Non-standard Boundary Research Articles

Robust and accurate numerical framework for multi-dimensional fractional-order telegraph equations using Jacobi/Jacobi-Romanovski spectral technique

This paper presents a novel spectral algorithm for the numerical solution of multi-dimensional fractional-order telegraph equations, a critical model used to capture the combined effects of diffusion and wave propagation. The core innovation of this work is the application of Jacobi-Romanovski polynomials as the basis functions for spectral discretization. These polynomials offer unique advantages, including the ability to handle nonstandard domains and boundary conditions, making them particularly suitable for partial differential equation (PDE) applications. A comprehensive error analysis is conducted, providing deep insights into the convergence rates and factors affecting the accuracy of the numerical solutions. Extensive numerical experiments further demonstrate the superior performance of the proposed spectral algorithm in solving a wide range of multi-dimensional fractional-order telegraph equation models. The results show a significant improvement in accuracy and computational efficiency compared to traditional numerical methods, such as finite difference or finite element techniques. This research advances the field of computational science by offering a robust, efficient, and versatile numerical framework for the precise solution of complex multi-dimensional PDEs.

A pressure-residual augmented GLS stabilized method for a type of Stokes equations with nonstandard boundary conditions

A pressure-residual augmented GLS stabilized method for a type of Stokes equations with nonstandard boundary conditions

The algorithmic resolution of spectral-element discretization for the time-dependent Stokes problem

We consider two algorithms for the resolution of the time-dependent Stokes problem with nonstandard boundary conditions by the domain-decomposition spectral-element method. The first algorithm (Elimination method) is based on the Uzawa method by decoupling the linear system, while the second algorithm (Global inversion) is based on the overall resolution of the system by the GMRES method. A detailed implementation is proposed and some numerical tests are carried out in two and three dimensions and where the domain is multiply connected.

Dimensional reduction of the S3/WZW duality

Recently proposed duality relates the critical level limit \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\widehat{k}\ o -2$$\\end{document} of \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\ ext{SU}}{\\left(2\\right)}_{\\widehat{k}}$$\\end{document} WZW models to a classical three-dimensional Einstein gravity on a sphere. In this paper, we propose a dimensional reduced version of this duality. The gravity side is reduced to a Jackiw-Teitelboim (JT) gravity on S2 with a non-standard boundary term, or a BF theory with SU(2) gauge symmetry. At least in low temperature limit, these two-dimensional gravity theories completely capture the original three-dimensional gravity effect. The CFT side is reduced to a certain complex Liouville quantum mechanics (LQM) with SU(2) gauge symmetry. Our proposal gives an interesting example of a holography without boundary. We also discuss a higher-spin generalization with SU(N) gauge symmetry.

Spectral element discretization of the time-dependent Stokes problem with nonstandard boundary conditions

This work deals with the spectral element discretization of the time-dependent Stokes problem in two- and three-dimensional domains. The boundary condition is defined on the normal component of the velocity and the tangential components of the vorticity. The discretization related to the time variable is processed by a Backward Euler method. We prove through a detailed numerical analysis the well-posedness of the full discrete problem.

Theoretical analysis for free vibration of submerged conical motor stator

The conical rim-driven motor stator is a structure with variable thickness and non-standard boundaries, which can be treated as an orthotropic conical shell. To prevent resonance and reduce electromagnetic noise, a modal analysis of the stator is necessary. This paper presents an analytical model for the free vibration of the conical stator in both onshore and underwater environments. Due to the relatively large thickness of the motor stator, the first-order shear deformation theory (FSDT) is employed. The fluid loading is calculated by discretizing the conical shell into a series of narrow cylindrical shells featuring diverse radii and thickness. The resulting equations are solved employing the transfer matrix method (TMM), which is capable of different types of boundary conditions (BCs). The convergence and accuracy of the proposed method are validated through a comparative analysis involving numerical simulations and outcomes derived from the thin shell theory. The importance of considering shear deformation and rotatory inertia in determining the natural frequency of motor stators is demonstrated. Finally, the impact of fluid loading on the vibration behavior of stators with various geometric parameters and boundary conditions are investigated and discussed. The proposed model presents a precise and convenient approach for analyzing the free vibration of submerged conical motor stators, and can be extended to conventional cylindrical motor stators through appropriate degradation procedures.

Overdamped dynamics of a falling inextensible network: Existence of solutions

We study the equations of overdamped motion of an inextensible triod with three fixed ends and a free junction under the action of gravity. The problem can be expressed as a system of PDEs that involves unknown Lagrange multipliers and non-standard boundary conditions related to the freely moving junction. It can also be formally interpreted as a gradient flow of the potential energy on a certain submanifold of the Otto–Wasserstein space of probability measures. We prove global existence of generalized solutions to this problem.

A second-gradient elasticity model and isogeometric analysis for the pantographic ortho-block

We present a continuum model for a three-dimensional pantographic structure that we call pantographic ortho-block. The model is based on the embedding of four families of elastic fibers in a three-dimensional domain. Similar to pantographic sheets, we find an energy density depending on the first and second-gradient of displacement resulting in a three-dimensional incomplete second-gradient material. For certain kinematic assumptions concerning perfect joints, we find two distinct zero-energy modes. Finally, we perform simulations based on isogeometric analysis to showcase standard and non-standard boundary conditions.

FROM PROPAGATION SYSTEMS TO TIME DELAYS AND BACK. CRITICAL CASES

The paper originates from the early ideas of A. D. Myshkis and his co-workers and of K. L. Cooke and his co-worker. These ideas send to a one-to-one correspondence between lossless and/or distortionless propagation described by nonstandard boundary value problems and a system of coupled differential and difference equations with deviated argument. In this way any property obtained for one mathematical ob­ject is automatically projected back on the other one. This approach is considered here for certain engineering applications. The common feature of these applications is the critical stability of the difference operator associated with the system with deviated argument obtained for each of the aforementioned applications. In fact the associated sys­tems are of neutral type and, according to the assumption of Hale, only strong stability of the difference operator ensures robust asymp­totic stability with respect to the delays. If the difference operator is in the critical case, the stability becomes fragile with respect to the delays. Based on some old results in the field, a conjecture concerning the (quasi)-critical modes of the system is stated; also a connection with the so called dissipative boundary conditions is suggested.

On One Normal Boundary Value Problem of the Field Theory

One normal problem of the field theory on a plane for a system of Poisson equations is studied. This problem is one of implementations of the general approach to formulation of boundary value problems of the field theory, which is based on studying the kernels of trace operators and kernels of trace functionals, which are taken as the main space of desired solutions. A peculiarity of this problem is that one of the boundary conditions imposed on the desired solution is nonlocal in nature. The considered boundary condition nonlocality is due to the fact that the base space in which the solution of the problem is sought is the kernel of the regular trace functional in the Sobolev space. The need to supplement the problem on the one-dimensional cokernel of the above-mentioned functional, which arises in this case, determines the second boundary condition, which is local (pointwise) in nature. That is why, in the framework of the duality of the Sobolev space and its conjugate space, the solution to the problem is the pair (u, с), where с is a constant that compensates for the one-dimensional “defect” of the problem. It should be noted that the theory of boundary value problems for vector fields on a plane differs essentially from the theory of boundary value problems for three-dimensional vector fields. One of the reasons for the difficulties in formulating three-dimensional boundary value problems is that a number of important surfaces cannot be made parallel to each other, which rules out the possibility to consider “tangential” boundary value problems, that is, problems with boundary tangential fields. In this respect, the theory of boundary value problems of plane vector fields differs from the three-dimensional theory for the better. On the other hand, the fact that a formula that would interrelating the Laplace operator with first-order field theory operations is not available on a plane is an obstacle to identifying new non-standard boundary value problems for plane fields. Revealing of this interrelation is a separate issue, the solution of which leads to the concept of the trace of first-order field theory operators as a singular functional above the space of traces of functions from the first-order Sobolev space. Thus, boundary value problems for plane fields are a separate section of the general theory of field boundary value problems. One of the non-standard problems in the theory of plane vector fields, which contains non-local boundary conditions, is considered. A theorem for existence and uniqueness of a weak solution of the problem posed is formulated and proved. The proof of the theorem includes a few steps: using the Galerkin method, the existence of a solution to the equation is established; the passage to the limit is made, and the vector function u(x) is found; the number α is determined, and the uniqueness of the function u and the number α is established. The main result of the work is that the problem posed is correct in nature.

DYNAMIC CHARACTERISTICS OF MIXTURE UNIFIED GRADIENT ELASTIC NANOBEAMS

The mixture unified gradient theory of elasticity is invoked for the rigorous analysis of the dynamic characteristics of elastic nanobeams. A consistent variational framework is established and the boundary-value problem of dynamic equilibrium enriched with proper form of the extra non-standard boundary conditions is detected. As a well-established privilege of the stationary variational theorems, the constitutive laws of the resultant fields cast as differential relations. The wave dispersion response of elastic nano-sized beams is analytically addressed and the closed form solution of the phase velocity is determined. The free vibrations of the mixture unified gradient elastic beam is, furthermore, analytically studied. The dynamic characteristics of elastic nanobeams is numerically evaluated, graphically illustrated, and commented upon. The efficacy of the established augmented elasticity theory in realizing the softening and stiffening responses of nano-sized beams is evinced. New numerical benchmark is detected for dynamic analysis of elastic nanobeams. The established mixture unified gradient elasticity model provides a practical approach to tackle dynamics of nano-structures in pioneering MEMS/NEMS.

Nonlinear flexure mechanics of mixture unified gradient nanobeams

The mixture unified gradient theory of elasticity is invoked for the nanoscopic study of the nonlinear flexure mechanics of nanobeams. A mixed variational framework is conceived based on ad hoc functional space of kinetic test fields, wherein a complete set of governing equations, classical and non-standard boundary conditions, and the constitutive relations are integrated into a solitary functional. The size-effect phenomenon associated with the nonlinear flexure of nanobeams are efficiently realized as the stress gradient theory, the strain gradient theory, and the classical elasticity theory are consistently unified within the framework of the mixture unified gradient elasticity. A viable numerical approach is introduced based on the conceived mixed variational framework while the autonomous series solution of the kinematic and kinetic field variables is implemented. As the kinetic field variables are directly determined rather than utilizing post-computations, the established consistent variational theory provides an advantageous means for efficacious approximate approaches. The nonlinear flexural response of the mixture unified gradient beams with kinematics constraints of practical interest in nano-mechanics is detected and thoroughly compared with the linear counterparts in terms of the characteristic parameters. The detected nonlinear flexural characteristics of nano-sized beams can pave the way ahead in the nonlinear mechanics of nano-structures.

Torsional deformation of an infinite elastic solid weakened by a penny-shaped crack in the presence of surface elasticity and Dugdale plastic zone

We consider the torsional deformation of an infinite three-dimensional isotropic elastic solid weakened by a penny-shaped crack whose boundary is enhanced by the incorporation of surface elasticity. Furthermore, we assume the presence of a Dugdale-type plastic zone around the crack front. The ensuing elastoplastic problem gives rise to a non-standard mixed boundary value problem which is then reduced to a hypersingular integral equation upon application of the Hankel transform. The hypersingular integral equation is subsequently solved numerically using orthogonal expansions revealing the effects of surface elasticity on the size of the plastic zone and crack-face torsional displacements. Our results demonstrate the effect of surface elasticity on the width of the plastic zone thus providing a significant shielding effect while increasing the load-bearing capacity of the solid in the vicinity of the defect. Our conclusions will be of particular interest when the presence of surface elasticity along a crack boundary is used in enhanced continuum-based models to study fracture at micro- and nano-scales, for example, in the prediction of fracture and failure in micro- and nano-structures.

On a new class of fractional calculus of variations and related fractional differential equations

This paper analyzes a class of fractional calculus of variations problems and their associated Euler-Lagrange (fractional differential) equations. Unlike the existing fractional calculus of variations which is based on the classical notion of fractional derivatives, the fractional calculus of variations considered in this paper is based on a newly developed notion of weak fractional derivatives and their associated fractional order Sobolev spaces. Since fractional derivatives are direction-dependent, using one-sided fractional derivatives and their combinations leads to new types of calculus of variations and fractional differential equations as well as nonstandard Neumann boundary operators. This paper establishes the well-posedness and regularities for a class of fractional calculus of variations problems and their Euler-Lagrange (fractional differential) equations. This is achieved first for one-sided Dirichlet energy functionals which lead to one-sided fractional Laplace equations, then for more general energy functionals which give rise to more general fractional differential equations.

Nonlocal gradient integral models with a bi-Helmholtz averaging kernel for functionally graded beams

In this paper, we develop the bi-Helmholtz averaging kernel-based variationally nonlocal strain and stress gradient (NStrainG and NStressG) integral models to study the size-dependent bending of functionally graded (FG) beams. The integral constitutive relation is transformed into a higher-order differential form while four non-standard constitutive boundary conditions are derived in detail. The Laplace transform technique is utilized to derive closed-form solutions. Interestingly, we find that, unlike the Helmholtz kernel case, the local stress contribution must be included in the bi-Helmholtz kernel-based NSrainG model to avoid the conflict between equilibrium and constitutive requirements. In the numerical simulations, a series of examples are presented to study the size-effects on the bending deformation of the beams with various boundary types. Numerical results show that the bi-Helmholtz kernel-based nonlocal gradient theories are capable of describing both softening and stiffening size-effects in small structures by suitably tuning the nonlocal and gradient relevant parameters involved in nonlocal gradient models. Moreover, the effect of FG parameter is also presented.

Stress-driven local/nonlocal mixture model for buckling and free vibration of FG sandwich Timoshenko beams resting on a nonlocal elastic foundation

We study the buckling and free vibration of functionally graded (FG) sandwich Timoshenko beams resting on an elastic foundation. In contrast to the majority of the literature on this subject, the behaviors of both the beam and elastic foundation are considered as nonlocal by applying the stress-driven strategy equipped with a bi-Helmholtz kernel. We find that in the presence of a nonlocal elastic foundation, the local/nonlocal mixture should be adopted in order to obtain a well-posed formulation of the problem at hand. The equations of motion and the standard boundary conditions are obtained by invoking Hamilton's principle. Each integro-differential constitutive law is transformed into an equivalently differential form equipped with four non-standard constitutive boundary conditions. The generalized differential quadrature method (GDQM) is then utilized to solve the corresponding eigenvalue problems numerically. Several comparative studies are conducted to validate the effectiveness of our solution. The numerical simulation results present the size-effect on the critical buckling loads and natural frequency of the beams with various boundary types, providing a new benchmark for further study of modeling small-scale beam structures using the bi-Helmholtz kernel-based stress-driven mixture model. Moreover, the influence of considering the size-dependency of the elastic foundation is also investigated.

Bootstrapping boundary-localized interactions II. Minimal models at the boundary

We provide evidence for the existence of non-trivial unitary conformal boundary conditions for a three-dimensional free scalar field, which can be obtained via a coupling to the m’th unitary diagonal minimal model. For large m we can demonstrate the existence of the fixed point perturbatively, and for smaller values we use the numerical conformal bootstrap to obtain a sharp kink that smoothly matches onto the perturbative predictions. The wider numerical analysis also yields universal bounds for the spectrum of any other boundary condition for the free scalar field. A second kink in these bounds hints at a second class of non-standard boundary conditions, as yet unidentified.

Unsteady non-Newtonian fluid flow with heat transfer and Tresca\u2019s friction boundary conditions

We consider an unsteady non-isothermal flow problem for a general class of non-Newtonian fluids. More precisely the stress tensor follows a power law of parameter p, namely sigma = 2 mu ( theta , upsilon , | D(upsilon ) |) |D( upsilon ) |^{p-2} D(upsilon ) - pi mathrm{Id} where θ is the temperature, π is the pressure, υ is the velocity, and D(upsilon ) is the strain rate tensor of the fluid. The problem is then described by a non-stationary p-Laplacian Stokes system coupled to an L^{1}-parabolic equation describing thermal effects in the fluid. We also assume that the velocity field satisfies non-standard threshold slip-adhesion boundary conditions reminiscent of Tresca’s friction law for solids. First, we consider an approximate problem (P_{delta }), where the L^{1} coupling term in the heat equation is replaced by a bounded one depending on a small parameter 0 < delta ll 1, and we establish the existence of a solution to (P_{delta }) by using a fixed point technique. Then we prove the convergence of the approximate solutions to a solution to our original fluid flow/heat transfer problem as δ tends to zero.

Generalized finite difference method for solving two-interval Sturm-Liouville problems with jump conditions

We consider a Sturm-Liouville problem defined on two disjoint intervals together with additional jump conditions across the common endpoint of these intervals. Based on Finite Difference Method (FDM) we have developed a new tecnique for solving such type nonstandard boundary value problems (BVP). To show applicability and effectiveness of the proposed generalization of FDM, we solved a simple but illustrative example. The obtained numerical solutions are graphically compared with the corresponding exact solutions.

On the analytical and meshless numerical approaches to mixture stress gradient functionally graded nano-bar in tension

The mixture stress gradient theory of elasticity is conceived via consistent unification of the classical elasticity theory and the stress gradient theory within a stationary variational framework. The boundary-value problem associated with a functionally graded nano-bar is rigorously formulated. The constitutive law of the axial force field is determined and equipped with proper non-standard boundary conditions. Evidences of well-posedness of the mixture stress gradient problems, defined on finite structural domains, are demonstrated by analytical analysis of the axial displacement field of structural schemes of practical interest in nano-mechanics. An effective meshless numerical approach is, moreover, introduced based on the proposed stationary variational principle while employing autonomous series solution of the kinematic and kinetic field variables. Suitable mathematical forms of the coordinate functions are set forth in terms of the modified Chebyshev polynomials, satisfying the required classical and non-standard boundary conditions. An excellent agreement between the numerical results of the axial displacement field of the functionally graded nano-bar and the analytical solution counterpart is confirmed on the entire span of the nano-sized bar, in terms of the mixture parameter and the stress gradient characteristic parameter. The effectiveness of the established meshless numerical approach, demonstrating a fast convergence rate and an admissible convergence region, is hence ensured. The established mixture stress gradient theory can effectively characterize the peculiar size-dependent response of functionally graded structural elements of advanced ultra-small systems.