Fiberwise linear differential operators

We define a new notion of fiberwise linear differential operator on the total space of a vector bundle E. Our main result is that fiberwise linear differential operators on E are equivalent to (polynomial) derivations of an appropriate line bundle over E ∗ {E^{\ast}} . We believe this might represent a first step towards a definition of multiplicative (resp. infinitesimally multiplicative) differential operators on a Lie groupoid (resp. a Lie algebroid). We also discuss the linearization of a differential operator around a submanifold.

Approximate Solution of Intuitionistic Fuzzy Differential Equations with the Linear Differential Operator by the Homotopy Analysis Method

In this work, the purpose is to discuss the homotopy analysis method (HAM) for the use of intuitionistic fuzzy differential equations with the linear differential operator. Furthermore, a numerical example is presented to shed light on the capability of the present method, and the numerical results illustrated by adopting the homotopy perturbation method (HPM) are compared with the exact solution to ensure the validity of our outcomes.

A method for calculating differential constitutive equation of linear viscoelastic composite model

The stress–strain behaviors of viscoelastic materials are often simulated using a model composed of various combinations of springs and dampers. With the increase in the number of springs and dampers, the viscoelastic characteristics of the model will approach those of the actual material. This study discusses how to obtain the differential constitutive equation of a viscoelastic model composed of any number of springs and dampers. First, the general viscoelastic model is regarded as the combination of various Kelvin units. The viscoelastic model is then transformed into a digraph. Based on the relationships between the independent path of the digraph and the strain equation of the viscoelastic model and between the closed enclosure and the stress equation, the derivation of the constitutive equation is transformed into operations involving the incidence matrix of the digraph. Finally, the coefficients of the linear differential operator of the constitutive equation of the viscoelastic model can be obtained by block matrix operations. This method is suitable for computer programming and has a certain significance for accurately constructing viscoelastic models of engineering materials.

Bit threads, Einstein’s equations and bulk locality

In the context of holography, entanglement entropy can be studied either by i) extremal surfaces or ii) bit threads, i.e., divergenceless vector fields with a norm bound set by the Planck length. In this paper we develop a new method for metric reconstruction based on the latter approach and show the advantages over existing ones. We start by studying general linear perturbations around the vacuum state. Generic thread configurations turn out to encode the information about the metric in a highly nonlocal way, however, we show that for boundary regions with a local modular Hamiltonian there is always a canonical choice for the perturbed thread configurations that exploits bulk locality. To do so, we express the bit thread formalism in terms of differential forms so that it becomes manifestly background independent. We show that the Iyer-Wald formalism provides a natural candidate for a canonical local perturbation, which can be used to recast the problem of metric reconstruction in terms of the inversion of a particular linear differential operator. We examine in detail the inversion problem for the case of spherical regions and give explicit expressions for the inverse operator in this case. Going beyond linear order, we argue that the operator that must be inverted naturally increases in order. However, the inversion can be done recursively at different orders in the perturbation. Finally, we comment on an alternative way of reconstructing the metric non-perturbatively by phrasing the inversion problem as a particular optimization problem.

Stieltjes Moment Sequences for Pattern-Avoiding Permutations

A small set of combinatorial sequences have coefficients that can be represented as moments of a nonnegative measure on $[0, \infty)$. Such sequences are known as Stieltjes moment sequences. They have a number of nice properties, such as log-convexity, which are useful to rigorously bound their growth constant from below. This article focuses on some classical sequences in enumerative combinatorics, denoted $Av(\mathcal{P})$, and counting permutations of $\{1, 2, \ldots, n \}$ that avoid some given pattern $\mathcal{P}$. For increasing patterns $\mathcal{P}=(12\ldots k)$, we recall that the corresponding sequences, $Av(123\ldots k)$, are Stieltjes moment sequences, and we explicitly find the underlying density function, either exactly or numerically, by using the Stieltjes inversion formula as a fundamental tool. We first illustrate our approach on two basic examples, $Av(123)$ and $Av(1342)$, whose generating functions are algebraic. We next investigate the general (transcendental) case of $Av(123\ldots k)$, which counts permutations whose longest increasing subsequences have length at most $k-1$. We show that the generating functions of the sequences $\, Av(1234)$ and $\, Av(12345)$ correspond, up to simple rational functions, to an order-one linear differential operator acting on a classical modular form given as a pullback of a Gaussian $\, _2F_1$ hypergeometric function, respectively to an order-two linear differential operator acting on the square of a classical modular form given as a pullback of a $\, _2F_1$ hypergeometric function. We demonstrate that the density function for the Stieltjes moment sequence $Av(123\ldots k)$ is closely, but non-trivially, related to the density attached to the distance traveled by a walk in the plane with $k-1$ unit steps in random directions. Finally, we study the challenging case of the $Av(1324)$ sequence and give compelling numerical evidence that this too is a Stieltjes moment sequence. Accepting this, we show how rigorous lower bounds on the growth constant of this sequence can be constructed, which are stronger than existing bounds. A further unproven assumption leads to even better bounds, which can be extrapolated to give an estimate of the (unknown) growth constant.

Ulam Stability of a Second Linear Differential Operator with Nonconstant Coefficients

In this paper, we obtain a result on Ulam stability for a second order differential operator acting on a Banach space. The result is connected to the existence of a global solution for a Riccati differential equation and some appropriate conditions on the coefficients of the operator.

Modified New Iterative Method for Solving Nonlinear Partial Differential Equations

In this paper, the iterative method, proposed by Gejji and Jafari in 2006, has been modified for solving nonlinear initial value problems. The Laplace transform was used in this modification to eliminate the linear differential operator in the differential equation. The convergence of the solution was discussed according to the modification proposed. To illustrate this modification some examples were presented.

Splines and fractional differential operators

Several classes of classical cardinal B-splines can be obtained as solutions of operator equations of the form [Formula: see text] where [Formula: see text] is a linear differential operator of integral order. In this paper, we consider classes of generalized B-splines consisting of cardinal polynomial B-splines of complex and hypercomplex orders and cardinal exponential B-splines of complex order and derive the fractional linear differential operators that are naturally associated with them. For this purpose, we also present the spaces of distributions onto which these fractional differential operators act.

New Approach for Solving Partial Differential Equations Based on Collocation Method

In this paper, a new approach for solving partial differential equations was introduced. The collocation method based on LA-transform and proposed the solution as a power series that conforming Taylor series. The method attacks the problem in a direct way and in a straightforward fashion without using linearization, or any other restrictive assumption that may change the behavior of the equation under discussion. Five illustrated examples are introduced to clarifying the accuracy, ease implementation and efficiency of suggested method. The LA-transform was used to eliminate the linear differential operator in the differential equation.