Endpoint Conditions Research Articles

Avoidance of the Lavrentiev gap for one-dimensional non-autonomous functionals with constraints

Abstract Consider a positive functional F ⁢ ( y ) := ∫ t T L ⁢ ( s , y ⁢ ( s ) , y ′ ⁢ ( s ) ) ⁢ 𝑑 s {F(y):=\int_{t}^{T}L(s,y(s),y^{\prime}(s))\,ds} , defined on the space of Sobolev functions W 1 , p ⁢ ( [ t , T ] ; ℝ n ) {W^{1,p}([t,T];{\mathbb{R}}^{n})} , where p ≥ 1 {p\geq 1} . This functional is minimized among functions y that may satisfy one or both endpoint conditions. The Lagrangian L is allowed to assume the value + ∞ {+\infty} . In numerous applications minimizers may not be explicit or even may not exist. In such circumstances, it is crucial to know that the infimum of F can be approximated using a sequence of Lipschitz functions that meet the given boundary conditions. However, there are instances where this approximation is not feasible, even with polynomial Lagrangians that meet Tonelli’s existence conditions: this situation is referred to as the Lavrentiev phenomenon. Some results are present in the literature if one requires the Lipschitz approximations to preserve just one endpoint constraint or when the Lagrangian is finite valued. The present paper deals with problems with two endpoint constraints and Lagrangians that are allowed to take extended values. As a byproduct, our Lipschitz approximations may preserve given state constraints. The extended-valued case is challenging since the phenomenon may occur even when the Lagrangian is constant on its effective domain, whose topology becomes relevant. Once assumed that the Lagrangian is radial convex on the rays of the last variable, our findings offer new insights, even when the Lagrangian L is real-valued, autonomous, and there are no state constraints.

Fractal Dimension of $$\alpha $$-Fractal Functions Without Endpoint Conditions

Fractal Dimension of $$\alpha $$-Fractal Functions Without Endpoint Conditions

Predicting thermoacoustic stability characteristics of longitudinal combustors using different endpoint conditions with a low Mach number flow

Combustion instability frequently occurs in propulsion and power generation systems. It is characterized by large-amplitude acoustic oscillations leading to undesirable consequences. Designing a stable combustor by predicting its stability characteristics is therefore essential. This study centers upon modeling a straight one-dimensional combustor with an acoustically compact heat source, low Mach numbers, and different end point conditions. To predict the stability characteristics, we examine six combustor configurations (open–closed, closed–closed, open–choked, closed–choked, open–open, and closed–open). A Galerkin expansion technique is implemented to capture the acoustic disturbances. The unsteady heat release is modeled using an N−τ formulation. The results show that steepening of the mean temperature gradient causes the eigenfrequency associated with an open outlet to increase more rapidly than that of a choked nozzle. Compared to a choked boundary, an open outlet generates higher eigenfrequencies and lower sound energy when coupled with an open inlet. Conversely, it triggers lower eigenfrequencies and higher sound energy using a closed inlet. The maximum possible growth of sound energy, Gmax, remains positively correlated with the inlet temperature, interaction index N, and inlet Mach number, but inversely proportional to the temperature gradient. The heat source extrema leading to the most and least amplified system energy seem to shift upstream, when the mean temperature gradient is successively increased. Their coordinates are similar in half-open tubes and exhibit a converse relation between the open–open and closed–choked tubes. At sufficiently low Mach numbers, the choked and closed outlets show equivalence in acoustic frequencies, transient energy evolution, and optimal heat source locations.

Error analysis of a residual-based Galerkin’s method for a system of Cauchy singular integral equations with vanishing endpoint conditions

In this paper, we develop Galerkin’s residual-based numerical scheme for solving a system of Cauchy-type singular integral equations of index minus N using Chebyshev polynomials of the first and second kind, where N is the total number of Cauchy-type singular integral equations in the system. Without theoretical analysis, a numerical scheme is not justified. Therefore, first, we prove the well-posedness of the system of Cauchy-type singular integral equations with the help of the compactness of an operator. Further, we derive a theoretical error bound and the order of convergence. Also, we show that the resulting system of equations obtained by applying the algorithm is well-posed together with an explicit representation of the solution in matrix form. Finally, we give some illustrative examples to validate the theoretical error bounds numerically.

Occurrence of gap for one-dimensional scalar autonomous functionals with one end point condition

Occurrence of gap for one-dimensional scalar autonomous functionals with one end point condition

The Akima's fitting method for quartic splines

For the Hermite type quartic spline interpolating on the partition knots and at the midpoint of each subinterval, we consider the estimation of the derivatives on the knots, and the values of these derivatives are obtained by constructing an algorithm of Akima's type. For computing the derivatives on endpoints are also considered alternatives that request optimal properties near the endpoints. The error estimate in the interpolation with this quartic spline is generally obtained in terms of the modulus of continuity. In the case of interpolating smooth functions, the corresponding error estimate reveal the maximal order of approximation O(h^3). A numerical experiment is presented for making the comparison between the Akima's cubic spline and the Akima's variant quartic spline havingdeficiency 2 and natural endpoint conditions.

Non-occurrence of gap for one-dimensional non-autonomous functionals

Let \(F(y){{:}{=}}\int \nolimits _t^TL(s, y(s), y'(s))\,ds\) be a positive functional defined on the space \(W^{1,p}([t,T]; {{\mathbb {R}}}^n)\) (\(p\ge 1\)) of Sobolev functions with, possibly, one or both end point conditions. It is important, especially for the applications, to be able to approximate the infimum of F with the values of F along a sequence of Lipschitz functions satisfying the same boundary condition(s). Sometimes this is not possible, i.e., the so called Lavrentiev phenomenon occurs. This is the case of the seemingly innocent Manià’s Lagrangian \(L(s,y,y')=(y^3-s)^2(y')^6\) and boundary data \(y(0)=0, y(1)=1\); nevertheless in this situation the phenomenon does not occur with just the end point condition \(y(1)=1\). The paper focuses about the different set of conditions that are needed to avoid the Lavrentiev phenomenon for problems depending on the number of end point conditions that are considered. Under minimal assumptions on the (possibly) extended value, Lagrangian, we ensure the non-occurrence of the Lavrentiev phenomenon with just one end point condition, thus extending a milestone result by Alberti and Serra Cassano to non-autonomous case. We then introduce an additional hypothesis, satisfied when the Lagrangian is bounded on bounded sets, in order to ensure the non-occurrence of the phenomenon when dealing with both end point conditions; the result gives some new light even in the autonomous case.

Foam-in-Vein: Characterisation of Blood Displacement Efficacy of Liquid Sclerosing Foams.

Sclerotherapy is among the least invasive and most commonly utilised treatment options for varicose veins. Nonetheless, it does not cure varicosities permanently and recurrence rates are of up to 64%. Although sclerosing foams have been extensively characterised with respect to their bench-top properties, such as bubble size distribution and half-life, little is known about their flow behaviour within the venous environment during treatment. Additionally, current methods of foam characterisation do not recapitulate the end-point administration conditions, hindering optimisation of therapeutic efficacy. Here, a therapeutically relevant apparatus has been used to obtain a clinically relevant rheological model of sclerosing foams. This model was then correlated with a therapeutically applicable parameter-i.e., the capability of foams to displace blood within a vein. A pipe viscometry apparatus was employed to obtain a rheological model of 1% polidocanol foams across shear rates of 6 s-1 to 400 s-1. Two different foam formulation techniques (double syringe system and Tessari) and three liquid-to-gas ratios (1:3, 1:4 and 1:5) were investigated. A power-law model was employed on the rheological data to obtain the apparent viscosity of foams. In a separate experiment, a finite volume of foam was injected into a PTFE tube to displace a blood surrogate solution (0.2% w/v carboxymethyl cellulose). The displaced blood surrogate was collected, weighed, and correlated with foam's apparent viscosity. Results showed a decreasing displacement efficacy with foam dryness and injection flowrate. Furthermore, an asymptotic model was formulated that may be used to predict the extent of blood displacement for a given foam formulation and volume. The developed model could guide clinicians in their selection of a foam formulation that exhibits the greatest blood displacement efficacy.

A singular Sturm–Liouville problem with limit circle endpoint and boundary conditions rationally dependent on the eigenparameter

A singular Sturm–Liouville problem with limit circle endpoint and boundary conditions rationally dependent on the eigenparameter

The convergence properties of a type II complete deficient quartic spline

A novel deficient complete quartic spline S having type II complete endpoint conditions S′′a=f′′a,S′′b=f′′b is constructed, with the expression given in the terms of the second derivative on the mesh nodes, and taking prescribed values on these grid nodes and at midpoints. The existence and uniqueness of this spline is proved, and the error estimates are provided. For less smooth interpolated functions, the interpolation error estimate is given in terms of the uniform modulus of continuity considering S′′a=S′′b=0 as endpoint conditions. Two numerical examples illustrate the geometric properties of this deficient quartic spline interpolation operator. The error estimates are obtained for S,S′,S′′, and S′′′ showing an optimal order of convergence O(h5).

Fractional variational problems on conformable calculus

In this paper, we deal with the variational problems defined by an integral that include fractional conformable derivative. We obtained the optimality results for variational problems with fixed end-point boundary conditions and variable end-point boundary conditions. Then, we studied on the variational problems with integral constraints and holonomic constraints, respectively.

Stratified heat transfer of magneto-tangent hyperbolic bio-nanofluid flow with gyrotactic microorganisms: Keller-Box solution technique

Abstract The purpose of the present investigation is to examine the heat, mass and microorganism concentration transfer rates in the magnetohydrodynamics (MHD) stratified boundary layer flow of tangent hyperbolic nanofluid past a linearly, uniform stretching surface comprising gyrotactic microorganisms as well as nanoparticles. The governing PDEs with relevant end point conditions are molded into a non-dimensional ordinary differential equation (ODE) form by means of the similarity transformation. The numerical solution of dimensionless problem is acquired within the frame of robust Keller-Box technique. The velocity, temperature, mass and motile microorganism density are investigated graphically within the context of different significant parameters. Numerical results have been inspected via plots and table (namely as the local Nusselt number, the local wall mass flux and the local microorganisms wall flux). This article proves that the energy, concentration and motile microorganism density reduce with increase in thermal, solutal and motile density stratification parameters. The asserted outcomes are beneficial to enhance the cooling and heating processes, energy generation, thermal machines, solar energy systems, industrial processes etc.

Application of some special operators on the analysis of a new generalized fractional Navier problem in the context of q-calculus

The key objective of this study is determining several existence criteria for the sequential generalized fractional models of an elastic beam, fourth-order Navier equation in the context of quantum calculus (q-calculus). The required way to accomplish the desired goal is that we first explore an integral equation of fractional order w.r.t. q-RL-integrals. Then, for the existence of solutions, we utilize some fixed point and endpoint conditions with the aid of some new special operators belonging to operator subclasses, orbital α-admissible and α-ψ-contractive operators and multivalued operators involving approximate endpoint criteria, which are constructed by using aforementioned integral equation. Furthermore, we design two examples to numerically analyze our results.

New Variational Problems with an Action Depending on Generalized Fractional Derivatives, the Free Endpoint Conditions, and a Real Parameter

This work presents optimality conditions for several fractional variational problems where the Lagrange function depends on fractional order operators, the initial and final state values, and a free parameter. The fractional derivatives considered in this paper are the Riemann–Liouville and the Caputo derivatives with respect to an arbitrary kernel. The new variational problems studied here are generalizations of several types of variational problems, and therefore, our results generalize well-known results from the fractional calculus of variations. Namely, we prove conditions useful to determine the optimal orders of the fractional derivatives and necessary optimality conditions involving time delays and arbitrary real positive fractional orders. Sufficient conditions for such problems are also studied. Illustrative examples are provided.

A Generalization of a Fractional Variational Problem with Dependence on the Boundaries and a Real Parameter

In this paper, we present a new fractional variational problem where the Lagrangian depends not only on the independent variable, an unknown function and its left- and right-sided Caputo fractional derivatives with respect to another function, but also on the endpoint conditions and a free parameter. The main results of this paper are necessary and sufficient optimality conditions for variational problems with or without isoperimetric and holonomic restrictions. Our results not only provide a generalization to previous results but also give new contributions in fractional variational calculus. Finally, we present some examples to illustrate our results.

Cubic spline fractal solutions of two-point boundary value problems with a non-homogeneous nowhere differentiable term

Fractal interpolation functions (FIFs) supplement and subsume all classical interpolants. The major advantage by the use of fractal functions is that they can capture either the irregularity or the smoothness associated with a function. This work proposes the use of cubic spline FIFs through moments for the solutions of a two-point boundary value problem (BVP) involving a complicated non-smooth function in the non-homogeneous second order differential equation. In particular, we have taken a second order linear BVP: y′′(x)+Q(x)y′(x)+P(x)y(x)=R(x) with the Dirichlet’s boundary conditions, where P(x) and Q(x) are smooth, but R(x) may be a continuous nowhere differentiable function. Using the discretized version of the differential equation, the moments are computed through a tridiagonal system obtained from the continuity conditions at the internal grids and endpoint conditions by the derivative function. These moments are then used to construct the cubic fractal spline solution of the BVP, where the non-smooth nature of y′′ can be captured by fractal methodology. When the scaling factors associated with the fractal spline are taken as zero, the fractal solution reduces to the classical cubic spline solution of the BVP. We prove that the proposed method is convergent based on its truncation error analysis at grid points. Numerical examples are given to support the advantage of the fractal methodology.

Halpern\u2013Ishikawa type iterative schemes for approximating fixed points of multi-valued non-self mappings

Let C be a nonempty closed convex subset of a real Hilbert space H and T: Crightarrow CB(H) be a multi-valued Lipschitz pseudocontractive nonself mapping. A Halpern–Ishikawa type iterative scheme is constructed and a strong convergence result of this scheme to a fixed point of T is proved under appropriate conditions. Moreover, an iterative method for approximating a fixed point of a k-strictly pseudocontractive mapping T: Crightarrow Prox(H) is constructed and a strong convergence of the method is obtained without end point condition. The results obtained in this paper improve and extend known results in the literature.

Early coagulation tests predict risk stratification and prognosis of COVID-19.

The ongoing outbreak of Coronavirus Disease 2019 (COVID-19) is hitting the world hard, but the relationship between coagulation disorders and COVID-19 is still not clear. This study aimed to explore whether early coagulation tests can predict risk stratification and prognosis. PubMed, Web of Science, Cochrane Library, and Scopus were searched electronically for relevant research studies published up to March 24, 2020, producing 24 articles for the final inclusion. The pooled standard mean difference (SMD) of coagulation parameters at admission were calculated to determine severe and composite endpoint conditions (ICU or death) in COVID-19 patients. Meta-analyses revealed that platelet count was not statistically related to disease severity and composite endpoint; elevated D-dimer correlated positively with disease severity (SMD 0.787 (0.277-1.298), P= 0.003, I2= 96.7%) but had no significant statistical relationship with composite endpoints. Similarly, patients with prolonged prothrombin time (PT) had an increased risk of ICU and increased risk of death (SMD 1.338 (0.551-2.125), P = 0.001, I2 = 92.7%). Besides, increased fibrin degradation products (FDP) and decreased antithrombin might also mean the disease is worsening. Therefore, early coagulation tests followed by dynamic monitoring is useful for recognizing coagulation disorders accompanied by COVID-19 and guiding timely therapy to improve prognosis.

Numerical simulation of fractional order optimal control problem

A simple and straightforward numerical method based on reflection operator is proposed in this paper for solving an optimal control problem (OCP) of continuous- time fractional order singular system (FOSS) with fixed final state end point condition by keeping final time fixed. Singular system (SS) dynamics is described by the fractional order differential equations (FDEs) in the sense of either Riemann-Liouville fractional derivative (RLFD) or Caputo fractional derivative (CFD). By using reflection operator we convert right RLFD to its equivalent left RLFD and then approximate the necessary conditions using Grünwald-Letnikov approximation (GLA). An example is illustrated to check the efficacy of proposed algorithm.

Eigenfunction orthogonality for one-dimensional acoustic systems with interior or end point conditions.

Some common exercises presented in introductory acoustics courses and texts illustrate solutions involving eigenvalues and eigenfunctions. Challenging extensions of these, even for one-dimensional (1D) systems, might involve a mass or spring loading the acoustic medium at an end point or at an interior point. These problems might be extended further by requiring that some given function be expanded in a series of the eigenfunctions, but such extended problems may lead to unexpected complications in regard to eigenfunction orthogonality. In this paper, Sturm-Liouville theory is used to develop a systematic method for predetermining eigenfunction orthogonality for 1D systems loaded at end points or interior points or having properties that change with jump discontinuities.