Semi-infinite Interval Research Articles

Extracting non-equilibrium material properties in visco-hyperelastic polymers based on analytical stress solution

This study extracts non-equilibrium material properties in the visco-hyperelastic polymers using an analytical stress solution. At first, the analytical stress solution of visco-hyperelastic polymers is obtained from finite-strain creep test data based on the generalized Maxwell’s rheological model. The solution uses the low-computational-cost exponential generalized pseudospectral method and analytically brings the stress components in terms of time and unknown non-equilibrium material properties of Maxwell’s elements. The solution invokes the Ogden hyperelastic model for the elastic element of the generalized Maxwell’s model. The exponential generalized pseudospectral method, based on exponential–rational Lagrange functions suitable for the semi-infinite time interval [Formula: see text] and the rational Chebyshev roots as the collocation points, is applied to derive the stress components. The non-equilibrium properties are derived by minimizing a residual, which is obtained by making the derivatives of the residual in terms of the non-equilibrium properties as zero. Notably, the proposed analytical method requires a small number of exponent–rational Lagrange basis functions and collocation points. Besides, the proposed method does not require integration, discretization, or linearization. Two case studies are considered: the moderate-stretched ETFE polymer with the neo-Hookean model as the elastic element and the large-stretched rubber with the six-parameter Ogden hyperelastic model as the elastic branch. The results are validated using uniaxial creep modeling in ANSYS and available reports in the literature.

Pulsating waves in a multidimensional reaction-diffusion system of epidemic type

In this work we investigate the existence of front propagation for a two-component reaction -diffusion system of epidemic type posed in a multi-dimensional periodic medium. This system is a spatially heterogeneous version of the well-known Kermack–McKendrick epidemic model with Fickian diffusion. We derive sufficient conditions for the existence of pulsating travelling waves propagating in an arbitrarily given unit direction. Specifically, we prove that the set of admissible wave speeds contains a semi-infinite interval. Then, for each direction of propagation and each admissible speed, there exists a pulsating travelling wave solution of the system which is globally bounded.

Quenching behavior and critical speed for a parabolic problem due to a moving nonlinear source

This article investigates an initial–boundary value problem on the semi-infinite interval for a parabolic equation with a moving nonlinear source. The study presents criteria for both finite-time quenching and global existence of the solution. It is shown that there exists a critical speed v∗ for the nonlinear source, ensuring global existence of the solution when the speed of the moving nonlinear source is greater than or equal to v∗, while finite-time quenching occurs when the speed is smaller than v∗. The formula to calculate the critical speed v∗ is also provided for a special case.

Receding Contact of Two Elastic Strips on Semi-Infinite Interval

Receding Contact of Two Elastic Strips on Semi-Infinite Interval

Orthonormal rational functions on a semi-infinite interval

In this paper we propose a novel family of weighted orthonormal rational functions on a semi-infinite interval. We write a sequence of integer-coefficient polynomials in several forms and derive their corresponding differential equations. These equations do not form Sturm-Liouville problems. We overcome this disadvantage by multiplying some factors, resulting in a sequence of irrational functions. We deduce various generating functions of this sequence and find the associated Sturm-Liouville problems, which bring orthogonality. Then we establish a Hilbert space of functions defined on a semi-infinite interval with an inner product induced by a weight function determined by the Sturm-Liouville problems mentioned above. The even subsequence of the irrational function sequence above forms an orthonormal basis for this space. The Fourier expansions of some power functions are deduced. The sequence of non-positive integer power functions forms a non-orthogonal basis for this Hilbert space. We give two examples, one example of Fourier expansion and one example of interpolation as applications.

Solving a class of Thomas–Fermi equations: A new solution concept based on physics-informed machine learning

This paper presents a novel physics-informed machine learning approach, designed to approximate solutions to a specific category of Thomas–Fermi differential equations. To tackle the inherent intricacies of solving Thomas–Fermi equations, we employ the quasi-linearization technique, which transforms the original non-linear problem into a series of linear differential equations. Our approach utilizes collocation least-squares support vector regression, leveraging fractional Chebyshev functions for finite interval simulations and fractional rational Chebyshev functions for semi-infinite intervals, enabling precise solutions for linear differential equations across varied spatial domains. These selections facilitate efficient simulations across both finite and semi-infinite intervals. Key contributions of our approach encompass its versatility, demonstrated through successful approximation of solutions for diverse Thomas–Fermi problem types, including those featuring non-local integral terms, Bohr radius boundary conditions, and isolated neutral atom boundary conditions defined on semi-infinite domains. Furthermore, our method exhibits computational efficiency, surpassing classical collocation methods by solving a sequence of positive definite linear equations or quadratic programming problems. Notably, our approach showcases precision, as evidenced by experiments, including the attainment of the initial slope of the renowned Thomas–Fermi equation with an impressive 39-digit precision.

Analysis of Exact Solution of Problem on Receding Contact of Two Elastic Strips on Semi-Infinite Interval

Analysis of Exact Solution of Problem on Receding Contact of Two Elastic Strips on Semi-Infinite Interval

An Efficient Quadrature Rule for Highly Oscillatory Integrals with Airy Function

In this work, our primary focus is on the numerical computation of highly oscillatory integrals involving the Airy function. Specifically, we address integrals of the form ∫0bxαf(x)Ai(−ωx)dx over a finite or semi-infinite interval, where the integrand exhibits rapid oscillations when ω≫1. The inherent high oscillation and algebraic singularity of the integrand make traditional quadrature rules impractical. In view of this, we strategically partition the interval into two segments: [0,1] and [1,b]. For integrals over the interval [0,1], we introduce a Filon-type method based on a two-point Taylor expansion. In contrast, for integrals over [1,b], we transform the Airy function into the first kind of Bessel function. By applying Cauchy’s integration theorem, the integral is then reformulated into several non-oscillatory and exponentially decaying integrals over [0,+∞), which can be accurately approximated by the generalized Gaussian quadrature rule. The proposed methods are accompanied by rigorous error analyses to establish their reliability. Finally, we present a series of numerical examples that not only validate the theoretical results but also showcase the accuracy and efficacy of the proposed method.

Fractional Boundary Layer Flow: Lie Symmetry Analysis and Numerical Solution

In this paper, we present a fractional version of the Sakiadis flow described by a nonlinear two-point fractional boundary value problem on a semi-infinite interval, in terms of the Caputo derivative. We derive the fractional Sakiadis model by substituting, in the classical Prandtl boundary layer equations, the second derivative with a fractional-order derivative by the Caputo operator. By using the Lie symmetry analysis, we reduce the fractional partial differential equations to a fractional ordinary differential equation, and, then, a finite difference method on quasi-uniform grids, with a suitable variation of the classical L1 approximation formula for the Caputo fractional derivative, is proposed. Finally, highly accurate numerical solutions are reported.

On a Hilbert Space Defined on a Semi-Infinite Interval with a Functional Theoretic Proof of a Relating Basis Theorem

On a Hilbert Space Defined on a Semi-Infinite Interval with a Functional Theoretic Proof of a Relating Basis Theorem

The Existence of a Unique Solution of Nonlinear Integral Equation in Fréchet Space

In this article, the existence of a unique solution on a semi-infinite interval for an integral equation in Fréchet space has been studied by applying a nonlinear alternative of the Leray-Schauder type for contraction maps.

Run-and-tumble motion in trapping environments

Complex or hostile environments can sometimes inhibit the movement capabilities of diffusive particles or active swimmers, who may thus become stuck in fixed positions. This occurs, for example, in the adhesion of bacteria to surfaces at the initial stage of biofilm formation. Here we analyze the dynamics of active particles in the presence of trapping regions, where irreversible particle immobilization occurs at a fixed rate. By solving the kinetic equations for run-and-tumble motion in one space dimension, we give expressions for probability distribution functions, focusing on stationary distributions of blocked particles, and mean trapping times in terms of physical and geometrical parameters. Different extensions of the trapping region are considered, from infinite to cases of semi-infinite and finite intervals. The mean trapping time turns out to be simply the inverse of the trapping rate for infinitely extended trapping zones, while it has a nontrivial form in the semi-infinite case and is undefined for finite domains, due to the appearance of long tails in the trapping time distribution. Finally, to account for the subdiffusive behavior observed in the adhesion processes of bacteria to surfaces, we extend the model to include anomalous diffusive motion in the trapping region, reporting the exact expression of the mean-square displacement.

Random attractors for a stochastic nonlocal delayed reaction–diffusion equation on a semi-infinite interval

Abstract The aim of this paper is to prove the existence and qualitative property of random attractors for a stochastic non-local delayed reaction–diffusion equation (SNDRDE) on a semi-infinite interval with a Dirichlet boundary condition at the finite end. This equation models the spatial–temporal evolution of the mature individuals for a two-stage species whose juvenile and adults both diffuse that lives on a semi-infinite domain and subject to random perturbations. By transforming the SNDRDE into a random evolution equation with delay, by means of a stationary conjugate transformation, we first establish the global existence and uniqueness of solutions to the equation, after which we show the solutions generate a random dynamical system. Then, we deduce uniform a priori estimates of the solutions and show the existence of bounded random absorbing sets. Subsequently, we prove the pullback asymptotic compactness of the random dynamical system generated by the SNDRDE with respect to the compact open topology, and hence obtain the existence of random attractors. At last, it is proved that the random attractor is an exponentially attracting stationary solution under appropriate conditions. The theoretical results are illustrated by application to the stochastic non-local delayed Nicholson’s blowfly equation.

A finite difference method on quasi-uniform grids for the fractional boundary-layer Blasius flow

In this paper, we propose a fractional formulation, in terms of the Caputo derivative, of the Blasius flow described by a non-linear two-point fractional boundary value problem on a semi-infinite interval. We develop a finite difference method on quasi-uniform grids, based on a suitable modification of the classical L1 approximation formula and show the consistency, the stability and the convergence. The numerical results confirm the theoretical ones. Comparisons with some recently proposed results are carried out to validate the accuracy of the obtained numerical results, and to show the efficiency and the reliability of the proposed numerical method.

Semi-infinite interval equilibrium problems: optimality conditions and existence results

This paper aims to obtain new Karush–Kuhn–Tucker optimality conditions for solutions to semi-infinite interval equilibrium problems with interval-valued objective functions. The Karush–Kuhn–Tucker conditions for the semi-infinite interval programming problem are particular cases of those found in this paper for constrained equilibrium problem. We illustrate this with some examples. In addition, we obtain solutions to the interval equilibrium problem in the unconstrained case. The results presented in this paper extend the corresponding results in the literature.

An automatic quadrature method for semi-infinite integrals of exponentially decaying functions and its Matlab code

For the integral of a function f over a semi-infinite interval [0,∞) with f(x) decaying exponentially as x→∞, an automatic quadrature scheme is constructed and implemented in Matlab. Three limit formulae of Clenshaw–Curtis-type rules in our recent work (Numerical Algorithms, vol.90, 2022, pp.3–30) are exploited that converge fast for analytic functions. Each formula is expressed by a weighted infinite sum of function values at nodes and so its truncated finite sum is used. The best of the three formulae in accuracy is used as an approximation whose error is estimated with other two formulae and no additional function evaluations, since nodes of the other formulae are included in those of the best one. Numerical results indicate that our automatic scheme is better in performance than the QUADPACK routine for several types of test integrals. Particularly, it copes quite well with oscillatory integrals including the sinusoidal function and/or the Bessel function that are known to be difficult to approximate.

Holographic measurement in CFT thermofield doubles

We extend the results of arXiv:2209.12903 by studying local projective measurements performed on subregions of two copies of a CFT2 in the thermofield double state and investigating their consequences on the bulk double-sided black hole holographic dual. We focus on CFTs defined on an infinite line and consider measurements of both finite and semi-infinite subregions. In the former case, the connectivity of the bulk spacetime is preserved after the measurement. In the latter case, the measurement of two semi-infinite intervals in one CFT or of one semi-infinite interval in each CFT can destroy the Einstein-Rosen bridge and disconnect the bulk dual spacetime. In particular, we find that a transition between a connected and disconnected phase occurs depending on the relative size of the measured and unmeasured subregions and on the specific Cardy state the measured subregions are projected on. We identify this phase transition as an entangled/disentangled phase transition of the dual CFT system by computing the post-measurement holographic entanglement entropy between the two CFTs. We also find that bulk information encoded in one CFT in the absence of measurement can sometimes be reconstructed from the other CFT when a measurement is performed, or can be erased by the measurement. Finally, we show that a purely CFT calculation of the Renyi entropy using the replica trick yields results compatible with those obtained in our bulk analysis.

New Aspects on the Solvability of a Multidimensional Functional Integral Equation with Multivalued Feedback Control

The current study demonstrates the existence of solutions to a multidimensional functional integral equation with multivalued feedback. We seek solutions for the multidimensional functional problem that is defined, continuous, and bounded on the semi-infinite interval. Our proof is based on the technique associated with measures of noncompactness by a given modulus of continuity in the space in BC(R+). Also, some sufficient conditions are investigated to demonstrate the asymptotic stability of the solutions to that multidimensional functional equation. Additionally, we give an example and some particular cases to illustrate our outcomes.

Достаточные условия существования Н∝ -наблюдателя состояния линейных непрерывных динамических систем

<p>The article deals with the problem of finding the observer of the state vector of linear continuous non-stationary dynamical systems with uncertainty of the initial conditions, limited external influences and measurement errors over a finite time interval. Sufficient conditions for the existence of an observer are formulated and proved on the basis of the expansion principle. Relationships are obtained for finding the parameters of the observer and the worst laws of change in external influences and measurement errors. As a limiting case, the problem of observer synthesis for stationary linear dynamical systems on a semi-infinite time interval is considered. Two applied problems of estimating the aircraft state vector based on the results of incomplete and inaccurate measurements are solved.</p>

Rasyonel Üslü Cebirsel ve Üstel Eşleme Yaklaşımı ile Thomas-Fermi Denklemi için İkinci Derece Doğruluklu Sonlu Farklar Yöntemi

Many problems based on natural sciences need to be solved by the scientists and engineers to serve the humanity. One of the well-known model in atomic universe is condensed into an equation, and called the Thomas-Fermi equation. It is a second order differential equation, which describes charge distributions of heavy, neutral atoms. No exact analytical solution has been found for the equation yet. In fact, strong nonlinearity, singular character and unbounded interval of the problem causes great difficulty to obtain an approximate numerical solution as well. In this paper, the Thomas-Fermi equation is solved using a second order finite difference method along with application of quasi-linearization method. Semi-infinite interval of the problem is converted into [0, 1) using two different coordinate transformations, namely algebraic and exponential mapping. Numerical order of accuracy has been checked using systematic mesh refinements and comparing the calculated initial slope y'(0). Calculated results for initial slope is found in good agreement with the results available in the literature. Lastly, accuracy is improved by the application of the Richardson extrapolation.