Computing and Communication Structure Design for Fast Mass-Parallel Numerical Solving PDE

Most systems in science, engineering, and biology are of partial differential systems (PDSs) modeled by partial differential equations. Many books about partial differential equations have been written by mathematicians and mainly address some fundamental mathematic backgrounds and discuss some mathematic properties of partial differential equations. Only a few books on PDSs have been written by engineers; however, these books have focused mainly on the theoretical stabilization analysis of PDSs, especially mechanical systems. This book investigates both robust stabilization control design and robust filter design and reference tracking control design in mechanical, signal processing, and control systems to fill a gap in the study of PDSs. Robust Engineering Designs of Partial Differential Systems and Their Applications offers some fundamental background in the first two chapters. The rest of the chapters focus on a specific design topic with a corresponding deep investigation into robust H∞ filtering, stabilization, or tracking design for more complex and practical PDSs under stochastic fluctuation and external disturbance. This book is aimed at engineers and scientists and addresses the gap between the theoretical stabilization results of PDSs in academic and practical engineering designs more focused on the robust H∞ filtering, stabilization, and tracking control problems of linear and nonlinear PDSs under intrinsic random fluctuation and external disturbance in industrial applications. Part I provides backgrounds on PDSs, such as Galerkin’s, and finite difference methods to approximate PDSs and a fuzzy method to approximate nonlinear PDSs. Part II examines robust H∞ filter designs for the robust state estimation of linear and nonlinear stochastic PDSs. And Part III treats robust H∞ stabilization and tracking control designs of linear and nonlinear PDSs. Every chapter focuses on an engineering design topic with both theoretical design analysis and practical design examples.

In the present study, the exact analytical solutions of the 2-dimensional nonlinear Klein-Gordon equation (NLKGE) is investigated using the 3-dimensional Laplace transform method in conjunction with the Daftardar-Gejji and Jafari Method (Iterative method). Through this method, the linear part of the problem is solved by using the 3-dimensional Laplace transform method, while the noise terms from the nonlinear part of the equation disappear through a successive iteration process of the Daftardar-Gejji and Jafari Method (DJM), where a single iteration gives the exact solution to the problem. Three test modeling problems from mathematical physics nonlinear Klein-Gordon equations are taken to confirm the performance and efficiency of the presented technique. For each illustrative example, the convergence of the novel iterative approach is shown. The findings also suggest that the proposed method could be applied to other types of nonlinear partial differential equation systems. HIGHLIGHTS Partial differential equations play a crucial role in the formulation of fundamental laws of nature and the mathematical analysis in different fields of studies The Klein-Gordon equation (KGE) is a nonlinear hyperbolic class of partial differential equations that occur in relativistic quantum mechanics and field theory, both of which are crucial to high-energy physicists The 3-dimensional Laplace transform method provides rapid convergence of the exact solution without any restrictive assumptions about the solution for linear differential equations The Laplace Transform method is often combined with other methods like the Daftardar-Gejji and Jafari (the new Iterative) method to solve complicated nonlinear differential equations The new iterative technique yields a series that can be summed up to get an analytical formula or used to build an appropriate approximation with a faster convergent series solution GRAPHICAL ABSTRACT

Preface

In recent years, machine learning methods have been used to solve partial differential equations (PDEs) and dynamical systems, leading to the development of a new research field called scientific machine learning, which combines techniques such as deep neural networks and statistical learning with classical problems in applied mathematics. In this paper, we present a novel numerical algorithm that uses machine learning and artificial intelligence to solve PDEs. Based on the Legendre-Galerkin framework, we propose an <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">unsupervised machine learning</i> algorithm that learns <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">multiple instances</i> of the solutions for different types of PDEs. Our approach addresses the limitations of both data-driven and physics-based methods. We apply the proposed neural network to general 1D and 2D PDEs with various boundary conditions, as well as convection-dominated <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">singularly perturbed PDEs</i> that exhibit strong boundary layer behavior.

Advection-diffusion equations are widely used in modeling a diverse range of problems. These mathematical models consist in a partial differential equation or system with initial and boundary conditions, which depend on the phenomena being studied. In the modeling, boundary conditions may be neglected and unnecessarily simplified, or even misunderstood, causing a model not to reflect the reality adequately, making qualitative and/or quantitative analyses more difficult. In this work we derive a general linear flux dependent boundary condition for advection-diffusion problems and show that it generates all possible boundary conditions, according to the outward flux on the boundary. This is done through an integral formulation, analyzing the total mass of the system. We illustrate the exposed cases with applications willing to clarify their meanings. Numerical simulations, by means of the Finite Difference Method, are used in order to exemplify the different boundary conditions' impact, making it possible to quantify the flux along the boundary. With qualitative and quantitative analysis, this work can be useful to researchers and students working on mathematical models with advection-diffusion equations.

ABSTRACTThis article examines the initial‐boundary value problem for a system of first‐order partial differential equations. Issues of existence and uniqueness of the solution in a broad sense are considered, while taking into account both periodic and multipoint conditions. The definition of a solution in a broad sense is introduced; the initial problem is reduced to the initial‐boundary value problem for ordinary differential equations. The two‐point boundary value problem for ordinary differential equations systems is studied by the Dzhumabaev method (parameterization method), which allows us to move on to the equivalent multipoint boundary value problem with functional parameters. An algorithm to find an approximate solution to the problem in a broad sense has been developed.

This work studies the structural properties and convergence approach of chance-constrained optimization of boundary-value elliptic partial differential equation systems (CCPDEs). The boundary conditions are random input functions deliberated from the boundary of the partial differential equation (PDE) system and in the infinite-dimensional reflexive and separable Banach space. The structural properties of the chance constraints studied in this paper are continuity, closedness, compactness, convexity, and smoothness of probabilistic uniform or pointwise state constrained functions and their parametric approximations. These are open issues even in the finite-dimensional Banach space. Thus, it needs finite-dimensional and smooth parametric approximation representations. We propose a convex approximation approach to nonconvex CCPDE problems. When the approximation parameter goes to zero from the right, the solutions of the relaxation and compression approximations converge asymptotically to the optimal solution of the original CCPDE. Due to the convexity of the problem, a global solution exists for the proposed approximations. Numerical results are provided to demonstrate the plausibility and applicability of the proposed approach.

One of the oldest and currently most promising application areas for quantum devices is quantum simulation. Popularised by Feynman in the early 1980s, it is important for the efficient simulation – compared to its classical counterpart – of one special partial differential equation (PDE): Schrodinger’s equation. This is possible because quantum devices themselves naturally obey Schrodinger’s equation. Just like with large-scale quantum systems, classical methods for other high-dimensional and large-scale PDEs often suffer from the curse-of-dimensionality, which a quantum treatment might in certain cases be able to mitigate. Aside from Schrodinger’s equation, can quantum simulators also efficiently simulate other PDEs? To enable the simulation of PDEs on quantum devices that obey Schrodinger’s equations, it is crucial to first develop good methods for mapping other PDEs onto Schrodinger’s equations. After a brief introduction to quantum simulation, I will address the above question in this talk by introducing a simple and natural method for mapping other linear PDEs onto Schrodinger’s equations. It turns out that by transforming a linear partial differential equation (PDE) into a higher-dimensional space, it can be transformed into a system of Schrodinger’s equations, which is the natural dynamics of quantum devices. This new method – called Schrodingerisation – thus allows one to simulate, in a simple way, any general linear partial differential equation and system of linear ordinary differential equations via quantum simulation. This simple methodology is also very versatile. It can be used directly either on discrete-variable quantum systems (qubits) or on analog/continuous quantum degrees of freedom (qumodes). The continuous representation in the latter case can be more natural for PDEs since, unlike most computational methods, one does not need to discretise the PDE first. In this way, we can directly map D-dimensional linear PDEs onto a (D + 1)-qumode quantum system where analog Hamiltonian simulation on (D + 1) qumodes can be used. It is the quantum version of analog computing and is more amenable to near-term realisation. I will show how this Schrodingerisation method can be applied to linear PDEs, systems of linear ODEs and also linear PDES with random coefficients, where the latter is important in the area of uncertainty quantification. Furthermore, these methods can be extended to solve problems in linear algebra by transforming iterative methods in linear algebra into the evolution of linear ODEs.

Exponential stability analysis of delayed partial differential equation systems: Applying the Lyapunov method and delay-dependent techniques

It is well known that the ordering of the unknowns on a finite-difference grid had effects on the computing efficiency of direct matrix solution methods. In reservoir simulation, diagonal ordering scheme (D2), alternating point ordering scheme (A3), and alternating diagonal ordering scheme (D4) have been found theoretically and experimentally to improve the performance of Gaussian type direct solution methods. This paper investigates the effects of D2, A3 and D4 ordering schemes on the performance of iterative matrix methods. The test model is the typical two-dimensional parabolic partial differential equation with Neumann type boundary conditions and the iterative method used is the point successive overrelaxation (PSOR). Ten different cases are presented.

Author index

On the variational iteration method and other iterative techniques for nonlinear differential equations

This paper discusses some special features of VLSI semiconductor devices due to the nonlinearity of the partial differential equations defining the electric potential and the density of holes and electrons flowing within the device. Currently, solutions for these nonlinear partial differential equations are computed with great speed on very large modern computers. However, a theoretical analytical treatment of these systems has not been systematically carried through. It is hoped that this analytical study may shed some light on some aspects of VLSI device operation that, until now, have required ingenious physical intuition and arguments for their understanding. In particular, the analytic approach has the potential to consider the following problems of importance in VLSI semiconductor devices: 1) arbitrarily shaped devices, 2) high space dimensional effects (especially those with curved boundaries), 3) nonuniqueness of solutions, 4) bifurcation phenomena, 5) instabilities and parameter dependence, and 6) latch up phenomena and high bias effects. I personally, believe there is much to be gained from an interplay between the impressive numerical simulation already attained and an analytic consideration of the equations themselves. Typical questions are (for example) (i) Electronic Instabilities. How do these arise from the boundary value problems themselves? What are the mathematical mechanisms involved? (ii) Structural Problems. Do the equations themselves have special structures that may aid the numerical simulation of their solutions? Another key point here is that the partial differential equation systems for semiconductors are another nonlinear generalization of Maxwell's equations of electromagnetism. As semiconductor devices become smaller, there is, of course, a greater need to rely on theoretical analysis. The immense success of Maxwell's mathematical electromagnetic theory and his own statistical mechanics ideas (here fused together) gives one confidence of the ultimate validity of the partial differential equation approach. In this article we shall limit ourselves to a discussion of the standard Van Roosbroeck equations (assuming the validity of the Einstein relations and Boltzmann statistics) and to questions of steady-state processes (i.e., time-independent problems). We hope to address other aspects in latter publications.

A unified method for extracting geometric shape features from binary image data using a steady-state partial differential equation (PDE) system as a boundary value problem is presented in this paper. The PDE and functions are formulated to extract the thickness, orientation, and skeleton simultaneously. The main advantage of the proposed method is that the orientation is defined without derivatives and thickness computation is not imposed a topological constraint on the target shape. A one-dimensional analytical solution is provided to validate the proposed method. In addition, two-dimensional numerical examples are presented to confirm the usefulness of the proposed method. Highlights A steady state partial differential equation for extraction of geometrical shape features is formulated. The functions for geometrical shape features are formulated by the solution of the proposed PDE. Analytical solution is provided in one-dimension. Numerical examples are provided in two-dimension.

The ability of hybrid nanomaterials to enhance heat transmission has attracted many academics to study the working fluid. This work examines the impressions of radiation efficiency on the convective Casson flow of hybrid Nano liquid across an angled moving plate with a magnetic impression using a mathematical solution. The main aim of this study is to increase the thermal efficiency of the fluid by using different categories of Cu and Al2O3 nanoparticles are combined with the base fluid water. A nonlinear partial differential equation system is created while keeping in mind some reasonable presumptions. Using the similarity transformation, the partial differential equations are changed into nonlinear ordinary differential equations. And it is then mathematically simplified with the bvp4c technique. Consequences of an exclusivity group of unique impacts on motion characteristics, shear stress, thermal field impressions, and heat transport are described clearly. The motion rised with increasing Casson fluid for stretching and shrinking surfaces. For radiation impression, an energy upsurge profile is visible. With an inclined plate, the shear rate decreases, and the buoyancy impression causes it to rise. This work provides new information about the behavior of heat plumes under magnetic fields, thermal radiation and flow, and has potential applications in enhancing cooling systems in industrial applications, modelling of oil reservoirs, and nuclear waste storage. Nanoparticles are used for cooling processors, cancer therapy, medicine, metal strips, automobile engines, welding equipment, fusion reactions, chemical reactions, and for cooling heat exchange mechanisms in various engineering devices due to their superior thermophysical properties.