Approximation Of Differential Equations Research Articles

A family of iterative methods to solve nonlinear problems with applications in fractional differential equations

In this work, first, a family of fourth‐order methods is proposed to solve nonlinear equations. The methods satisfy the Kung‐Traub optimality conjecture. By developing the methods into memory methods, their efficiency indices are increased. Then, the methods are extended to the multi‐step methods for finding the solutions to systems of problems. The formula for the order of convergence of the multi‐step iterative methods is , where is the step number of the methods. It is clear that computing the Jacobian matrix derivative evaluation and its inversion are expensive; therefore, we compute them only once in every cycle of the methods. The important feature of these multi‐step methods is their high‐efficiency index. Numerical examples that confirm the theoretical results are performed. In applications, some nonlinear problems related to the numerical approximation of fractional differential equations (FDEs) are constructed and solved by the proposed methods.

L 2 convergence of smooth approximations of stochastic differential equations with unbounded coefficients

The aim of this article is to obtain convergence in mean in the uniform topology of piecewise linear approximations of stochastic differential equations (SDEs) with C 1 drift and C 2 diffusion coefficients with uniformly bounded derivatives. Convergence analyses for such Wong-Zakai approximations most often assume that the coefficients of the SDE are uniformly bounded. Almost sure convergence in the unbounded case can be obtained using now standard rough path techniques, although L q convergence appears yet to be established and is of importance for several applications involving Monte-Carlo approximations. We consider L 2 convergence in the unbounded case using a combination of traditional stochastic analysis and rough path techniques. We expect our proof technique extend to more general piecewise smooth approximations.

A Broken FEEC Framework for Electromagnetic Problems on Mapped Multipatch Domains

We present a framework for the structure-preserving approximation of partial differential equations on mapped multipatch domains, extending the classical theory of finite element exterior calculus (FEEC) to discrete de Rham sequences which are broken, i.e., fully discontinuous across the patch interfaces. Following the Conforming/Nonconforming Galerkin (CONGA) schemes developed in Campos Pinto and Sonnendrücker (Math Comput 85:2651–2685, 2016) and Campos Pinto and Güçlü (Broken-FEEC discretizations and Hodge Laplace problems. arXiv:2109.02553, 2022), our approach is based on: (i) the identification of a conforming discrete de Rham sequence with stable commuting projection operators, (ii) the relaxation of the continuity constraints between patches, and (iii) the construction of conforming projections mapping back to the conforming subspaces, allowing to define discrete differentials on the broken sequence. This framework combines the advantages of conforming FEEC discretizations (e.g. commuting projections, discrete duality and Hodge–Helmholtz decompositions) with the data locality and implementation simplicity of interior penalty methods for discontinuous Galerkin discretizations. We apply it to several initial- and boundary-value problems, as well as eigenvalue problems arising in electromagnetics. In each case our formulations are shown to be well posed thanks to an appropriate stabilization of the jumps across the interfaces, and the solutions are extremely robust with respect to the stabilization parameter. Finally we describe a construction using tensor-product splines on mapped cartesian patches, and we detail the associated matrix operators. Our numerical experiments confirm the accuracy and stability of this discrete framework, and they allow us to verify that expected structure-preserving properties such as divergence or harmonic constraints are respected to floating-point accuracy.

A dynamic hysteresis model of heat and mass transfer for hygrothermal bio-based materials

The hygrothermal behaviour of porous building materials is significantly determined by the moisture and heat transfer phenomena. However, most of existing models used to predict this phenomenon neglect moisture hysteresis effect and dynamic capillarity effect, limiting their accuracy in assessing the efficiency and sustainability of bio-based building materials. To bridge this gap, the model presented in this paper considers both dynamic and hysteresis effects. The Finite Element Method (FEM) is adopted for the discrete approximation of the partial differential equations governing heat and moisture transfer. The model is validated with respect to experiment performed on bio-based material subjected to thermo-hygrical boundaries conditions. Results show unequivocally that dynamic capillarity effect and moisture hysteresis effect should be considered simultaneously to quantify hygrothermal behaviour of porous building materials. This critical insight underscores the significance of this research, offering a comprehensive model for evaluating the efficiency and sustainability of these materials by addressing the often-neglected aspects of moisture behaviour, i.e., dynamic and hysteresis effects. This model is marked by its precision and thorough consideration of essential factors, making it an indispensable contribution to the field and advances the understanding of the hygrothermal behaviour of bio-based building materials in real conditions. It would help in designing optimized building envelop.

Reduced Lagrange multiplier approach for non-matching coupling of mixed-dimensional domains

Many physical problems involving heterogeneous spatial scales, such as the flow through fractured porous media, the study of fiber-reinforced materials, or the modeling of the small circulation in living tissues — just to mention a few examples — can be described as coupled partial differential equations defined in domains of heterogeneous dimensions that are embedded into each other. This formulation is a consequence of geometric model reduction techniques that transform the original problems defined in complex three-dimensional domains into more tractable ones. The definition and the approximation of coupling operators suitable for this class of problems is still a challenge. We develop a general mathematical framework for the analysis and the approximation of partial differential equations coupled by non-matching constraints across different dimensions, focusing on their enforcement using Lagrange multipliers. In this context, we address in abstract and general terms the well-posedness, stability, and robustness of the problem with respect to the smallest characteristic length of the embedded domain. We also address the numerical approximation of the problem and we discuss the infsup stability of the proposed numerical scheme for some representative configuration of the embedded domain. The main message of this work is twofold: from the standpoint of the theory of mixed-dimensional problems, we provide general and abstract mathematical tools to formulate coupled problems across dimensions. From the practical standpoint of the numerical approximation, we show the interplay between the mesh characteristic size, the dimension of the Lagrange multiplier space, and the size of the inclusion in representative configurations interesting for applications. The latter analysis is complemented with illustrative numerical examples.

Stochastic differential equation approximations of generative adversarial network training and its long-run behavior

AbstractThis paper analyzes the training process of generative adversarial networks (GANs) via stochastic differential equations (SDEs). It first establishes SDE approximations for the training of GANs under stochastic gradient algorithms, with precise error bound analysis. It then describes the long-run behavior of GAN training via the invariant measures of its SDE approximations under proper conditions. This work builds a theoretical foundation for GAN training and provides analytical tools to study its evolution and stability.

Convergence Rates for the Truncated Euler-Maruyama Method for Nonlinear Stochastic Differential Equations

In this paper, our main aim is to investigate the strong convergence rate of the truncated Euler-Maruyama approximations for stochastic differential equations with superlinearly growing drift coefficients. When the diffusion coefficient is polynomially growing or linearly growing, the strong convergence rate of arbitrarily close to one half is established at a single time T or over a time interval [0,T], respectively. In both situations, the common one-sided Lipschitz and polynomial growth conditions for the drift coefficients are not required. Two examples are provided to illustrate the theory.

Finding analytical approximations for discrete, stochastic, individual-based models of ecology

Discrete time, spatially extended models play an important role in ecology, modelling population dynamics of species ranging from micro-organisms to birds. An important question is how ’bottom up’, individual-based models can be approximated by ’top down’ models of dynamics. Here, we study a class of spatially explicit individual-based models with contest competition: where species compete for space in local cells and then disperse to nearby cells. We start by describing simulations of the model, which exhibit large-scale discrete oscillations and characterize these oscillations by measuring spatial correlations. We then develop two new approximate descriptions of the resulting spatial population dynamics. The first is based on local interactions of the individuals and allows us to give a difference equation approximation of the system over small dispersal distances. The second approximates the long-range interactions of the individual-based model. These approximations capture demographic stochasticity from the individual-based model and show that dispersal stabilizes population dynamics. We calculate extinction probability for the individual-based model and show convergence between the local approximation and the non-spatial global approximation of the individual-based model as dispersal distance and population size simultaneously tend to infinity. Our results provide new approximate analytical descriptions of a complex bottom-up model and deepen understanding of spatial population dynamics.

Efficient and accurate nonlinear model reduction via first-order empirical interpolation

We present a model reduction approach that extends the original empirical interpolation method to enable accurate and efficient reduced basis approximation of parametrized nonlinear partial differential equations (PDEs). In the presence of nonlinearity, the Galerkin reduced basis approximation remains computationally expensive due to the high complexity of evaluating the nonlinear terms, which depends on the dimension of the truth approximation. The empirical interpolation method (EIM) was proposed as a nonlinear model reduction technique to render the complexity of evaluating the nonlinear terms independent of the dimension of the truth approximation. We introduce a first-order empirical interpolation method (FOEIM) that makes use of the partial derivative information to construct an inexpensive and stable interpolation of the nonlinear terms. We propose two different FOEIM algorithms to generate interpolation points and basis functions. We apply the FOEIM to nonlinear elliptic PDEs and compare it to the Galerkin reduced basis approximation and the EIM. Numerical results are presented to demonstrate the performance of the three reduced basis approaches.

A deep learning feed-forward neural network framework for the solutions to singularly perturbed delay differential equations

In this paper, we explore a deep learning feedforward artificial neural network (ANN) framework as a numerical tool for approximating the solutions to singularly perturbed delay differential equations (SPDDE). Our approach trains the network with fewer uniform data points along with linear interpolation as a dependent variable. More importantly, the exact solution is not used for training the network. A mean square or Euclidean norm type total loss function is utilized to assess the deep learning network’s performance. In the training and testing phase, our network adjusts itself to fit the tangent and the curvature by minimizing the total loss function and fine-tuning the network’s hyperparameters during the backpropagation stage. Our focus in this contribution is to investigate an answer to the question- “whether neural networks can learn to solve the differential equations with both delay and boundary layer behavior”? Traditional numerical methods are known to perform poorly in approximating the solutions to SPDDE due to boundary layer behavior, which is the sharp change in the gradient of the solution inside a region of very small width. Our proposed network architecture is used to investigate the above question for three varieties of SPDDE. The numerical results demonstrate that the fine-tuned adaptive deep learning architecture can effectively approximate the solution to SPDDE for varieties of delay and perturbation parameters. It is expected that the current method can be extendable to approximate 2D convection-diffusion partial differential equations with a boundary layer.

Investigation of the sedimentation characterization of magnetorheological fluids

Magnetorheological fluids (MRFs) are smart materials that respond to an external magnetic field by changing their rheological properties. A main drawback of MR fluids is sedimentation which may render the fluids unusable over time. In this study, we introduce a system for the investigation of the sedimentation of magnetorheological fluids in the gravitational field. First of all, differential equation approximations describing the initial stage and the post-saturated stage of the sedimentation are proposed to correlate the experimental results. The custom-developed apparatus is explained, which incorporates a measuring sensor with a lock-in amplifier, a vertically positioned test tube, and a motorized z-axis translating stage. The measurement principle is based on the change of mutual induction between the primary and secondary coils of a sensor, according to the volume fraction-dependent susceptibility. The sensor's coils are driven by a lock-in amplifier, which is able to sense the minor settling distances. The motorized translating axis provides the opportunity for the analysis of the concentration profile of the sample column along its entire length and the device is able to work in fully automated mode. The performance of the device is validated by measurements with Lord MR fluids, for which the sedimentation constant and ratio metrics are provided.

Wong-Zakai approximations for stochastic differential equations with path-dependent coefficients

In this paper, we shall prove an L p ( p > 2 ) L^{p}(p>2) strong convergence rate of the non-adapted Wong-Zakai approximation for stochastic differential equations with path-dependent coefficients under d ∞ d_{\infty } -Lipschitz conditions.

Using analytical methods for finding the approximate solutions to fractional differential equations

This essay focuses on studying the nonlinear fractional integral equation. Various methods, including Akbari-Ganji's Method (AGM), Homotopy Perturbation Method (HPM), and Vibrational Iteration Method (VIM), are utilized to obtain its solution. We introduce an innovative approach to obtain rough approximations for fractional differential equations. These equations play a significant role in the field of fluid dynamics and find widespread application. In this article, we have used analytical methods to check the correctness of the answers. Maple mathematical software is used to solve all fractional equations. Ordinary equations and fractional differential equations have connections to entropy, wavelets, and other related concepts. To demonstrate the method, a few examples are employed, chosen for their accuracy and simplicity of implementation. The solutions are explained using convergent series. According to the calculations performed with analytical methods on the fractional integral equations of the nonlinear oscillator, Due to the swing movements of the oscillator, as the swing movement's increase, the velocity gradient becomes an upward trend.

Approximation of distribution-independent and distribution-dependent stochastic differential equations with singular drifts

This work is concerned with convergence rate of Euler–Maruyama scheme for distribution-independent and distribution-dependent stochastic differential equations, where the drifts involved are Dini continuous and unbounded. Via introducing a new Zvonkin-type’s transformation, we investigate convergence rate of Euler–Maruyama scheme for stochastic differential equations with singular coefficients which allows the drifts to be unbounded. Moreover, via the analysis of interacting particle systems, we show an approximation issue on a class of distribution-dependent stochastic differential equations with singular drifts. More specifically, propagation of chaos and convergence rate of the Euler–Maruyama scheme associated with the interacting particle systems are investigated.

Numerical Approximations of a Class of Nonlinear Second-Order Boundary Value Problems using Galerkin-Compact Finite Difference Method

In this study, we examine numerical approximations for 2nd-order linear-nonlinear differential equations with diverse boundary conditions, followed by the residual corrections of the first approximations. We first obtain numerical results using the Galerkin weighted residual approach with Bernstein polynomials. The generation of residuals is brought on by the fact that our first approximation is computed using numerical methods. To minimize these residuals, we use the compact finite difference scheme of 4th-order convergence to solve the error differential equations in accordance with the error boundary conditions. We also introduce the formulation of the compact finite difference method of fourth-order convergence for the nonlinear BVPs. The improved approximations are produced by adding the error values derived from the approximations of the error differential equation to the weighted residual values. Numerical results are compared to the exact solutions and to the solutions available in the published literature to validate the proposed scheme, and high accuracy is achieved in all cases.

An application of the multiplicative Sewing Lemma to the high order weak approximation of stochastic differential equations

We introduce a variant of the multiplicative Sewing Lemma in [Gerasimovičs, Hocquet, Nilssen; J. Funct. Anal. 281 (2021)] which yields arbitrary high order weak approximations to stochastic differential equations, extending the cubature approximation on Wiener space introduced by Lyons and Victoir. Our analysis allows to derive stability estimates and explicit weak convergence rates. As a particular example, a cubature approximation for stochastic differential equations driven by continuous Gaussian martingales is given.

Error analysis for a Crouzeix–Raviart approximation of the p-Dirichlet problem

Abstract In the present paper, we examine a Crouzeix–Raviart approximation for non-linear partial differential equations having a (p, δ)-structure for some p ∈ (1, ∞) and δ ⩾ 0. We establish a priori error estimates, which are optimal for all p ∈ (1, ∞) and δ ⩾ 0, medius error estimates, i.e., best-approximation results, and a primal–dual a posteriori error estimate, which is both reliable and efficient. The theoretical findings are supported by numerical experiments.

Improving shape formation under conditions of plane tensile stress

Thin-walled axisymmetric truncated parts made of sheet billets are actively used in rocket and aerospace engineering. Improvement to their shape formation, based on directed material thickness change will ensure the production of parts with minimum thickness variation. This will also enable aviation and space industry enterprises to attain leading positions, as well as reduce labor costs. This work studies the possibility of obtaining thin-walled axisymmetric parts of truncated tapered shape using one of the methods of sheet metal stamping under flat tensile stress conditions (flanging). The mechanism was identified and the analysis of the stress-strain state of the billet during deformation was carried out. This takes into account the minimizing of the difference between the specified and technologically possible thicknesses. A mathematical model was developed to consider the shaping method based on the process of flanging. Theoretical studies were based on the principles of the plastic deformation theory of sheet materials. This was achieved by the following factors: approximate differential equations of force equilibrium; equations of constraint; plasticity conditions; and fundamental constitutive relations under given initial and boundary conditions. The process of flanging was simulated using the LS-DYNA software package with the following initial data of a conical billet made of 12Kh18N10T steel: cone angle 16.4°, thickness Sbillet = 0.3 mm. The aim was to eliminate errors in designing a tool for future implementation of the method on a manufactured die tooling, as well as to confirm the theoretical conclusions on the selection of technological parameters and achieve minimal thickness variation. The steps of computer modeling are presented, indicating the main process parameters such as material model, mechanical characteristics of the workpiece material, type of elements, kinematic loads, conditions of contact interaction of elements with each other, etc.

Numerical solutions of stochastic functional differential equations with impulsive perturbations and Markovian switching

This paper is focused on analysis of Euler–Maruyama-type approximations for stochastic functional differential equations with impulsive perturbations and Markovian switching. The motivation stems from a wide range of applications. The paper establishes mean square convergence of the approximations under a local Lipschitz condition and a linear growth condition. In addition, this work ascertains the rates of convergence of the numerical solutions under the global Lipschitz condition. Finally, the performance of the algorithm is demonstrated through a numerical example.

Investigation of Heat and Mass Transfer in the Process of Fuel Preparation from Biomass Particles with High Moisture

Currently, in a number of countries an urgent task for development of fuel and energy complexes is to increase the share of generation by involving solid fuels in circulation. Among such projects, those that allow the disposal of waste from various industries are particularly significant. Expired food products in this context are represented as a renewable local energy resource. However, such products require serious activities to prepare them for incineration or other type of high-temperature processing in order to obtain energy. The purpose of the present work is to improve methods of preparing fuel from recycled carrot fruits (unsuitable for use in the food sector). During the fuel preparation of carrots, the drying stage is limiting for the rational organization of its processing in boilers. In addition, the drying stage is extremely energy-consuming, so reliable prediction of its kinetics largely determines the efficiency of the entire technological process. In the course of the study, the following tasks were solved: a numerical method was developed for describing the processes of internal and external heat and mass transfer problems using an explicit difference approximation of differential equations of heat and mass transfer; parametric identification of the proposed one-dimensional mathematical model was performed using empirical dependencies known from literature sources; empirical verification of the proposed mathematical model was carried out by comparing the calculated forecasts obtained with the results of their own field experiments. The fact that the proposed mathematical model and the results of the full-scale experiment are independent, while the calculated forecasts and experimental data are in good agreement, makes us possible to consider the proposed calculation method as a reliable scientific basis for a computer method for calculating of heat and mass transfer processes when organizing the preparation of fuel from carrot fruits.