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We provide sparse estimates for gradients of solutions to divergence form elliptic partial differential equations in terms of the source data. We give a general result of Meyers (or Gehring) type, a result for linear equations with VMO coefficients and a result for linear equations with Dini continuous coefficients. In addition, we provide an abstract theorem conditional on PDE estimates available. The linear results have the full range of weighted estimates with Muckenhoupt weights as a consequence. [ABSTRACT FROM AUTHOR]

A class of semilinear elliptic differential equations was investigated in this study. By constructing the inverse function, using the method of upper and lower solutions and the principle of comparison, the existence of the maximum positive solution and the minimum positive solution was explored. Furthermore, the uniqueness of the positive solution and its asymptotic estimation at the origin were evaluated. The results show that the asymptotic estimation is similar to that of the corresponding boundary blowup problems. Compared with the conclusions of Wei's work in 2017, the asymptotic behavior of the solution only depends on the asymptotic behavior of b (x) at the origin and the asymptotic behavior of g at infinity. [ABSTRACT FROM AUTHOR]

In control theory, typically a nominal model is assumed based on which an optimal control is designed and then applied to an actual (true) system. This gives rise to the problem of performance loss because of the mismatch between the true and assumed models. A robustness problem in this context is to show that the error because of the mismatch between a true and an assumed model decreases to zero as the assumed model approaches the true model. We study this problem when the state dynamics of the system are governed by controlled diffusion processes. In particular, we discuss continuity and robustness properties of finite and infinite horizon α-discounted/ergodic optimal control problems for a general class of nondegenerate controlled diffusion processes as well as for optimal control up to an exit time. Under a general set of assumptions and a convergence criterion on the models, we first establish that the optimal value of the approximate model converges to the optimal value of the true model. We then establish that the error because of the mismatch that occurs by application of a control policy, designed for an incorrectly estimated model, to a true model decreases to zero as the incorrect model approaches the true model. We see that, compared with related results in the discrete-time setup, the continuous-time theory lets us utilize the strong regularity properties of solutions to optimality (Hamilton–Jacobi–Bellman) equations, via the theory of uniformly elliptic partial differential equations, to arrive at strong continuity and robustness properties. Funding: The research of S. Yüksel was partially supported by the Natural Sciences and Engineering Research Council of Canada (NSERC). [ABSTRACT FROM AUTHOR]

Using the Pohozaev-type arguments, we prove the nonexistence results for sub-elliptic problems with critical Sobolev-Hardy exponents and Hardy-type potentials on the Carnot group. [ABSTRACT FROM AUTHOR]

In this article, exact solutions of the Biswas–Arshed equation are obtained using the extended Weierstrass transformation method (EWTM). This method is widely used in solid-state physics, electrodynamics, and mathematical physics, and it yields exact solution functions involving trigonometric, rational trigonometric, Weierstrass elliptic, wave, and rational functions. The process involves expanding the solution functions of an elliptic differential equation into finite series by transforming them into Weierstrass functions. Furthermore, it generates parametric solutions for nonlinear algebraic equation systems, which are particularly useful in mathematical physics. These solutions are derived using the Mathematica package program. To analyze the behavior of these determined solution functions, the article employs separate two- and three-dimensional graphs showing the real and imaginary components, along with contour and density graphs. These visuals aid in comprehending the physical characteristics exhibited by these solution functions. [ABSTRACT FROM AUTHOR]

Algebraic multigrid (AMG) methods are among the most efficient solvers for linear systems of equations and they are widely used for the solution of problems stemming from the discretization of Partial Differential Equations (PDEs). A severe limitation of AMG methods is the dependence on parameters that require to be fine-tuned. In particular, the strong threshold parameter is the most relevant since it stands at the basis of the construction of successively coarser grids needed by the AMG methods. We introduce a novel deep learning algorithm that minimizes the computational cost of the AMG method when used as a finite element solver. We show that our algorithm requires minimal changes to any existing code. The proposed Artificial Neural Network (ANN) tunes the value of the strong threshold parameter by interpreting the sparse matrix of the linear system as a gray scale image and exploiting a pooling operator to transform it into a small multi-channel image. We experimentally prove that the pooling successfully reduces the computational cost of processing a large sparse matrix and preserves the features needed for the regression task at hand. We train the proposed algorithm on a large dataset containing problems with a strongly heterogeneous diffusion coefficient defined in different three-dimensional geometries and discretized with unstructured grids and linear elasticity problems with a strongly heterogeneous Young's modulus. When tested on problems with coefficients or geometries not present in the training dataset, our approach reduces the computational time by up to 30%. [ABSTRACT FROM AUTHOR]

We deal with probabilistic numerical solutions for linear elliptic equations with Neumann boundary conditions in a Lipschitz domain, by using a probabilistic numerical scheme introduced by Milstein and Tretyakov based on new numerical layer methods. [ABSTRACT FROM AUTHOR]

We investigate a class of parametric elliptic semilinear partial differential equations of second order with homogeneous essential boundary conditions, where the coefficients and the right-hand side (and hence the solution) may depend on a parameter. This model can be seen as a reaction-diffusion problem with a polynomial nonlinearity in the reaction term. The efficiency of various numerical approximations across the entire parameter space is closely related to the regularity of the solution with respect to the parameter. We show that if the coefficients and the right-hand side are analytic or Gevrey class regular with respect to the parameter, the same type of parametric regularity is valid for the solution. The key ingredient of the proof is the combination of the alternative-to-factorial technique from our previous work [1] with a novel argument for the treatment of the power-type nonlinearity in the reaction term. As an application of this abstract result, we obtain rigorous convergence estimates for numerical integration of semilinear reaction-diffusion problems with random coefficients using Gaussian and Quasi-Monte Carlo quadrature. Our theoretical findings are confirmed in numerical experiments. [ABSTRACT FROM AUTHOR]

In this work, we establish existence results for the following fourth-order Kirchhoff-type elliptic problem with Hardy potential M(∫Ω |∆u| pdx) ∆²pu - a |x|p|u|p-2u = λf(x, u), in Ω, u = ∆u = 0, on ∂Ω. Precisely, by using the classical Hardy inequality and critical point theory, we prove the existence of multiple weak solutions for the fourth-order Kirchhoff-type elliptic problem with Hardy potential. [ABSTRACT FROM AUTHOR]

For various kinds of parabolic and elliptic partial differential and differential-difference equations, results on the stabilization of solutions are presented. For the Cauchy problem for parabolic equations, the stabilization is treated as the existence of a limit as the time unboundedly increases. For the half-space Dirichlet problem for parabolic equations, the stabilization is treated as the existence of a limit as the independent variable orthogonal to the boundary half-plane unboundedly increases. In the classical case of the heat equation, the necessary and sufficient condition of the stabilization consists of the existence of the limit of mean values of the initial-value (boundary-value) function over balls as the ball radius tends to infinity. For all linear problems considered in the present paper, this property is preserved (including elliptic equations and differential-difference equations). The Wiener Tauberian theorem is used to establish this property. To investigate the differential-difference case, we use the fact that translation operators are Fourier multipliers (as well as differential operators), which allows one to use a standard Gel'fand-Shilov operational scheme. For all quasilinear problems considered in the present paper, the mean value from the stabilization criterion is changed: It undergoes a monotonic map, which is explicitly constructed for each investigated nonlinear boundary-value problem. [ABSTRACT FROM AUTHOR]

In this paper, we give a computationally optimal relaxed scalar auxiliary variable (SAV) approach for solving gradient flow systems, in which we only need to solve an elliptic partial differential equation with constant coefficients at each time step. In addition to being applicable to various types of auxiliary variables, there are several advantages of our methods, including: (i) by modifying the optimization procedure in correction of auxiliary variables into a linear programming problems, the original hard-to-solve nonlinear optimization problem arising from previous relaxed SAV approach introduced in Jiang et al. (2022) [28] can be avoided; (ii) the resulting method yields some novel linear unconditionally energy stable schemes, in which backward Euler and Crank–Nicolson formulas are used to discretize the time so that the accuracy can reach the first- and second-order, respectively; (iii) comparing with the baseline SAV approach, the discrete energy in the energy dissipation law is closer to the original energy. Finally, ample numerical results demonstrate the accuracy, efficiency and energy stability of the proposed scheme. [ABSTRACT FROM AUTHOR]

We employ the Craig-Wayne-Bourgain (CWB) method to construct quasi-periodic solutions for the nonlinear delayed perturbation equations. The linearized equation at each step of the iterations is a differential-difference equation on the torus. We shall combine the techniques of Green's function estimate and the reducibility method in KAM theory to solve the linear equation, which generalizes the applicability of the CWB method. As an application, we study the positive quasi-periodic solutions for a class of Lotka-Volterra equations with quasi-periodic coefficients and time delay. [ABSTRACT FROM AUTHOR]

In the current research, we delve into the intricate realm of bifurcation analysis for Föppl–von Kármán plates, employing a precise numerical tool. This innovative numerical approach melds the power of spectral discretization with the prowess of a high-order continuation method-based Taylor series development (HODC). It is worth noting that combining the high-order continuation method with such discretization techniques offers an efficient path-following approach, complete with adaptive step lengths, capable of tackling a wide array of nonlinear problems. Despite the extensive applications of nonlinear elasticity, the spectral method remains relatively uncharted territory within this context. However, our deep-rooted understanding and expertise in the field drive us to embrace this method alongside high-order development continuation for bifurcation analysis of Föppl–von Kármán plates. The governing equations governing thin elastic plates experiencing significant elastic deflections manifest as a pair of coupled nonlinear differential equations, famously known as the von Kármán (vK) equations, presented in a strong form with two principal unknowns: deflection (w) and the Airy stress function (F). Leveraging Chebyshev decomposition matrices, we approximate these fourth-order elliptic nonlinear partial differential equations. Subsequently, we harness high-order development continuation techniques to morph these nonlinear systems into linear ones. Our rigorous evaluation and validation of this numerical approach's precision and performance come to fruition through a comprehensive buckling analysis encompassing multiple illustrative examples. [ABSTRACT FROM AUTHOR]

In this paper, a new extension of the perturbed nonlinear Chen-Lee-Liu equation as a variable coefficients model represents optical pulse propagation in a monomode fiber is studied. The direct similarity reduction method is used to transform the Chen-Lee-Liu model with variable coefficients to a nonlinear ordinary differential equation and the Jacobi elliptic expansion method is used to solve the reduced equation and as a result, novel optical solitons, periodic, and singular waves have arisen. Then, to prove the physical existence and importance of the presented variable coefficients Chen-Lee-Liu model, another similarity technique is used to transform the Chen-Lee-Liu model to the generalized derivative Schrödinger equation which has known bright and dark soliton solutions. Finally, a graphical representation of the obtained wave solutions according to different structures of the variable coefficients is presented. [ABSTRACT FROM AUTHOR]

Copyright of Gaceta de la Real Sociedad Matematica Espanola is the property of Real Sociedad Matematica Espanola and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

In this paper, we consider, from a numerical point of view, a two-temperature poro-thermoelastic problem. The model is written as a coupled linear system of hyperbolic and elliptic partial differential equations. An existence result is proved and energy decay properties are recalled. Then we introduce a fully discrete approximation by using the finite element method and the implicit Euler scheme. Some a priori error estimates are obtained, from which the linear convergence of the approximation is deduced under an appropriate additional regularity. Finally, some numerical simulations are performed to demonstrate the accuracy of the approximation, the decay of the discrete energy and the behaviour of the solution depending on a constitutive parameter. [ABSTRACT FROM AUTHOR]

Nine point sixth order compact numerical approximations are suggested to solve 2D nonlinear elliptic partial differential equations (NLEPDEs) and for the estimation of normal derivatives on a uniform rectangular grid subject to Dirichlet boundary conditions. We deliberate error analysis and reveal that, under specific conditions, our method converges to the sixth order. In addition, we extend our technique to vector form in order to solve the system of NLEPDEs. In application, we discuss nine-point compact sixth order approximations for bi- and tri-harmonic elliptic boundary value problems. Numerical experiments are carried out on several benchmark problems including bi- and tri-harmonic equations, and verified the sixth order convergence of the proposed methods. [ABSTRACT FROM AUTHOR]

Transformations are much used to connect complicated nonlinear differential equations to simple equations with known exact solutions. Two examples of this are the Hopf–Cole transformation and the simple equations method. In this article, we follow an idea that is opposite to the idea of Hopf and Cole: we use transformations in order to transform simpler linear or nonlinear differential equations (with known solutions) to more complicated nonlinear differential equations. In such a way, we can obtain numerous exact solutions of nonlinear differential equations. We apply this methodology to the classical parabolic differential equation (the wave equation), to the classical hyperbolic differential equation (the heat equation), and to the classical elliptic differential equation (Laplace equation). In addition, we use the methodology to obtain exact solutions of nonlinear ordinary differential equations by means of the solutions of linear differential equations and by means of the solutions of the nonlinear differential equations of Bernoulli and Riccati. Finally, we demonstrate the capacity of the methodology to lead to exact solutions of nonlinear partial differential equations on the basis of known solutions of other nonlinear partial differential equations. As an example of this, we use the Korteweg–de Vries equation and its solutions. Traveling wave solutions of nonlinear differential equations are of special interest in this article. We demonstrate the existence of the following phenomena described by some of the obtained solutions: (i) occurrence of the solitary wave–solitary antiwave from the solution, which is zero at the initial moment (analogy of an occurrence of particle and antiparticle from the vacuum); (ii) splitting of a nonlinear solitary wave into two solitary waves (analogy of splitting of a particle into two particles); (iii) soliton behavior of some of the obtained waves; (iv) existence of solitons which move with the same velocity despite the different shape and amplitude of the solitons. [ABSTRACT FROM AUTHOR]

Generalizing tensor-product splines to smooth functions whose control nets can outline more general topological polyhedra, bi-cubic polyhedral-net splines form a first-order differentiable piecewise polynomial space. Each polyhedral control net node is associated with one bi-cubic function. A polyhedral control net admits grid-, star-, n -gon-, polar- and three types of T-junction configurations. Analogous to tensor-product splines, polyhedral-net splines can both model curved geometry and represent higher-order functions on the geometry. This paper explores the use of polyhedral-net splines for solving elliptic partial differential equations on curved smooth free-form surfaces without additional meshing. [ABSTRACT FROM AUTHOR]

Recently, the localized oscillatory radial basis functions collocation method (L-ORBFs) has been introduced to solve elliptic partial differential equations in 2D with a large number of computational nodes. The research clearly shows that the L-ORBFs is very convenient and useful for solving large-scale problems, but this method is numerically less accurate. In this paper, we propose a numerical scheme to improve the accuracy of the L-ORBFs by adding low-degree polynomials in the localized collocation process. The numerical results validate that the proposed numerical scheme is highly accurate and clearly outperforms the results of the L-ORBFs. [ABSTRACT FROM AUTHOR]

We consider a stochastic production planning problem with regime switching. There are k ≥ 1 regimes corresponding to different economic cycles. The problem is to minimize the production costs and analyze the problem by the value function approach. Our main contribution is to show that the optimal production is characterized by an exact solution of an elliptic system of partial differential equations. A verification result is given for the determined solution. [ABSTRACT FROM AUTHOR]

This article, titled "On the Estimate of Integral Norm of Solution of the Dirichlet Problem for Quasilinear Elliptic Equation," explores the estimation of the integral norm of the solution to the Dirichlet problem for a quasilinear elliptic equation. The author investigates the localization effect and the presence of dead cores in the solution. The article introduces a theorem that offers an estimate for the integral norm of the solution based on specific conditions. The proof of the theorem involves the use of inequalities and Young's inequality. The text is a mathematical article that discusses estimates and convergence of solutions to a family of problems, providing theorems, equations, and references to support its claims. The article acknowledges the contributions of anonymous referees and references other relevant literature in the field. [Extracted from the article]

Elliptic boundary value problems (BVPs) are widely used in various scientific and engineering disciplines that involve finding solutions to elliptic partial differential equations subject to certain boundary conditions. This article introduces a novel approach for solving elliptic BVPs using an artificial neural network (ANN)-based radial basis function (RBF) collocation method. In this study, the backpropagation neural network is employed, enabling learning from training data and enhancing accuracy. The training data consist of given boundary data from exact solutions and the radial distances between exterior fictitious sources and boundary points, which are used to construct RBFs, such as multiquadric and inverse multiquadric RBFs. The distinctive feature of this approach is that it avoids the discretization of the governing equation of elliptic BVPs. Consequently, the proposed ANN-based RBF collocation method offers simplicity in solving elliptic BVPs with only given boundary data and RBFs. To validate the model, it is applied to solve two- and three-dimensional elliptic BVPs. The results of the study highlight the effectiveness and efficiency of the proposed method, demonstrating its capability to deliver accurate solutions with minimal data input for solving elliptic BVPs while relying solely on given boundary data and RBFs. [ABSTRACT FROM AUTHOR]

Copyright of Gaceta de la Real Sociedad Matematica Espanola is the property of Real Sociedad Matematica Espanola and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

The study of optimal control problems under uncertainty plays an important role in scientific numerical simulations. This class of optimization problems is strongly utilized in engineering, biology and finance. In this paper, a stochastic gradient method is proposed for the numerical resolution of a nonconvex stochastic optimization problem on a Hilbert space. We show that, under suitable assumptions, strong or weak accumulation points of the iterates produced by the method converge almost surely to stationary points of the original optimization problem. Measurability and convergence rates of a stationarity measure are handled, filling a gap for applications to nonconvex infinite dimensional stochastic optimization problems. The method is demonstrated on an optimal control problem constrained by a class of elliptic semilinear partial differential equations (PDEs) under uncertainty. [ABSTRACT FROM AUTHOR]

In recent years, differential equations have become the method of choice to compute multi-loop Feynman integrals. Whenever they can be cast into canonical form, their solution in terms of special functions is straightforward. Recently, progress has been made in understanding the precise canonical form for Feynman integrals involving elliptic polylogarithms. In this article, we make use of an algorithmic approach that proves powerful to find canonical forms for these cases. To illustrate the method, we reproduce several known canonical forms from the literature and present examples where a canonical form is deduced for the first time. Together with this article, we also release an update for INITIAL, a publicly available Mathematica implementation of the algorithm. [ABSTRACT FROM AUTHOR]

This paper suggests reduced-order modeling using the Galerkin proper orthogonal decomposition (POD) to find approximate solutions for parabolic partial differential equations. We first transform a parabolic partial differential equation to the frequency-dependent elliptic equations using the Fourier integral transform in time. Such a frequency-domain method enables efficiently implementing a parallel computation to approximate the solutions because the frequency-variable elliptic equations have independent frequencies. Then, we introduce reduced-order modeling to determine approximate solutions of the frequency-variable elliptic equations quickly. A set of snapshots consists of the finite element solutions of the frequency-variable elliptic equations with some selected frequencies. The solutions are approximated using the general basis of the high-dimensional finite element space in a Hilbert space. reduced-order modeling employs the Galerkin POD for the snapshot subspace spanned by a set of snapshots. An orthonormal basis for the snapshot space can be easily computed using the spectral decomposition of the correlation matrix of the snapshots. Additionally, using an appropriate low-order basis of the snapshot space allows approximating the solutions of the frequency-variable elliptic equations quickly, where the approximate solutions are used for the inverse Fourier transforms to determine the approximated solutions in the time variable. Several numerical tests based on the finite element method are presented to asses the efficient performances of the suggested approaches. [ABSTRACT FROM AUTHOR]

The purpose of this study is to investigate the role that a deep learning approach could play in computational mechanics. In this paper, a convolutional neural network technique based on modified loss function is proposed as a surrogate of the finite element method (FEM). Several surrogate-based physics-informed neural networks (PINNs) are developed to solve representative boundary value problems based on elliptic partial differential equations (PDEs). According to the authors' knowledge, the proposed method has been applied for the first time to solve boundary value problems with elliptic partial differential equations as the governing equations. The results of the proposed surrogate-based approach are in good agreement with those of the conventional FEM. It is found that modification of the loss function could improve the prediction accuracy of the neural network. It is demonstrated that to some extent, the deep learning approach could replace the conventional numerical method as a significant surrogate model. [ABSTRACT FROM AUTHOR]

We investigate the half-space Dirichlet problem with summable boundary-value functions for an elliptic equation with an arbitrary amount of potentials undergoing translations in arbitrary directions. In the classical case of partial differential equations, the half-space Dirichlet problem for elliptic equations attracts great interest from researchers due to the following phenomenon: the solutions acquire qualitative properties specific for nonstationary (more exactly, parabolic) equations. In this paper, such a phenomenon is studied for nonlocal generalizations of elliptic differential equations, more exactly, for elliptic differential-difference equations with nonlocal potentials arising in various applications not covered by the classical theory. We find a Poisson-like kernel such that its convolution with the boundary-value function satisfies the investigated problem, prove that the constructed solution is infinitely smooth outside the boundary hyperplane, and prove its uniform power-like decay as the timelike independent variable tends to infinity. [ABSTRACT FROM AUTHOR]

Partial differential equations are used to model fluid flow in porous media. Neural networks can act as equation solution approximators by basing their forecasts on training samples of permeability maps and their corresponding two-point flux approximation solutions. This paper illustrates how convolutional neural networks of various architecture, depth and parameter configurations manage to forecast solutions of the Darcy’s flow equation for various domain sizes. [ABSTRACT FROM AUTHOR]

The Helmholtz equation as an elliptic partial differential equation possesses many applications in the time-harmonic wave propagation phenomena, such as the acoustic cavity and radiation wave. In this paper, we establish a numerical method based on the orthonormal shifted discrete Chebyshev polynomials for finding complex solution of this equation. The presented method transforms the Helmholtz equation into an algebraic system of equations that can be easily solved. Four practical examples are examined to show the accuracy of the proposed technique. [ABSTRACT FROM AUTHOR]

Author Contributions Conceptualization, X.Y. and R.R.; methodology, X.Y. and R.R.; software, X.Y. and R.R.; validation, X.Y. and R.R.; formal analysis, X.Y. and R.R.; investigation, X.Y. and R.R.; resources, X.Y. and R.R.; data curation, X.Y. and R.R.; writing - original draft preparation, X.Y. and R.R.; writing - review and editing, X.Y. and R.R.; visualization, X.Y. and R.R.; supervision, X.Y. and R.R.; project administration, X.Y. and R.R.; funding acquisition, X.Y. and R.R. All authors have read and agreed to the published version of the manuscript. Well-posedness, dynamics, and control of nonlinear differential system with an initial-boundary value involve many mathematical, physical, and engineering problems. So, in [[1]], Xinggui Li, Xinsong Yang and Ruofeng Rao proposed the boundedness assumptions on the variables. Analytical Theory We may begin with a research paper by Xinggui Li and the Guest Editors Xinsong Yang and Ruofeng Rao [[1]]: "Impulsive Stabilization on Hyper-Chaotic Financial System under Neumann Boundary". [Extracted from the article]

We identify two families of ten-point Feynman diagrams that generalize the elliptic double box, and show that they can be expressed in terms of the same class of elliptic multiple polylogarithms to all loop orders. Interestingly, one of these families can also be written as a dlog form. For both families of diagrams, we provide new 2ℓ-fold integral representations that are linearly reducible in all but one variable and that make the above properties manifest. We illustrate the simplicity of this integral representation by directly integrating the three-loop representative of both families of diagrams. These families also satisfy a pair of second-order differential equations, making them ideal examples on which to develop bootstrap techniques involving elliptic symbol letters at high loop orders. [ABSTRACT FROM AUTHOR]

We present a loop-by-loop method for computing the differential equations of Feynman integrals using the recently developed dual form formalism. We give explicit prescriptions for the loop-by-loop fibration of multi-loop dual forms. Then, we test our formalism on a simple, but non-trivial, example: the two-loop three-mass elliptic sunrise family of integrals. We obtain an ε-form differential equation within the correct function space in a sequence of relatively simple algebraic steps. In particular, none of these steps relies on the analysis of q-series. Then, we discuss interesting properties satisfied by our dual basis as well as its simple relation to the known ε-form basis of Feynman integrands. The underlying K3-geometry of the three-loop four-mass sunrise integral is also discussed. Finally, we speculate on how to construct a "good" loop-by-loop basis at three-loop. [ABSTRACT FROM AUTHOR]

• Mathematical model to describe drug delivery to the posterior segment of the eye. • Efficacy of intravitreal (IVI), suprachoroidal (SCS) and subretinal (SRS) injections enhanced by iontophoresis. • Protocols based on the use of a series of electric pulses at fixed times, can lead to an optimized SRS or SCS release. • In case of choroidal, retinal or vitreous diseases, SRS injection is the right option. Drug and gene delivery to the eye, namely to the posterior segment of the eye, is one of the most challenging problems for ophthalmologists and pharmacologists. The reason lies in the fact that the eye is protected by multiple barriers that prevent the permeation of xenobiotics and consequently prevent drugs from permeating ocular tissues. Intravitreal injection (IVI), a procedure that releases the drug directly into the vitreous, has become the gold standard for the treatment of posterior diseases. However, due to its invasiveness, several complications can occur. Medical and pharmaceutical researchers are continuously looking for new compounds, new access routes and new ways of enhancing the release. In recent years, two less invasive routes, subretinal space (SRS) and suprachoroidal space (SCS), have received considerable attention, to target the posterior segment of the eye. The aim of the paper is to present a mathematical model that simulates the coupling of an electric field with SRS or SCS injections, and to show how drug distribution compares with the gold standard, IVI. The model is represented by coupled systems of parabolic and elliptic partial differential equations. The ocular barriers are described with a certain detail. A priori estimates establish qualitative properties of the total mass released. Numerical simulations, in different scenarios, illustrate the comparative behaviour of the three treatments for short and long times. The evolution of mean drug concentration in the different ocular tissues, with and without electrical assistance, is studied. [ABSTRACT FROM AUTHOR]

Given input–output pairs of an elliptic partial differential equation (PDE) in three dimensions, we derive the first theoretically rigorous scheme for learning the associated Green's function G. By exploiting the hierarchical low-rank structure of G, we show that one can construct an approximant to G that converges almost surely and achieves a relative error of O (Γ ϵ - 1 / 2 log 3 (1 / ϵ) ϵ) using at most O (ϵ - 6 log 4 (1 / ϵ)) input–output training pairs with high probability, for any 0 < ϵ < 1 . The quantity 0 < Γ ϵ ≤ 1 characterizes the quality of the training dataset. Along the way, we extend the randomized singular value decomposition algorithm for learning matrices to Hilbert–Schmidt operators and characterize the quality of covariance kernels for PDE learning. [ABSTRACT FROM AUTHOR]

For the purpose of solving elliptic partial differential equations, we suggest a new approach using an h-adaptive local discontinuous Galerkin approximation based on Sinc points. The adaptive approach, which uses Poly-Sinc interpolation to achieve a predetermined level of approximation accuracy, is a local discontinuous Galerkin method. We developed an a priori error estimate and demonstrated the exponential convergence of the Poly-Sinc-based discontinuous Galerkin technique, as well as the adaptive piecewise Poly-Sinc method, for function approximation and ordinary differential equations. In this paper, we demonstrate the exponential convergence in the number of iterations of the a priori error estimate derived for the local discontinuous Galerkin technique under the condition that a reliable estimate of the precise solution of the partial differential equation at the Sinc points exists. For the purpose of refining the computational domain, we employ a statistical strategy. The numerical results for elliptic PDEs with Dirichlet and mixed Neumann-Dirichlet boundary conditions are demonstrated to validate the adaptive greedy Poly-Sinc approach. [ABSTRACT FROM AUTHOR]

In this paper, we consider a polynomial-augmented radial basis function collocation method (RBFCM) for solving fourth-order multi-dimensional elliptic partial differential equations (PDEs). Instead of choosing the RBF centres inside the problem domain, we use fictitious centres that can be located outside it to deal with the double boundary conditions of the fourth-order PDEs, and to also improve the computational accuracy. Furthermore, additional polynomial constraints applied to the approximation matrix of the RBFCM using fictitious centres, significantly alleviate the sensitivity of the method to the variation of the shape parameter, whereby it makes the RBFCM more practical for solving partial differential equations. We also apply the proposed method to a fourth-order elliptic PDE with random forcing terms (input data of the model). The input data are assumed to depend on a finite number of random variables at the collocation points. Numerical experiments are presented to demonstrate the effectiveness of the proposed method in terms of the selection of the shape parameter. In addition, the numerical results show that the proposed method achieves a higher accuracy than the conventional RBFCM. [ABSTRACT FROM AUTHOR]

Building on previous research that emphasized the importance of focusing on resonance frequencies for a fundamental understanding of the thrust and complex internal flow phenomena of the SparkJet actuators, this study theoretically derives analytic formulae for several important resonance frequencies. The research addresses the typical configuration of the SparkJet actuators, which consists of a cylindrical cavity and orifice connected by a conical converging nozzle. While the resonance frequencies of the SparkJet actuators can be obtained by solving the eigenvalue problem presented in previous studies, this eigenvalue problem, despite being a typical boundary value problem in the form of an elliptic partial differential equation, is challenging to solve using conventional numerical methods such as iterative methods, because the eigenvalue is included as an unknown in the operator. Consequently, Boundary Element Method (BEM) or methods using the Green functions have been proposed to obtain numerical solutions, but these still require handling large matrix data, resulting in significant computational costs and memory consumption. To overcome this, the current study first simplifies the eigenvalue problem using the conformal mapping presented in previous research. Then, referencing prior studies that claim that three types of resonance frequencies (Helmholtz resonance frequency and resonance frequencies representing streamwise or radial directional oscillation) significantly affect the thrust of the SparkJet actuators, characteristics of these frequencies are mathematically defined. Using these mathematical characteristics, the study derives the asymptotic and approximate solutions of the simplified eigenvalue problem, from which the resonance frequencies are obtained. The analytic formulae proposed in this study theoretically explain the already known geometric tendencies of the resonance frequencies and reveal new geometric factors influencing resonance frequencies, which were previously unknown. This approach is expected to facilitate obtaining several important resonance frequencies of the SparkJet actuator more promptly and accurately and to provide a deeper understanding of the nature of the complex oscillation phenomena inside the actuator. • Analytic formulae for several resonance frequencies of the SparkJet actuator are theoretically derived. • An asymptotic solution for the Helmholtz resonance frequency is derived. • A transcendental equation for the resonance frequencies representing streamwise directional oscillation is de- rived. • A simple analytic expression for the resonance frequencies representing radial directional oscillation is derived. • These analytic formulae can precisely predict the resonance frequency of the SparkJet actuator. [ABSTRACT FROM AUTHOR]

We evaluate the maxcut of the two loops sunrise amplitude with three different masses by direct use of the loop momenta in the Minkoskean (as opposed to the usual Euclidean) continuous dimension regularisation, obtaining in that way six related but different functions expressed in the form of one dimensional finite integrals. We then consider the 4th order homogeneous equation valid for the maxcut, and show that for arbitrary dimension d the six functions do satisfy the equation separately. We further discuss the d = 2 , 3 , 4 cases, verifying that only four of them are linearly independent. The equal mass limit is also shortly considered. [ABSTRACT FROM AUTHOR]

We combine concepts from multilevel solvers for partial differential equations (PDEs) with neural network based deep learning and propose a new methodology for the efficient numerical solution of high-dimensional parametric PDEs. An in-depth theoretical analysis shows that the proposed architecture is able to approximate multigrid V-cycles to arbitrary precision with the number of weights only depending logarithmically on the resolution of the finest mesh. As a consequence, approximation bounds for the solution of parametric PDEs by neural networks that are independent on the (stochastic) parameter dimension can be derived. The performance of the proposed method is illustrated on high-dimensional parametric linear elliptic PDEs that are common benchmark problems in uncertainty quantification. We find substantial improvements over state-of-the-art deep learning-based solvers. As particularly challenging examples, random conductivity with high-dimensional non-affine Gaussian fields in 100 parameter dimensions and a random cookie problem are examined. Due to the multilevel structure of our method, the amount of training samples can be reduced on finer levels, hence significantly lowering the generation time for training data and the training time of our method. [ABSTRACT FROM AUTHOR]

In this paper, we consider the following concave-convex semilinear elliptic system with double critical exponents: ∫-Δu = |u|2∗-2u + α/2* |u|α-2|v|βu + λ|u|q-2u, in Ω, -Δv = |v|2*-2v + β/2* |u|α|v|β-2v + μ|v|q-2v, in Ω, u, v > 0, in Ω, u = v = 0, on ∂Ω, where Ω ⊂ RN(N ≥ 3) is a bounded domain with smooth boundary, λ, μ > 0, 1 < q < 2, α > 1, β > 1, α + β = 2* = 2N/N-2. By the Nehari manifold method and variational method, we obtain two positive solutions which improves the recent results in the literature. [ABSTRACT FROM AUTHOR]

In this paper, an efficient and straightforward numerical procedure is constructed to solve multi-dimensional linear and nonlinear elliptic partial differential equations (PDEs). Although the numerical procedure for the constraint-type fictitious time integration method overcomes the numerical stability problem, the parameter's definition, numerical accuracy and computational efficiency have not been resolved, and the lack of initial guess values results in reduced computational efficiency. Therefore, the normalized two-point boundary value solution of the Lie-group shooting method is proposed and considered in the numerical procedure to avoid the problem of the initial guess value. Then, a space-time variable, including the minimal fictitious time step and convergence rate factor, is introduced to study the relationship between the initial guess value and convergence rate factor. Some benchmark numerical examples are tested. As the results show, this numerical procedure using the normalized boundary value solution can significantly converge within one step, and the numerical accuracy is better than that demonstrated in the previous literature. [ABSTRACT FROM AUTHOR]

Optimal control problems are a class of optimisation problems with partial differential equations as constraints. These problems arise in many application areas of science and engineering. The finite element method was used to transform the optimal control problems of an elliptic partial differential equation into a system of linear equations of saddle point form. The main focus of this paper is to characterise and exploit the structure of the coeficient matrix of the saddle point system to build an eficient numerical process. These systems are of large dimension, block, sparse, indefinite and ill conditioned. The numerical solution of saddle point problems is a computational task since well known numerical schemes perform poorly if they are not properly preconditioned. The main task of this paper is to construct a preconditioner the mimic the structure of the system coeficient matrix to accelerate the convergence of the generalised minimal residual method. Explicit expression of the eigenvalue and eigenvectors for the preconditioned matrix are derived. The main outcome is to achieve optimal convergence results in a small number of iterations with respect to the decreasing mesh size h and the changes in fi the regularisation problem parameters. The numerical results demonstrate the effectiveness and performance of the proposed preconditioner compared to the other existing preconditioners and confirm theoretical results. [ABSTRACT FROM AUTHOR]

Here and hereafter, we identify HT ht with a 1-form (not relabelled). Thus HT ht is constant. Nomenclature Throughout HT ht is a simply-connected bounded domain with outward unit normal vectorfield HT ht . Therefore, by Eq. (3), HT ht equals zero. [Extracted from the article]

Let Σ be a C³ compact symmetric convex hypersurface in ℝ8. We prove that when Σ carries exactly four geometrically distinct closed characteristics, then there are at least two irrationally elliptic closed characteristics on Σ. [ABSTRACT FROM AUTHOR]

In this paper, based on the combination of the Jacobi elliptic equation and the concept of the simple equation method, we introduce a new approach for solving fractional partial differential equations, where the fractional derivative is defined in the sense of the conformable fractional derivative. By use of a nonlinear transformation, the proved chain rule and the properties of fractional calculus, certain fractional partial differential equation can be converted into another ordinary differential equation of integer order. With general solutions of the Jacobi elliptic equation, a series of exact solutions for the ordinary differential equation can be obtained subsequently based on the homogeneous balance principle and with the aid of mathematical software. As for applications of this approach, we apply it to seek exact solutions for the space fractional (2+1)- dimensional breaking soliton equations and the space-time fractional BBM equation. As a result, abundant solitary wave solutions, periodic wave solutions, rational function solutions and Jacobi elliptic function solutions are successfully found. [ABSTRACT FROM AUTHOR]

This paper develops a two-scale asymptotic homogenization method for periodic composite Kirchhoff plates. In this method, a three-dimensional (3D) periodic plate problem is simplified as a Kirchhoff plate problem, which is governed by a fourth-order uniformly elliptic partial differential equation (PDE) with periodically oscillating coefficients. Then, a two-scale solution in an asymptotic expansion form is presented for the PDE, and it is found that the first-order perturbed displacement in the asymptotic solution is zero. Additionally, periodic boundary and normalization constraint conditions are proposed to determine the unique solution to unit cell problems. Moreover, standard finite element formulations for calculating the perturbed displacements are derived from the principle of virtual work. Physical interpretations of the influence functions are presented by analyzing the properties of self-balanced quasi-load vectors used for solving the influence functions. Numerical comparisons show that the present method is physically acceptable and highly accurate. [ABSTRACT FROM AUTHOR]