Methods For Partial Differential Equations Research Articles

High-Gain-Observer-Based Predictive Output Feedback for Nonlinear Systems With Large Input-Delays

Abstract The predictor feedback has been demonstrated to be quite effective in large delay compensation. However, few researches in the field of predictor feedback for large delays focused on output feedback control (OFC). This paper develops the previous work to design high-gain-observer-based predictive output feedback for nonlinear systems with large delays. Two methods are employed for large delay compensation: the backstepping-based partial differential equation (PDE) method and the reduction-based ordinary differential equation (ODE) method. It appears that, for continuous-time control, the first method leads to simpler linear matrix inequality (LMI) conditions and deal with larger delays, whereas the second method is easily exploited for sampled-data implementation under continuous-time measurement. Lyapunov–Krasovskii method is presented to guarantee the exponential stability of the nonlinear systems under predictor-based controllers. Through a simulation example of pendulum, the proposed methods are demonstrated to be efficient when the input delays are too large for the system to be stabilized without a predictor.

Galerkin Spectral Method of Stochastic Partial Differential Equations Driven by Multivariate Poisson Measure

A considerable body of prior research has been dedicated to devising efficient and high‐order numerical methods for solving stochastic partial differential equations (SPDEs) driven by discrete or continuous random variables. The majority of these efforts have concentrated on introducing numerical approaches for SPDEs involving independent random variables, with limited exploration into solving SPDEs driven by dependent and discrete random variables. In this paper, we introduce a Galerkin spectral method tailored for addressing SPDEs driven by multivariate Poisson variables. Our novel numerical method is applied to solve the one‐ and two‐dimensional stochastic Poisson equations, as well as the one and two‐dimensional stochastic heat equation, stochastic KdV equation, and stochastic Burgers equation, respectively. Notably, all the presented stochastic equations are governed by two dependent Poisson random variables.

Prescribed-Time Network-Based Deployment of Nonlinear Multiagent Systems: A Discrete-Space PDE Method.

This article attempts to design the prescribed-time time-varying deployment schemes for first-order and second-order nonlinear multiagent systems (MASs). We assume that all agents can obtain the information of their current and final relative positions with their neighbors, and the final absolute velocities (as well as their current and final relative velocities, the final absolute accelerations for the second-order MASs) through a communication network, whereas two boundary agents are able to obtain their current and final absolute positions (as well as their current and final absolute velocities for the second-order MASs). The neighbor relationship of all agents is described by a spatial variable and two static-feedback controllers are introduced, which can be expressed as a second-order space difference of the spatial variable. Then, the deployment of MASs can be transformed into the stabilization of discrete-space partial differential equation (PDE) systems. Three virtual agents are introduced to constitute the Dirchlet and Neumann boundary conditions. Several algebraic inequality criteria are derived to guarantee that the prescribed-time time-varying deployment can be achieved within a prescribed time under the Dirchlet and mixed boundary conditions. Unlike the published results, our results are derived based on the discrete-space PDE systems instead of continuous-space PDE systems, which is consistent with the discrete spatial distribution of agents. Finally, two numerical examples are given to illustrate the effectiveness of our results.

A super-localized generalized finite element method

AbstractThis paper presents a novel multi-scale method for elliptic partial differential equations with arbitrarily rough coefficients. In the spirit of numerical homogenization, the method constructs problem-adapted ansatz spaces with uniform algebraic approximation rates. Localized basis functions with the same super-exponential localization properties as the recently proposed Super-Localized Orthogonal Decomposition enable an efficient implementation. The method’s basis stability is enforced using a partition of unity approach. A natural extension to higher order is presented, resulting in higher approximation rates and enhanced localization properties. We perform a rigorous a priori and a posteriori error analysis and confirm our theoretical findings in a series of numerical experiments. In particular, we demonstrate the method’s applicability for challenging high-contrast channeled coefficients.

An Adaptive ANOVA Stochastic Galerkin Method for Partial Differential Equations with High-dimensional Random Inputs

An Adaptive ANOVA Stochastic Galerkin Method for Partial Differential Equations with High-dimensional Random Inputs

Mathematical Modeling of Dynamic Supply Chains Subject to Demand Fluctuations

This research work aims to develop the mathematical modeling for a class of dynamic supply chains. Demand fluctuation corresponds to product demand volatility, which increases or decreases over a given time frame. Industrial engineering practitioners should consider the function that applied mathematical modeling plays in providing approximations of solutions that may be used in simulations and technical implementations at the strategic, tactical, and operational levels of an organization. In order to achieve proper results, two mathematical models are presented in this paper: In addition to a finite-dimensional system of Ordinary Differential Equations (ODEs) for coupled dynamic pricing, production rate, and inventory level, which properly integrates Lyapunov stability analysis of the dynamical system and simulations, there is an infinite-dimensional Partial Differential Equation (PDE) production level modeling system available. Infinite and finite-dimensional systems incorporate a dynamic pricing approach in the mathematical modeling. The main research goal of this work is to explore the dynamic nature of supply chains applying PDE and ODE methods, with proper analytical analysis and simulations for both systems.

Preface

The 2023 7th International Conference on Mechanics, Mathematics and Applied Physics (ICMMAP 2023) took place in Dalian, China during 8th to 10th September 2023. The conferences in this series started in 2017 as an academic conference and gradually expanded to include wider topic fields as well. Now the International Conference on Mechanics, Mathematics and Applied Physics has been developed into an annual international conference based in China with topics encompassing the whole area of mechanics, mathematics and applied physics.The aim of the Conference was to provide a platform for the exploration of both fundamental topics and new applications of research fields related to mechanics, mathematics and applied physics, bring different scientific communities together, and thus facilitate the contacts between science, technology and industry. All the attendees were welcomed to give comments and raise questions to the speeches and presentations during the Conference.More than 100 participants attended the Conference which consisted of ten keynote speeches, as well as plenty of oral and poster contributions. Among them, Prof. Zhenxin Liu (Dalian University of Technology, China), who is interested in the researches on Stochastic Dynamical Systems and Stochastic Differential Equations, delivered a keynote speech on The Averaging Principle for SDEs with Monotone Coefficients. And Prof. Chongjun Li (Dalian University of Technology, China) made a report on Revised SBFEM on Arbitrary Polygons and Faceted Polyhedrons with the Second Order Completeness by Elimination of Bubble Functions. He focuses his research on Numerical Approximation, Computational Geometry, Multivariate Spline Theory, and Spline Finite Elements. And the other speakers performed their speeches one by one, sharing research experience and professional insights on specific topics. It is really a great opportunity to promote academic communication and collaboration worldwide.This volume gathers papers from regular contributions of ICMMAP 2023; each contribution submitted for publication has been peer reviewed and the Editors are very grateful to the referees for their careful support in improving the original manuscripts. The manuscripts accepted for publication cover different topics of mechanics, mathematics and applied physics: Computational Mechanics, Multiphase Fluid Mechanics, Partial Differential Equation, Simulation and Monte-Carlo Methods, Nuclear Electronics, etc.It is a great honor to present this volume of Journal of Physics: Conference Series and we deeply thank the authors for their enthusiastic and high-grade contribution.Finally, we would like to thank the Conference Chair, the Publication Chair, the members of the Technical Program Committee as well as Organizing Committee, the conference secretariat and the support from Dalian Jiaotong University that allowed the success of ICMMAP 2023.The Committee of ICMMAP 2023List of Committee Member are avaible in this pdf.

Frobenius methods for analytic second order linear partial differential equations

The main subject of this text is the study of analytic second order linear partial differential equations. We aim to solve the classical equations and some more, in the real or complex analytical case. This is done by introducing methods inspired by the Frobenius method for second order linear ordinary differential equations. We introduce a notion of Euler type partial differential equation. To such a PDE we associate an indicial conic, which is an affine plane curve of degree two. Then comes the concept of regular singularity and finally convergence theorems, which must necessarily take into account the type of PDE (parabolic, elliptical or hyperbolic) and a nonresonance condition. This condition gives a new geometric interpretation of the original condition between the roots of the original Frobenius theorem for second order ODEs. The interpretation is something like, a certain reticulate has or not vertices on the indexical conic. Finally, we retrieve the solution of all the classical PDEs by this method (heat diffusion, wave propagation and Laplace equation), and also increase the class of those that have explicit algorithmic solution to far beyond those admitting separable variables. The last part of the text is dedicated to the construction of PDE models for the classical ODEs like Airy, Legendre, Laguerre, Hermite and Chebyshev by two different means. One model is based on the requirement that the restriction of the PDE to lines through the origin must be the classical ODE model. The second is based on the idea of having symmetries on the PDE model and imitating the ODE model. We study these PDEs and obtain their solutions, obtaining for the framework of PDEs some of the classical results, like existence of polynomial solutions (Laguerre, Hermite and Chebyshev polynomials).

Convergence of adaptive two-grid weak Galerkin finite element methods for semilinear elliptic differential equations

In this paper, we investigate the convergence of an adaptive two-grid weak Galerkin (ATGWG) finite element method for second order semilinear elliptic partial differential equations (PDEs). First, we propose an ATGWG method and then prove that the sum of the energy error and the error estimator of ATGWG method between two consecutive adaptive loops is a contraction. The weak Galerkin (WG) elements (Pj(T),Pℓ(∂T),RTj(T)) (Wang and Ye, 2013) are studied in this paper and numerical experiments based on the lowest order case with j=l=0 are provided to support the theoretical results.

MetaPINNs: Predicting soliton and rogue wave of nonlinear PDEs via the improved physics-informed neural networks based on meta-learned optimization

Efficiently solving partial differential equations (PDEs) is a long-standing challenge in mathematics and physics research. In recent years, the rapid development of artificial intelligence technology has brought deep learning-based methods to the forefront of research on numerical methods for partial differential equations. Among them, physics-informed neural networks (PINNs) are a new class of deep learning methods that show great potential in solving PDEs and predicting complex physical phenomena. In the field of nonlinear science, solitary waves and rogue waves have been important research topics. In this paper, we propose an improved PINN that enhances the physical constraints of the neural network model by adding gradient information constraints. In addition, we employ meta-learning optimization to speed up the training process. We apply the improved PINNs to the numerical simulation and prediction of solitary and rogue waves. We evaluate the accuracy of the prediction results by error analysis. The experimental results show that the improved PINNs can make more accurate predictions in less time than that of the original PINNs.

Roadmap of the Multiplier Method for Partial Differential Equations

This review paper gives an overview of the method of multipliers for partial differential equations (PDEs). This method has made possible a lot of solutions to PDEs that are of interest in many areas such as applied mathematics, mathematical physics, engineering, etc. Looking at the history of the method and synthesizing the newest developments, we hope to give it the attention that it deserves to help develop the vast amount of work still needed to understand it and make the best use of it. It is also an interesting and a relevant method in itself that could possibly give interesting results in areas of mathematics such as modern algebra, group theory, topology, etc. The paper will be structured in such a manner that the last review known for this method will be presented to understand the theoretical framework of the method and then later work done will be presented. The information of four recent papers further developing the method will be synthesized and presented in such a manner that anyone interested in learning this method will have the most relevant information available and have all details cited for checking.

An extreme learning machine-based method for computational PDEs in higher dimensions

We present two effective methods for solving high-dimensional partial differential equations (PDE) based on randomized neural networks. Motivated by the universal approximation property of this type of networks, both methods extend the extreme learning machine (ELM) approach from low to high dimensions. With the first method the unknown solution field in d dimensions is represented by a randomized feed-forward neural network, in which the hidden-layer parameters are randomly assigned and fixed while the output-layer parameters are trained. The PDE and the boundary/initial conditions, as well as the continuity conditions (for the local variant of the method), are enforced on a set of random interior/boundary collocation points. The resultant linear or nonlinear algebraic system, through its least squares solution, provides the trained values for the network parameters. With the second method the high-dimensional PDE problem is reformulated through a constrained expression based on an Approximate variant of the Theory of Functional Connections (A-TFC), which avoids the exponential growth in the number of terms of TFC as the dimension increases. The free field function in the A-TFC constrained expression is represented by a randomized neural network and is trained by a procedure analogous to the first method. We present ample numerical simulations for a number of high-dimensional linear/nonlinear stationary/dynamic PDEs to demonstrate their performance. These methods can produce accurate solutions to high-dimensional PDEs, in particular with their errors reaching levels not far from the machine accuracy for relatively lower dimensions. Compared with the physics-informed neural network (PINN) method, the current method is both cost-effective and more accurate for high-dimensional PDEs.

Application of the GKM of to some nonlinear partial equations

In this manuscript, the strain wave equation, which plays an important role in describing different types of wave propagation in microstructured solids and the (2+1) dimensional Bogoyavlensky Konopelchenko equation, is defined in fluid mechanics as the interaction of a Riemann wave propagating along the $y$-axis and a long wave propagating along the $x$-axis, were studied. The generalized Kudryashov method (GKM), which is one of the solution methods of partial differential equations, was applied to these equations for the first time. Thus, a series of solutions of these equations were obtained. These found solutions were compared with other solutions. It was seen that these solutions were not shown before and were presented for the first time in this study. The new solutions of these equations might have been useful in understanding the phenomena in which waves are governed by these equations. In addition, 2D and 3D graphs of these solutions were constructed by assigning certain values and ranges to them.

Rational reparameterization of unstructured quadrilateral meshes for isogeometric analysis with optimal convergence

Unstructured quadrilateral meshes are widely used in representing surfaces of arbitrary topology. However, in connection with isogeometric analysis, unstructured meshes suffer from sub-optimal convergence rates in extraordinary regions, especially for solving higher-order PDEs (partial differential equations). In this work, a rational reparameterization method is proposed to achieve optimal convergence rates for isogeometric analysis using Catmull-Clark surfaces. The scheme lifts the convergence of analysis solutions by tuning the weights of control vertices of the input geometry in extraordinary regions, which are often neglected in constructing analysis-suitable parameterizations. Preferred tuning parameters with optimal convergence rates in L2-norm are determined statistically based on a test suite composed of typical testing solutions with a range of parameters. Numerical experiments confirm the validity of the proposed method for PDEs up to the fourth-order. Comparisons with the original Catmull-Clark scheme and a state-of-the-art tuned subdivision scheme also show the superiority of the proposed method in terms of both optimal convergence and reduced absolute errors of resulting analysis solutions.

Reduced basis stochastic Galerkin methods for partial differential equations with random inputs

We present a reduced basis stochastic Galerkin method for partial differential equations with random inputs. In this method, the reduced basis methodology is integrated into the stochastic Galerkin method, resulting in a significant reduction in the cost of solving the Galerkin system. To reduce the main cost of matrix-vector manipulation involved in our reduced basis stochastic Galerkin approach, the secant method is applied to identify the number of reduced basis functions. We present a general mathematical framework of the methodology, validate its accuracy and demonstrate its efficiency with numerical experiments.

A spectral shifted gegenbauer collocation method for fractional pantograph partial differential equations and its error analysis

A spectral shifted gegenbauer collocation method for fractional pantograph partial differential equations and its error analysis

On the Modified Numerical Methods for Partial Differential Equations Involving Fractional Derivatives

This paper provides both analytical and numerical solutions of (PDEs) involving time-fractional derivatives. We implemented three powerful techniques, including the modified variational iteration technique, the modified Adomian decomposition technique, and the modified homotopy analysis technique, to obtain an approximate solution for the bounded space variable ν. The Laplace transformation is used in the time-fractional derivative operator to enhance the proposed numerical methods’ performance and accuracy and find an approximate solution to time-fractional Fornberg–Whitham equations. To confirm the accuracy of the proposed methods, we evaluate homogeneous time-fractional Fornberg–Whitham equations in terms of non-integer order and variable coefficients. The obtained results of the modified methods are shown through tables and graphs.

Efficient Iterative Arbitrary High-Order Methods: an Adaptive Bridge Between Low and High Order

AbstractWe propose a new paradigm for designing efficient p-adaptive arbitrary high-order methods. We consider arbitrary high-order iterative schemes that gain one order of accuracy at each iteration and we modify them to match the accuracy achieved in a specific iteration with the discretization accuracy of the same iteration. Apart from the computational advantage, the newly modified methods allow to naturally perform the p-adaptivity, stopping the iterations when appropriate conditions are met. Moreover, the modification is very easy to be included in an existing implementation of an arbitrary high-order iterative scheme and it does not ruin the possibility of parallelization, if this was achievable by the original method. An application to the Arbitrary DERivative (ADER) method for hyperbolic Partial Differential Equations (PDEs) is presented here. We explain how such a framework can be interpreted as an arbitrary high-order iterative scheme, by recasting it as a Deferred Correction (DeC) method, and how to easily modify it to obtain a more efficient formulation, in which a local a posteriori limiter can be naturally integrated leading to the p-adaptivity and structure-preserving properties. Finally, the novel approach is extensively tested against classical benchmarks for compressible gas dynamics to show the robustness and the computational efficiency.

Randomized neural network with Petrov–Galerkin methods for solving linear and nonlinear partial differential equations

We present a new approach for solving partial differential equations (PDEs) based on randomized neural networks and the Petrov–Galerkin method, which we call the RNN-PG methods. This method uses randomized neural networks to approximate unknown functions and allows for a flexible choice of test functions, such as finite element basis functions, Legendre or Chebyshev polynomials, or neural networks. We apply the RNN-PG methods to various problems, including Poisson problems with primal or mixed formulations, and time-dependent problems with a space–time approach. This paper is adapted from the work originally posted on arXiv.com by the same authors (arXiv:2201.12995, Jan 31, 2022). The new ingredients include non-linear PDEs such as Burger’s equation and a numerical example of a high-dimensional heat equation. Numerical experiments show that the RNN-PG methods can achieve high accuracy with a small number of degrees of freedom. Moreover, RNN-PG has several advantages, such as being mesh-free, handling different boundary conditions easily, solving time-dependent problems efficiently, and solving high-dimensional problems quickly. These results demonstrate the great potential of the RNN-PG methods in the field of numerical methods for PDEs.

CONVERGENCE AND STABILITY OF GALERKIN FINITE ELEMENT METHOD FOR HYPERBOLIC PARTIAL DIFFERENTIAL EQUATION WITH PIECEWISE CONTINUOUS ARGUMENTS OF ADVANCED TYPE

This paper deals with the convergence and stability of Galerkin finite element method for a hyperbolic partial differential equations with piecewise continuous arguments of advanced type. First of all, we obtain the expression of analytic solution by the method of separation variable, then the sufficient conditions for stability are obtained. Semidiscrete and fully discrete schemes are derived by Galerkin finite element method, and their convergence are both analyzed in L2-norm. Moreover, the stability of the two schemes are investigated. The semidiscrete scheme can achieve unconditionally stability. The sufficient conditions of stability for fully discrete scheme are derived under which the analytic solution is asymptotically stable. Finally, some numerical experiments are presented to illustrate the theoretical results.