Functions For Partial Differential Equations Research Articles

A deep learning approach for solving diffusion-induced stress in large-deformed thin film electrodes

Understanding the effects of large deformation on electrochemical performance of electrodes, such as silicon and tin, in lithium-ion battery has stimulated great interest in analyzing stress evolution during electrochemical cycling and in developing mechanochemical models for the stress analysis. As the complexity of mechanochemical models continues to increase, there is an urgent need to develop computational methods to reduce computational costs and increase computational efficiency. In this work, we propose a physics-inspired neural network (PINN) based on the DeepXDE deep learning library in analyzing diffusion-induced stresses (DIS) in a thin film electrode with large deformation, in which a loss function, which is associated with the loss functions of partial differential equations (PDEs) in the domain and the initial/boundary conditions for the mechanochemical problem, is introduced. Using the proposed physics-inspired neural network, the mechanical equations, including geometric, constitutive and equilibrium equations, in the framework of finite deformation theory as well as the mass transport equation are solved to obtain the stress evolution in the large-deformed thin-film electrode. The numerical results are in accord with the results obtained from finite element method. This work provides a unique approach for deep learning to solve the coupling mechanochemical problems in energy storage.

An investigation of global radial basis function collocation methods applied to Helmholtz problems

Global radial basis function (RBF) collocation methods with inifinitely smooth basis functions for partial differential equations (PDEs) work in general geometries, and can have exponential converg ...

Treating Transcendental Functions in Partial Differential Equations Using the Variational Iteration Method with Bernstein Polynomials

We aim through this paper to present an improved variational iteration method (VIM) based on Bernstein polynomials (BP) approximations to be used with transcendental functions. The key benefits gained from this modification are to reach stable and fairly accurate results and, at the same time, to expand the unknown function’s domain in partial differential equations (PDEs). The proposed approach introduces the Bernstein polynomials in the transcendental functions of nonlinear PDEs. A number of examples were included in order to expound the method’s capacity and reliability. From the results, we conclude that the VIM with BP is a powerful mathematical tool that can be applied to solve nonlinear PDEs.

GREEN'S FUNCTIONS OF PARTIAL DIFFERENTIAL EQUATIONS WITH INVOLUTIONS

In this paper we develop a way of obtaining Green's functions of partial differential equations with linear involutions by reducing the equation to a higher-order PDE without involutions. The developed theory is applied to a model of heat transfer in a conducting plate which is bent in half.

Regular functions with values in a noncommutative algebra using Clifford analysis

We construct a noncommutative algebra C(2) that is a subalgebra of the Pauli matrices of M(2;C), and investigate the properties of solutions with values in C(2) of the inhomogeneous Cauchy-Riemann system of partial differential equations with coefficients in the associated Pauli matrices. In addition, we construct a commutative subalgebra C(4) of M(4;C), obtain some properties of biregular functions with values in C(2) on in C2 x C2, define a J-regular function of four complex variables with values in C(4), and examine some properties of J-regular functions of partial differential equations.

Numerical modelling of coupled heat, air and moisture transfer in building envelopes

This paper reports on numerical modeling of heat, air, and moisture transfer through multilayered walls. Building materials are often subjected to temporal climatic variations, which can induce a transfer of heat and moisture through the walls of the building and the foundation soil. These materials are generally considered as porous media. The coupled heat, air and moisture transfer in building materials is of paramount importance in the construction area. In this way, a mathematical model has been elaborated and validated using a benchmark example. Here, we aim to determine the energy losses. The capillary pressure is considered as potential moisture which represents both the transport of vapor and liquid phases of the water. Basing on basic functions of partial differential equations (PDE’s), one can convert certain measurable properties of porous media as coefficients depending on the temperature and the capillary pressure. The results obtained compare favourably with other available in the literature.

Static and Dynamic Problems for Anisotropic Inhomogeneous Shells with Variable Parameters and Their Numerical Solution (Review)

Studies on the static and dynamic deformation of isotropic and anisotropic elastic shell-like bodies of complex shape performed using classical and refined problem statements are reviewed. To solve two-dimensional boundary-value problems and eigenvalue problems, use is made of a nontraditional discrete-continuum approach based on the spline-approximation of the unknown functions of partial differential equations with variable coefficients. This enables reducing the original problem to a system of one-dimensional problems solved with the discrete-orthogonalization method. An analysis is made of numerical results on the distribution of stress and displacement fields and dynamic characteristics depending on the loading and boundary conditions, geometrical and mechanical parameters of elastic bodies. Emphasis is placed on the accuracy of the results

On a theory of nonlocal elasticity of bi-Helmholtz type and some applications

A theory of nonlocal elasticity of bi-Helmholtz type is studied. We employ Eringen’s model of nonlocal elasticity, with bi-Helmholtz type kernels, to study dispersion relations, screw and edge dislocations. The nonlocal kernels are derived analytically as Green functions of partial differential equations of fourth order. This continuum model of nonlocal elasticity involves two material length scales which may be derived from atomistics. The new nonlocal kernels are nonsingular in one-, two- and three-dimensions. Furthermore, the nonlocal elasticity of bi-Helmholtz type improves the one of Helmholtz type by predicting a dispersion relationship with zero group velocity at the end of the first Brillouin zone. New solutions for the stresses and strain energy of screw and edge dislocations are found.

Numerical Methods for the Determination of Material Properties in Soil Science*

This article deals with the determination of nonlinear coefficient functions in partial differential equations in the field of soil science. We consider two examples to illustrate the numerical determination of nonlinear coefficient functions. In detail, these are the determination of the sorption characteristic of a chemical and the determination of the unsaturated hydraulic properties of a porous medium from measurements obtained from suitable column experiments. This inverse problem is treated by minimizing a least square functional. To cope with the ill-posedness, we apply a parametrization of the unknown nonlinear coefficient function, which is defined by an appropriate interpolation. The parametrization does not use a priori assumptions, which are not justified by physical properties. This kind of parametrization permits a hierarchical approach in the number of the degrees of freedom used. According to the hierarchical structure, we integrate the determination of the coefficient functions into a multi-level procedure. The investigation of the stability of the parametrization is based on the singular values of the sensitivity matrix.