Second-order Partial Differential Equations Research Articles

A fuzzy based computational model to analyze the influence of mitochondria, buffer, and ER fluxes on cytosolic calcium distribution in neuron cells.

A free calcium ion in the cytosol is essential for many physiological and physical functions. Also, it is known as a second messenger as the quantity of free calcium ions is an essential part of brain signaling. In this work, we have attempted to study calcium signaling in the presence of mitochondria, buffer, and endoplasmic reticulum fluxes. Small organelles called mitochondria are found in the nervous system and are involved in several cellular functions, including energy production, response to stress, calcium homeostasis regulation, and pathways leading to cell death. It has been discovered that buffer, endoplasmic reticulum, and mitochondria significantly affect calcium signaling. To investigate how various circumstances impact the quantity of calcium in the cytosol, a mathematical model ofa second-order linear partial differential equation with fuzzy boundary conditionshas been developed. Systems having ambiguous or imprecise boundaryvalues can be effectively modeled and simulated with the help of fuzzy boundary conditions. Models can provide more dependable and instructive outcomes and become adaptable to real-world circumstances by implementing fuzzy logic into boundary conditions. In this paper, we observed the Fuzzy Laplace Transform to solve variable coefficient fuzzy differential equations using triangular fuzzy numbers. It isnoted that maintaining the delicate calcium ion balance, which controls essential cellular functions, depends on the buffer affinity. Also, neurodegenerative illnesses like Alzheimer's, Parkinson's, etc. are linked to disruptions in the control of components such as buffers, mitochondria, and the endoplasmic reticulum.

PARALLEL PERFORMANCE OF THE EXPLICIT FDTD METHODS APPLIED ON 3D ACOUSTIC WAVE EQUATION

The 3D acoustic wave equation is a second-order linear hyperbolic partial differential equation. It is widely studied in acoustics, fluid dynamics, electromagnetism, and seismology within both science and engineering fields. Among the various numerical methods, the finite difference method is considered to be the simpler to understand and easy to implement. Based on the discretize schemes finite difference time domain method described by the explicit scheme in which spatial second order derivatives are evaluated at the previous time step. The solution of 3D acoustic wave equation may be required on the high-resolution mesh points consequently more computational resources are required. In this study parallel algorithm for the numerical solution of 3D acoustic wave equation is proposed and designed by using data parallel approach and message passing schemes. The algorithm is implemented on shared memory parallel system using MATLAB parallel computing mode. The parallel performance of the designed algorithm is analyzed on different mesh sizes and time steps. It is revealed that the proposed algorithm may reduce computational time up to 3 times as compared to sequential solution algorithm. The proposed parallel algorithm remains more efficient on workers while for the efficiency of the algorithm drops because of the high communication time. The results of the proposed research could be utilized in the large-scale simulations of 3D acoustic wave equations and may enable to simulate the acoustic wave pressure at more refined meshes on high performance computing systems.

A mathematical foundation for QUMOND

We link the QUMOND theory with the Helmholtz–Weyl decomposition and introduce a new formula for the gradient of the Mondian potential using singular integral operators. This approach allows us to demonstrate that, under very general assumptions on the mass distribution, the Mondian potential is well-defined, once weakly differentiable, with its gradient given through the Helmholtz–Weyl decomposition. Furthermore, we establish that the gradient of the Mondian potential is an Lp vector field. These findings lay the foundation for a rigorous mathematical analysis of various issues within the realm of QUMOND. Further, we prove that the once weakly differentiable Mondian potential solves a second-order partial differential equation in distribution sense. Thus, the question arises whether the potential has second-order derivatives. We affirmatively answer this question in the situation of spherical symmetry, although our investigation reveals that the regularity of the second derivatives is weaker than anticipated. We doubt that a similarly general regularity result can be proven without symmetry assumptions. In conclusion, we explore the implications of our results for numerous problems within the domain of QUMOND.

SOLVABILITY OF NONLINEAR BOUNDARY VALUE PROBLEMS FOR DIFFERENTIAL EQUILIBRIUM EQUATIONS OF NON-FLAT TIMOSHENKO TYPE SHELLS OF NON-ZERO GAUSSIAN CURVATURE IN ISOMETRIC COORDINATES

We study the solvability of a boundary value problem for a system of five nonlinear second-order partial differential equations under given nonlinear boundary conditions, which describes the equilibrium state of elastic non-flat inhomogeneous isotropic shells with loose edges in the framework of the Timoshenko shear model, referred to isometric coordinates.The boundary value problem is reduced to a nonlinear operator equation with respect to generalized displacements in Sobolev space, the solvability of which is established using the principle of compressed mappings.

Proximal Galerkin: A Structure-Preserving Finite Element Method for Pointwise Bound Constraints

AbstractThe proximal Galerkin finite element method is a high-order, low iteration complexity, nonlinear numerical method that preserves the geometric and algebraic structure of pointwise bound constraints in infinite-dimensional function spaces. This paper introduces the proximal Galerkin method and applies it to solve free boundary problems, enforce discrete maximum principles, and develop a scalable, mesh-independent algorithm for optimal design with pointwise bound constraints. This paper also introduces the latent variable proximal point (LVPP) algorithm, from which the proximal Galerkin method derives. When analyzing the classical obstacle problem, we discover that the underlying variational inequality can be replaced by a sequence of second-order partial differential equations (PDEs) that are readily discretized and solved with, e.g., the proximal Galerkin method. Throughout this work, we arrive at several contributions that may be of independent interest. These include (1) a semilinear PDE we refer to as the entropic Poisson equation; (2) an algebraic/geometric connection between high-order positivity-preserving discretizations and certain infinite-dimensional Lie groups; and (3) a gradient-based, bound-preserving algorithm for two-field, density-based topology optimization. The complete proximal Galerkin methodology combines ideas from nonlinear programming, functional analysis, tropical algebra, and differential geometry and can potentially lead to new synergies among these areas as well as within variational and numerical analysis. Open-source implementations of our methods accompany this work to facilitate reproduction and broader adoption.

Linear fuzzy partial differential equations for A-linearly correlated fuzzy processes

In this paper, we deal with linear fuzzy partial differential equations (FPDEs) whose solutions correspond to A-linearly correlated fuzzy processes. More precisely, we investigate general second-order linear fuzzy partial differential equations and provide solutions for two essential cases: the fuzzy advection equation and the fuzzy wave-like equation with fuzzy velocity terms. One of the advantages of our approach is that these FPDEs can be converted into a classical system of partial differential equations (PDEs). Thus, any suitable and appropriate method to solve PDEs can be applied to solve this classical system and, eventually leading to the derivation of a fuzzy solution. Finally, we provide some examples to demonstrate our results.

Multi-level physics informed deep learning for solving partial differential equations in computational structural mechanics

Physics-informed neural network has emerged as a promising approach for solving partial differential equations. However, it is still a challenge for the computation of structural mechanics problems since it involves solving higher-order partial differential equations as the governing equations are fourth-order nonlinear equations. Here we develop a multi-level physics-informed neural network framework where an aggregation model is developed by combining multiple neural networks, with each one involving only first-order or second-order partial differential equations representing different physics information such as geometrical, constitutive, and equilibrium relations of the structure. The proposed framework demonstrates a remarkable advancement over the classical neural networks in terms of the accuracy and computation time. The proposed method holds the potential to become a promising paradigm for structural mechanics computation and facilitate the intelligent computation of digital twin systems.

Longitudinal wave propagation in FG rods under impact force

A well-known fact is that one-dimensional wave analysis is the theoretical basis of the famous Hopkinson bar dynamic testing technique. The current one-dimensional wave theory is mostly confined to the slender rods of isotropic materials. It is not easy to obtain an analytical solution to the wave equation of an anisotropic rod. In this work, rods with both elastic modulus and density graded in the length direction are presented and analyzed. The one-dimensional variable coefficient wave equation corresponding to the functionally graded rod is constructed and converted into a second-order variable coefficient partial differential equation using the Laplace approach. Then, the details of solving the partial differential equation of the second-order variable coefficients are given. It is worth noting that here we construct a variable coefficient equation that satisfies the form of Euler's equation. It is still difficult to obtain analytical solutions for other equations that do not satisfy this form. Subsequently, studies of the wave propagation characteristics of rods with different graded configurations are carried out. The theoretical results show that the wave propagation behavior and post-impact vibration of the rod are significantly influenced by the graded configuration. It is possible to adjust not only the impact response at the end but also the impact response in the middle of the rod by optimizing the design of the rod's graded configuration.

Kolmogorov equations on the space of probability measures associated to the nonlinear filtering equation: the viscosity approach

. We study the backward Kolmogorov equation on the space of probability measures associated to the Kushner-Stratonovich equation of nonlinear filtering. We prove existence and uniqueness in the viscosity sense and, in particular, we provide a comparison theorem. In the context of stochastic filtering, it is natural to consider measure-valued processes that satisfy stochastic differential equations. In the literature, a classical way to address this problem is by assuming that these measure-valued processes admit a density. Our approach is different and we work directly with measures. Thus, the backward Kolmogorov equation we study is a second-order partial differential equation of parabolic type on the space of probability measures with compact support. In the literature, only few results are available on viscosity solutions for this kind of problems and in particular the uniqueness is a very challenging issue. Here we find a viscosity solution and then we prove that it is unique via comparison theorem.

INTEGRAL DESCRIPTION AND SOLUTION OF THE TILE TYPE PROBLEM FOR ONE OF THE FOURTH ORDER DIFFERENTIAL EQUATIONS OF MIXED GENERATION

Partial differential equations were developed mainly at the end and beginning of the twentieth century in connection with the solution of physical problems. Many domestic and foreign scientists have been studying this scientific direction. Second-order partial differential equations have been widely studied, but higher-order partial differential equations have been studied relatively little. Higher order differential equations of mixed type are comparatively less studied. From this point of view, the study and investigation of differential equations of mixed type of higher order is relevant. In this article, for the first time, the solution to a degenerate fourth-order mixed type differential equation is expressed using solutions to two second-order differential equations. After this, for degenerate differential equations of mixed type (1) of the fourth order, the integral representation of the solution is determined explicitly using arbitrary functions. Some found integral representations of the solution are used to solve Cauchy type problems in the characteristic domain. The solution to the Cauchy type problem is found in explicit form.

Bending analysis of thin plates with variable stiffness resting on elastic foundation via a two-network strategy physics-informed neural network method

A physics-informed neural network method based on a two-network strategy is introduced to address the bending problem of thin plates with variable stiffness resting on an elastic foundation. This problem is governed by a fourth-order partial differential equation (PDE), and the use of a one-network strategy to solve the PDE directly may lead to convergence issues due to singular points. Following the principles of Kirchhoff plate theory, the governing PDE is equivalently transformed into four second-order PDEs. A two-network strategy is employed for solution. We present numerical examples under various load conditions, plate geometries, foundation types, thicknesses, and material properties. The obtained results are validated against finite element method (FEM) solutions and literature. Discussions on alternative network strategies were conducted, and this study reveals that the presented two-network strategy have proven to be effective and robust. It also facilitates easier imposition of boundary conditions.

The role of conjugacy in the dynamics of time of arrival operators

The construction of time of arrival (TOA) operators canonically conjugate to the system Hamiltonian entails finding the solution of a specific second-order partial differential equation called the time kernel equation (TKE). In this paper, we provide an exact analytic solution of the TKE for a special class of potentials satisfying a specific separability condition. The solution enables us to investigate the time evolution of the eigenfunctions of the conjugacy-preserving TOA operators (CPTOA) and show that they exhibit unitary arrival at the intended arrival point at a time equal to their corresponding eigenvalues. We also compare the dynamics between the TOA operators constructed by quantization and those independent of quantization for specific interaction potentials. We find that the CPTOA operator possesses smoother and sharper unitary dynamics over the Weyl-quantized one within numerical accuracy.

Inverse Coefficient Problem for the 2D Wave Equation with Initial and Nonlocal Boundary Conditions

In this paper, we consider direct and inverse problems for the two-dimensional wave equation. The direct problem is an initial boundary value problem for this equation with nonlocal boundary conditions. In the inverse problem, it is required to find the time-variable coefficient at the lower term of the equation. The classical solution of the direct problem is presented in the form of a biorthogonal series in eigenvalues and associated functions, and the uniqueness and stability of this solution are proven. For solution to the inverse problem, theorems of existence in local, uniqueness in global, and an estimate of conditional stability are obtained. The problems of determining the right-hand sides and variable coefficients at the lower terms from initial boundary value problems for second-order linear partial differential equations with local boundary conditions have been studied by many authors. Since the nonlinearity is convolutional, the unique solvability theorems in them are proven in a global sense. In the works, the method of separation of variables is used to find the classical solution of the direct problem in the form of a biorthogonal series in terms of eigenfunctions and associated functions. The nonlocal integral condition is used as the overdetermination condition with respect to the solution of the direct problem. The direct problem reduces to equivalent integral equations of the Fourier method. To establish integral inequalities, the generalized Gronwall-Bellman inequality is used. We obtain an a priori estimate of the solution in terms of an unknown coefficient, that are useful for studying the inverse problem.

Anisotropic functionally graded nano-beam models and closed-form solutions in plane gradient elasticity

This study delves into the investigation of exact analytical solutions for the plane stress and displacement fields within linear homogeneous anisotropic nano-beam models of gradient elasticity. It focuses on solving the Helmholtz equation, which encompasses a second-order non-homogeneous linear partial differential equation in plane gradient elasticity theory, utilizing polynomial series-type solutions. The analysis centers on the utilization of gradient Airy stress functions to derive stress fields in the gradient theory. The research yields closed-form analytical solutions for Airy stress functions, stress, and strain fields in both classical and gradient theories. The study considers five distinct types of two-dimensional functionally graded cantilever beams with various boundary conditions: a cantilever anisotropic nano-beam subjected to a concentrated force at the free end, a cantilever anisotropic nano-beam under a uniform load, a simple anisotropic nano-beam under a uniform load, a propped cantilever anisotropic nano-beam under a uniform load, and a fixed end cantilever anisotropic nano-beam under a uniform load. General analytical solutions for the gradient stress and displacement fields of two-dimensional and one-dimensional anisotropic nano-beams under different boundary conditions are provided. The study showcases significant strain gradient size effects at the nano-scale through the derived analytical solutions for anisotropic beams. Additionally, it demonstrates that the strain gradient theory results for the limit case of the gradient coefficient c precisely align with results for isotropic and anisotropic materials in elasticity theory and classical theory. Furthermore, real-world applications are discussed, considering stress and displacement fields in real anisotropic materials such as TiSi2 single crystals and orthotropic materials, which are special cases of anisotropic materials like wood and epoxy as documented in the literature.

Numerical analysis of a nonsmooth quasilinear elliptic control problem: I. Explicit second-order optimality conditions

In this paper, we derive explicit second-order necessary and sufficient optimality conditions of a local minimizer to an optimal control problem for a quasilinear second-order partial differential equation with a piecewise smooth but not differentiable nonlinearity in the leading term. The key argument rests on the analysis of level sets of the state. Specifically, we show that if a function vanishes on the boundary and its the gradient is different from zero on a level set, then this set decomposes into finitely many closed simple curves. Moreover, the level sets depend continuously on the functions defining these sets. We also prove the continuity of the integrals on the level sets. In particular, Green’s first identity is shown to be applicable on an open set determined by two functions with nonvanishing gradients. In the second part to this paper, the explicit sufficient second-order conditions will be used to derive error estimates for a finite-element discretization of the control problem.

An efficient localized Trefftz method for the simulation of two-dimensional sloshing behaviors

This study proposes a new method for effectively and accurately simulating sloshing in a two-dimensional numerical tank. This technique uses the localized Trefftz method (LTM), a meshless numerical approach. Sloshing in a numerical tank is mathematically described as a space-time boundary-value problem rooted in potential flow theory. This occurrence is governed by a second-order partial differential equation and two nonlinear free-surface boundary conditions. In this study, the moving boundary problem undergoes discretization in time and space by using the LTM and the explicit Euler method, respectively. After discretization with the explicit Euler method, the elevation of the free surface is updated, and a boundary-value problem is generated at each time step. This boundary-value problem can be effectively examined using the recently developed LTM, which eliminates the need for complex meshing. Contrary to conventional methods, such as the method of fundamental solutions and the boundary element method, the LTM generates sparse matrices, allowing large-scale numerical simulations. Four numerical examples are presented to demonstrate the clarity and accuracy of the proposed meshless approach.

Robust domain decomposition methods for high-contrast multiscale problems on irregular domains with virtual element discretizations

Our research focuses on the development of domain decomposition preconditioners tailored for second-order elliptic partial differential equations. Our approach addresses two major challenges simultaneously: i) effectively handling coefficients with high-contrast and multiscale properties, and ii) accommodating irregular domains in the original problem, the coarse mesh, and the subdomain partition. The robustness of our preconditioners is crucial for real-world applications, such as the efficient and accurate modeling of subsurface flow in porous media and other important domains.The core of our method lies in the construction of a suitable partition of unity functions and coarse spaces utilizing local spectral information. Leveraging these components, we implement a two-level additive Schwarz preconditioner. We demonstrate that the condition number of the preconditioned systems is bounded with a bound that is independent of the contrast. Our claims are further substantiated through selected numerical experiments, which confirm the robustness of our preconditioners.

Hutchinson Trace Estimation for high-dimensional and high-order Physics-Informed Neural Networks

Physics-Informed Neural Networks (PINNs) have proven effective in solving partial differential equations (PDEs), especially when some data are available by seamlessly blending data and physics. However, extending PINNs to high-dimensional and even high-order PDEs encounters significant challenges due to the computational cost associated with automatic differentiation in the residual loss function calculation. Herein, we address the limitations of PINNs in handling high-dimensional and high-order PDEs by introducing the Hutchinson Trace Estimation (HTE) method. Starting with the second-order high-dimensional PDEs, which are ubiquitous in scientific computing, HTE is applied to transform the calculation of the entire Hessian matrix into a Hessian vector product (HVP). This approach not only alleviates the computational bottleneck via Taylor-mode automatic differentiation but also significantly reduces memory consumption from the Hessian matrix to an HVP’s scalar output. We further showcase HTE’s convergence to the original PINN loss and its unbiased behavior under specific conditions. Comparisons with the Stochastic Dimension Gradient Descent (SDGD) highlight the distinct advantages of HTE, particularly in scenarios with significant variability and variance among dimensions. We further extend the application of HTE to higher-order and higher-dimensional PDEs, specifically addressing the biharmonic equation. By employing tensor-vector products (TVP), HTE efficiently computes the colossal tensor associated with the fourth-order high-dimensional biharmonic equation, saving memory and enabling rapid computation. The effectiveness of HTE is illustrated through experimental setups, demonstrating comparable convergence rates with SDGD under memory and speed constraints. Additionally, HTE proves valuable in accelerating the Gradient-Enhanced PINN (gPINN) version as well as the Biharmonic equation. Overall, HTE opens up a new capability in scientific machine learning for tackling high-order and high-dimensional PDEs.

On the problem of solvability of nonlinear boundary value problems for shallow isotropic shells of Timoshenko type in isometric coordinates

The solvability of a boundary value problem for a system, which describes the equilibrium state of elastic shallow inhomogeneous isotropic shells with loose edges referred to isometric coordinates in the Timoshenko shear model and consists of five non-linear second-order partial differential equations under given non-linear boundary conditions, is studied. The boundary value problem is reduced to a nonlinear operator equation for generalized displacements in Sobolev space, the solvability of this equation is established with the help of the contraction mapping principle.

A second-order phase field-lattice Boltzmann model with equation of state inputting for two-phase flow containing soluble surfactants

In this paper, we propose a phase field-lattice Boltzmann (LB) model with an equation of state (EOS) inputting for two-phase flow containing soluble surfactants. In this model, both the order parameter for the phase field and the surfactant concentration are described by second-order partial differential equations, along with Navier–Stokes equations for the flow field. Changes in surfactant concentration do not affect the order parameter distribution; hence, an unwanted sharpening effect cannot arise. Most importantly, in the existing models, the EOS of surface tension is determined by posterior simulation tests instead of being directly set as an input parameter before the simulations. Hence, it is difficult to determine the model parameters in practical applications. To address this issue, we systematically develop a fully analytical EOS for surface tension based on the Gibbs–Duhem equation. Subsequently, an approximate explicit form for EOS is provided by utilizing the Jacobi–Gauss quadrature rule. Furthermore, a multiple-relaxation-time LB scheme is utilized to numerically solve the governing equations of three physical fields. Two benchmark examples are simulated to validate the accuracy of the present model. The consistency between the numerical results and the analytical EOS is verified. Moreover, the dynamics of droplets with surfactant in simple shear flow is investigated, unveiling the profound impact of various factors, such as surfactant bulk concentration, capillary number, and viscosity ratio, on single droplet deformation and two equal-sized droplets interaction. A detailed exploration of the fluid mechanism involved in two-phase flow with soluble surfactants is presented.