S( d dt )w(0)=Ta are also satisfied for a (or any) given a. This problem is addressed in Section 6. We show how our results generalize to behaviors defined by equations of arbitrary order classical properties of behaviors defined by first order systems such as E d dt w+Aw=f .

Conditions for Instability of One Class of Nonlinear Nonautomomous Dynamic Systems

Consideration was given to the conditions for instability of the equilibrium states of a nonlinear nonautonomous dynamic systems obeying an ordinary vector differential equation of arbitrary order whose right-hand side f(x, t) satisfies the following conditions: (i) for any t ≥ 0, div f(x, t) > 0 almost everywhere on the set H that is a neighborhood of the equilibrium point of the system x = 0 and (ii) \lim_{t \to \infty} div f(x, t) = 0 at any point x ∈ H. The equilibrium states of such systems can be both stable and unstable. For one class of these systems, sufficient instability conditions were given, which enables one to carry out studies using only the information about the right-hand side of the system.

Symmetry Analysis of Differential Equations Using MATHLIE

A general procedure for solving ordinary differential equations of arbitrary order is discussed. The method used is based on symmetries of a differential equation. The known symmetries allow the derivation of first integrals of the equation. The knowledge of at least r symmetries of an ordinary differential equation of order n with r ≥ n is the basis for deriving the solution. Our aim is to show that Lie's theory is a useful tool for solving ordinary differential equations of higher orders. Bibliography: 12 titles.

Stability of Classes of Mappings and Holder Continuity of Higher Derivatives of Elliptic Solutions to Systems of Nonlinear Differential Equations

Nirenberg published the following well-known result in 1954: Let a function z be a twice continuously differentiable solution to a nonlinear second-order elliptic equation. Suppose that the function F defining the equation is continuous and has continuous first-order partial derivatives with respect to all of its arguments (i.e., independent together with z and the symbols of all first- and second-order partial derivatives of z). Then the partial derivatives of z are locally Holder continuous. Simultaneously with Nirenberg, Morrey obtained an analogous result for elliptic systems of second-order nonlinear equations. In this article, we get the same result for the higher derivatives of elliptic solutions to systems of nonlinear partial differential equations of arbitrary order and a rather general shape. The proof is based on the results of the author's recent research on the study of the stability phenomena in the Cl-norm of classes of mappings.

Bounded Oscillation of Nonlinear Neutral Differential Equations of Arbitrary Order

The paper is concerned with oscillation properties of n-th order neutral differential equations of the form $$[x(t) + cx(\tau (t))]^{(n)} + q(t)f\left( {x(\sigma (t))} \right) = 0,{\text{ }}t \geqslant t_0 >0,$$ where c is a real number with \(\left| c \right| \ne 1,q \in C\left( {\left[ {t_0 ,\infty } \right),\mathbb{R}} \right),f \in C(\mathbb{R},\mathbb{R}),\tau ,\sigma \in C\left( {\left[ {t_0 ,\infty } \right),\mathbb{R}_ + } \right)\) with τ(t) < t and \(\mathop {\lim }\limits_{t \to \infty } \tau (t) = \mathop {\lim }\limits_{t \to \infty } \sigma (t) = \infty \).Sufficient conditions are established for the existence of positive solutions and for oscillation of bounded solutions of the above equation. Combination of these conditions provides necessary and sufficient conditions for oscillation of bounded solutions of the equation. Furthermore, the results are generalized to equations in which c is a function of t and a certain type of a forcing term is present.

Smooth Solutions for a Class of Nonlinear Parabolic Evolution Equations

The purpose of the paper is to introduce a new linearization method for local well-posedness of nonlinear evolution equations. This approach is based on an implicit function theorem of Nash–Moser type. The technique is illustrated by an application to a general class of fully nonlinear parabolic partial differential equations of arbitrary order on ℝ n . The estimates required by the Nash–Moser technique are derived for the higher order Sobolev norms of the solutions of the linearized parabolic equation using semigroup theory and elliptic theory. In particular, a priori estimates, resolvent estimates and commutator estimates are involved. The general method based on a combination of Nash–Moser techniques with semigroup theory is applicable to other problems and has already been used to prove short-time solvability for some nonlinear Schrodinger type equation. This approach might be useful in other situations as well since it compensates for a loss of derivatives in the estimates of the solutions of the linearized equation.

Essential self-adjointness of symmetric linear relations associated to first order systems

The purpose of this note is to present several criteria for essential self-adjointness. The method is based on ideas due to Shubin. This note is divided into two parts. The first part deals with symmetric first order systems on the line in the most general setting. Such a symmetric first order system of differential equations gives rise naturally to a symmetric linear relation in a Hilbert space. In this case even regularity is nontrivial. We will announce a regularity result and discuss criteria for essential self-adjointness of such systems. A byproduct of the regularity result is a short proof of a result due to Kogan and Rofe-Beketov (V. Kogan and F. Rofe-Beketov: “On square-integrable solutions of symmetric systems of differential equations of arbitrary order”, Proc. Roy. Soc. Edinburgh Sect. A 74 (1974/75), 5-40): the so-called formal deficiency indices of a symmetric first order system are locally constant on ℂ∖ℝ. The regularity and its corollary are based on joint work with Mark Malamud. Details will be published elsewhere. In the second part we consider a complete riemannian manifold, M, and a first order differential operator, D:C 0 ∞ (E)→C 0 ∞ (F), acting between sections of the hermitian vector bundles E,F. Moreover, let V:C ∞ (E)→L loc ∞ (E) be a self-adjoint zero order differential operator. We give a sufficient condition for the Schrödinger operator H=D t D+V to be essentially self-adjoint. This generalizes recent work of I. Oleinik (I. Oleinik: “On the essential self-adjointness of the Schrödinger operator on a complete Riemannian manifold”, Math. Notes 54 (1993), 934-939), (I. Oleinik: “On the connection of the classical and quantum mechanical completeness of a potential at infinity on complete riemannian manifolds”, Math. Notes 55 (1994), 380-386), (I. Oleinik: “On the essential self-adjointness of the general second order elliptic operators”, Proc. Amer. Math. Soc. 127 (1999), 889-900), M. Shubin (M. Shubin: “Classical and quantum completeness for the Schrödinger operators on non-compact manifolds”, In: B. Booß-Bavnbek and K. P. Wojciechowski (eds.), “Geometric aspects of partial differential equations” (Roskilde, 2000), vol. 242 of Contemp. Math. Amer. Math. Soc., Providence, RI (1999), pp. 257-269), (M. Shubin: “Essential self-adjointness for magnetic Schrödinger operators on non-compact manifolds”, In: Seminaire: Équations aux Dérivées Partielles, 2000-1999). École Polytech., Palaiseau (1999), pp. Exp. No. XV, 24), and M. Braverman (M. Braverman: “On self-adjointness of a Schrödinger operator on differential forms”, Proc. Amer. Math. Soc. 126 (2000), 617-623). We essentially use the method of Shubin. Our presentation shows that there is a close link between Shubin’s self-adjointness condition for the Schrödinger operator and Chernoff’s self-adjointness condition for powers of first order operators. We also discuss non-elliptic operators. However, in this case we need an additional assumption. We conjecture that the additional assumption turns out to be obsolete in general. The criteria we are going to present in the first and second part of this note are very closely related. In fact, after we had done the second part, we saw that the theory can be extended to symmetric linear relations associated to symmetric first order systems.

Variationally trivial Lagrangians and locally variational differential equations of arbitrary order

We continue our investigation of the Lagrangian formalism on jet bundle extensions using Fock space methods. We are able to provide the most general form of a variationally trivial Lagrangian of arbitrary order and we also give a generic expression for the most general locally variational differential equation. As anticipated in the literature, these expressions involve some special combinations of the highest order derivatives, called hyper-Jacobians.

Nonlinear functional differential equations of arbitrary orders

Nonlinear functional differential equations of arbitrary orders