Differential Equations Of Infinite Order Research Articles

Topological defects in nonlocal field theories

In this paper we study for the first time topological defects in the context of nonlocal field theories in which Lagrangians contain infinite-order differential operators. In particular, we analyze domain walls. Despite the complexity of non-linear infinite-order differential equations, we are able to find an approximate analytic solution. We first determine the asymptotic behavior of the nonlocal domain wall close to the vacua. Then, we find a linearized nonlocal solution by perturbing around the well-known local ‘kink’, and show that it is consistent with the asymptotic behavior. We develop a formalism to study the solution around the origin, and use it to verify the validity of the linearized solution. We find that nonlocality makes the width of the domain wall thinner, and the energy per unit area smaller as compared to the local case. For the specific domain wall solution under investigation we derive a theoretical constraint on the energy scale of nonlocality which must be larger than the corresponding symmetry-breaking scale. We also briefly comment on other topological defects like string and monopole.

On the Generalized Cauchy Problem for One Class of Differential Equations of Infinite Order

We establish the solvability of a nonlocal multipoint (in time) problem regarded as a generalization of the Cauchy problem for the evolution equation with pseudodifferential operator (differentiation operator of infinite order) with initial conditions from the space of generalized functions of the ultradistribution type.

Linear operators, the Hurwitz zeta function and Dirichlet L-functions

At the 1900 International Congress of Mathematicians, Hilbert claimed that the Riemann zeta function is not the solution of any algebraic ordinary differential equation its region of analyticity [5]. In 2015, Van Gorder addresses the question of whether the Riemann zeta function satisfies a non-algebraic differential equation and constructs a differential equation of infinite order which zeta satisfies [7]. However, as he notes in the paper, this representation is formal and Van Gorder does not attempt to claim a region or type of convergence. In this paper, we show that Van Gorder's operator applied to the zeta function does not converge pointwise at any point in the complex plane. We also investigate the accuracy of truncations of Van Gorder's operator applied to the zeta function and show that a similar operator applied to zeta and other L-functions does converge.

Generalized symplectic Schur functions and SUC hierarchy

In this paper, we first define a generalization of the symplectic Schur functions and their vertex operator realizations. By means of the vertex operators, we obtain a series of non-linear partial differential equations of infinite order, the symplectic UC hierarchy (called the SUC hierarchy); we regard it as an extension of the symplectic KP hierarchy.

Application measure of noncompactness and operator type contraction for solvability of an infinite system of differential equations in lp-space

The aim of this paper is to obtain existence results for an infinite system of second order differential equations in the sequence space lp for 1 ? p < ? with the help of a technique associated with measures of noncompactness and contractive condition of operator type. We also provide some illustrative examples in support of our existence theorems.

Bound states of Newton’s equivalent finite square well

In this paper, a one-parameter family of Newton’s equivalent Hamiltonians (NEH) for finite square well potential is analyzed in order to obtain bound state energy spectrum and wave functions. For a generic potential, each of the NEH is classically equivalent to one another and to the standard Hamiltonian yielding Newton’s equations. Quantum mechanically, however, they are expected to be different from each other. The Schrödinger’s equation coming from each NEH with finite square well potential is an infinite order differential equation. The matching conditions, therefore, demand the wave functions to be infinitely differentiable at the well boundaries. To handle this, we provide a way to consistently truncate these conditions. It turns out as expected that bound state energy spectrum and wave functions are dependent on the parameter [Formula: see text] which is used to characterize different NEH. As [Formula: see text], the energy spectrum coincides with that from the standard quantum finite square well.

Fixed Point and Common Fixed Point Results of \(D_F\)-Contractions via Measure of Non-compactness with Applications

In this paper, we study a new contraction mapping inspired by the concept of \(F\)-contraction, which was recently introduced by Wardowski [20]. We find common fixed points for a sequence of mappings by introducing \(D_F\)-contractive operators in Banach space using the concept of measure of non-compactness. As an application, we prove some results on the existence of solutions for a system of an infinite fractional order differential equations in the space \(c\), where space \(c\) consists of real sequences having the finite limits.

On Fredholm Solvability of the Dirichet Problem for Linear Differential Equations of Infinite Order

We propose a new approach to investigation of solvability of the Dirichlet problem for differential equations of infinite order. Namely, by using the embedding theorems obtained by the author in previous papers for the energy spaces, corresponding to operators of infinite order, the initial differential operator of infinite order is expressed as a sum of the main and subordinate operators of infinite order. The conditions, under which the above Dirichlet problems are soluble, are established by using the main term of the corresponding differential operator.

Nonlocal Problem Multipoint in Time for a Class of Partial Differential Equations of Infinite Order

We establish the correct solvability of a nonlocal problem multipoint in time for the evolutionary equation with differential operator of infinite order.

Some new class of special functions suggested by the confluent hypergeometric function

In the present work, we introduce the function representing a rapidly convergent power series which extends the well-known confluent hypergeometric function \(_1F_1[z]\) as well as the integral function \( f(z) = \sum \nolimits _{n=1}^\infty \frac{z^n}{n!^n} \) considered by Sikkema (Differential operators and equations, P. Noordhoff N. V., Djakarta, 1953). We introduce the corresponding differential operators and obtain infinite order differential equations, for which these new special functions are the eigen functions. First we establish some properties, as the order zero of these entire (integral) functions, integral representations, differential equations involving a kind of hyper-Bessel type operators of infinite order. Then we emphasize on the special cases, especially the corresponding analogues of the exponential, circular and hyperbolic functions, called here as \({\ell }\)-H exponential function, \({\ell }\)-H circular and \({\ell }\)-H hyperbolic functions. At the end, the graphs of these functions are plotted using the Maple software.

Linear and nonlinear abstract differential equations of high order

Abstract The nonlocal boundary value problems for linear and nonlinear degenerate abstract differential equations of arbitrary order are studied. The equations have the variable coefficients and small parameters in principal part. The separability properties for linear problem, sharp coercive estimates for resolvent, discreetness of spectrum and completeness of root elements of the corresponding differential operator are obtained. Moreover, optimal regularity properties for nonlinear problem is established. In application, the separability and spectral properties of nonlocal boundary value problem for the system of degenerate differential equations of infinite order is derived.

A Variational Principle Approach for Vibration of Non-Uniform Nanocantilever Using Nonlocal Elasticity Theory

In this article free vibration of a nanocantilever with nonuniform cross section is studied using nonlocal elasticity within the scope of continuum mechanics. Based on an exact variational principle approach, an asymptotic partial differential equation of infinite order is derived along with the corresponding boundary conditions. These equations involve essential higher-order differential terms which, if neglected, have previously led to some rather intriguing observations and conclusions. A reduced sixth-order differential equation is, then, solved for a nanocantilever by applying finite element method using quintic spline interpolation functions. The finite element model developed will be of practical use and reference to physicists and engineers, alike, in the analysis and design of more complicated nanostructures.The paper also resolves some of the strange observations from similar studies like: (i) the non-existence of real eigenvalues (for the first and second modes) for nonlocal parameter (e0a/L)>0.62; and (ii) the existence of a height ratio where frequency becomes independent of size effects, this ratio was defined as the ‘critical height ratio’. It is clear from this study that these observations were a result of using a governing equation and boundary conditions which were not exact thus leading to erroneous observations and conclusions.

Solutions of infinite order differential equations without the grouping phenomenon and a generalization of the Fabry–Pólya theorem

Let {λn}n=1∞ be a sequence of distinct complex numbers diverging to infinity so that |λn|≤|λn+1| for all n∈N, and let {μn}n=1∞ be a sequence of positive integers. Consider the setΛ:={λ1,λ1,…,λ1︸μ1-times,λ2,λ2,…,λ2︸μ2-times,…,λk,λk,…,λk︸μk-times,…}. Subject to the condition μnlog⁡|λn|/|λn|→0 as n→∞, we prove that all non-entire Taylor–Dirichlet series of the form∑n=1∞(∑k=0μn−1cn,kzk)eλnz,cn,k∈C, have a convex natural boundary if and only if Λ is an interpolating variety for the space of entire functions of infraexponential type A|z|0. Our result is in the spirit of the Fabry–Pólya gap results.We also prove that if Λ is the zero set of some F∈A|z|0 but not an interpolating variety, it is still possible for the solutions of the differential equation of infinite order F(d/dz)f=0 to admit a Taylor–Dirichlet series representation, that is, a representation without groupings.

The Existence and Uniqueness of Solutions for a Class of Nonlinear Fractional Differential Equations with Infinite Delay

We prove the existence and uniqueness of solutions for two classes of infinite delay nonlinear fractional order differential equations involving Riemann-Liouville fractional derivatives. The analysis is based on the alternative of the Leray-Schauder fixed-point theorem, the Banach fixed-point theorem, and the Arzela-Ascoli theorem inΩ={y:(−∞,b]→ℝ:y|(−∞,0]∈ℬ}such thaty|[0,b]is continuous andℬis a phase space.

An Addendum to Theorems of A. F. Leont’ev and L. Carleson on an Infinite Order Differential Equation on a Real Interval

In this article we give a necessary and sufficient condition in order for a function to be a solution for a special case of an infinite order differential equation that was investigated by L. Carleson and A. F. Leont’ev.

Representation of multipliers on spaces of real analytic functions

We consider multipliers on the spaces of real analytic functions of one variable, i.e., maps for which monomials are eigenvectors. We prove representation theorems in terms of analytic functionals and in terms of holomorphic functions. We characterize Euler differential operators among multipliers. Then we characterize when such operators are surjective or have a continuous linear right inverse on the space of real analytic functions over an interval not containing zero. In particular we solve the problem when Euler differential equation of infinite order has a solution in the space of real analytic functions on an interval not containing zero.

Vibration Analysis of Axially Compressed Nanobeams and its Critical Pressure Using a New Nonlocal Stress Theory

The transverse vibration of a nanobeam subject to initial axial compressive forces based on nonlocal elasticity theory is investigated. The effects of a small nanoscale parameter at molecular level unavailable in classical mechanics theory are presented and analyzed. Explicit solutions for natural frequency, vibration mode shapes are derived through two different methods: separation of variables and multiple scales. The respective numerical solutions are in close agreement. Validity of the models and approaches presented in the work are verified. Unlike the previous studies for a nonlocal nanostructure, this paper adopts the effective nonlocal bending moment instead of the pure traditional nonlocal bending moment. The analysis yields an infinite-order differential equation of motion which governs the vibrational behaviors. For practical analysis and as examples, an eight-order governing differential equation of motion is solved and the results are discussed. The paper presents a complete nonlocal nanobeam model and the results may be helpful for the application and design of various nano-electro-mechanical devices, e.g. nano-drivers, nano-oscillators, nano-sensors, etc., where a nanobeam acts as a basic element.

Testing for Equivalence of Means under Heteroskedasticity by an Approximate Solution of a Partial Differential Equation of Infinite Order

Abstract. A test for two‐sided equivalence of means has been developed under the assumption of normally distributed populations with heterogeneous variances. Its rejection region is limited by functions ±h that depend on the empirical variances. h is stated implicitly by a partial differential equation, an exact solution of which would provide a test that is exactly similar at the boundary of the null hypothesis of non‐equivalence. h is approximated by Taylor series up to third powers in the reciprocal number of degrees of freedom. This suffices to obtain error probabilities of the first kind that are very close to a nominal level of α = 0.05 at the boundary of the null hypothesis. For more than 10 data points in each group, they range between 0.04995 and 0.05005, and are thus much more precise than those obtained by other authors.

Schwinger pair production in space- and time-dependent electric fields: Relating the Wigner formalism to quantum kinetic theory

The non-perturbative electron-positron pair production (Schwinger effect) is considered for space- and time-dependent electric fields $\vec{E}(\vec{x},t)$. Based on the Dirac-Heisenberg-Wigner (DHW), formalism we derive a system of partial differential equations of infinite order for the sixteen irreducible components of the Wigner function. In the limit of spatially homogeneous fields the Vlasov equation of quantum kinetic theory (QKT) is rediscovered. It is shown that the quantum kinetic formalism can be exactly solved in the case of a constant electric field $E(t)=E_0$ and the Sauter-type electric field $E(t)=E_0\operatorname{sech}^2(t/\tau)$. These analytic solutions translate into corresponding expressions within the DHW formalism and allow to discuss the effect of higher derivatives. We observe that spatial field variations typically exert a strong influence on the components of the Wigner function for large momenta or for late times.

Problem with nonlocal two-point condition in time for a homogeneous partial differential equation of infinite order with respect to space variables

We specify a class of unique solvability of a problem with nonlocal boundary condition for a homogeneous partial differential equation of the first order with respect to time and of infinite order with respect to space variables with constant complex coefficients. In the class of quasipolynomials of special form, we indicate formulas for the construction of a solution of the problem that require a finite number of differentiation operations of analytically given functions. For the case where there exists a nonunique solution of the problem, we present an algorithm of the construction of its partial solution.