Linear Partial Differential Equations Research Articles

Hyers-Ulam stability of a general linear partial differential equation

In this paper we will study Hyers-Ulam stability for a general linear partial differential equation of first order in a Banach space.

A new simultaneously compact finite difference scheme for high-dimensional time-dependent PDEs

This paper presents a new compact finite difference scheme for solving linear high-dimensional time-dependent partial differential equations (PDEs). Despite the different spatial and time conditions in time-dependent problems, we propose a new compact finite difference scheme simultaneously both in time and space with arbitrary order accuracy. The merit of the proposed method is that the approximation of partial derivatives are derived simultaneously at all grid points. Also, by substituting the partial derivatives in the linear time-dependent PDEs a linear system of equations is derived. To show the efficiency and applicability of the proposed method, the fourth, sixth, and eighth-order simultaneously compact finite difference schemes are used for solving parabolic and convection–diffusion equations which have both time and spatial partial derivatives. The numerical results show the accuracy of the proposed method.

Operator-norm resolvent estimates for thin elastic periodically heterogeneous rods in moderate contrast

We provide resolvent asymptotics as well as various operator-norm estimates for the system of linear partial differential equations describing thin infinite elastic rods with material coefficients that rapidly oscillate along the rod. The resolvent asymptotics is derived simultaneously with respect to the rod thickness and the period of material oscillations, which are taken to be of the same order. The analysis is carried out separately on two invariant subspaces pertaining to the out-of-line and in-line displacements, under the assumption on material symmetries as well as in the general case when these two types of displacements are intertwined.

Evolution of Nonlinear Wave Pulses in the sine-Gordon Equation Theory

We give the solution of the Whitham modulation equations for envelopes of pulses evolving according to the sine-Gordon equation. The Whitham equations are interpreted as the equations of relativistic hydrodynamics and their solving is reduced by the hodograph method to solving a linear partial differential equation. We describe the class of solutions of this equation with separation of variables and illustrate the theory by an example of the nonlinear wave packet evolution accompanied by its shrinking and decrease of the number of oscillations in the Whitham nonlinear region.

GKZ-system of the 2-loop self energy with 4 propagators

Applying the system of linear partial differential equations derived from the Mellin–Barnes representation and the Miller transformation, we present the GKZ-system of the Feynman integral of the 2-loop self energy diagram with 4 propagators. The codimension of the derived GKZ-system equals the number of independent dimensionless ratios among the external momentum squared and virtual mass squared. In total 536 hypergeometric functions are obtained in the neighborhoods of the origin and infinity, in which 30 linearly independent hypergeometric functions whose convergent regions have nonempty intersection constitute a fundamental solution system in a proper subset of the whole parameter space.

Stability Analysis of Second-Order Linear PDEs on Time Scales

This paper presents the stability analysis of a class of second-order linear partial differential equations (PDEs) on time scales with diffusion operator and first-order partial derivative. According to the time scale theory, the Lyapunov functional method, and some inequality techniques, sufficient conditions for exponential stability are strictly obtained, and the results are generalized for that where both the discrete-time and continuous-time cases are considered jointly. In addition, the theoretical results are applied to exponential synchronization of reaction-diffusion neural networks (RDNNs). Simulation examples are given to verify the feasibility of our results.

Geometric Aspects of Shape Optimization

We present a review of known results in shape optimization from the point of view of Geometric Analysis. This paper is devoted to the mathematical aspects of the shape optimization theory. We focus on the theory of gradient flows of objective functions and their regularizations. Shape optimization is a part of calculus of variations which uses the geometry. Shape optimization is also related to the free boundary problems in the theory of Partial Differential Equations. We consider smooth perturbations of geometrical domains in order to develop the shape calculus for the analysis of shape optimization problems. There are many applications of such a framework, in solid and fluid mechanics as well as in the solution of inverse problems. For the sake of simplicity we consider model problems, in principle in two spatial dimensions. However, the methods presented are used as well in three spatial dimensions. We present a result on the convergence of the shape gradient method for a model problem. To our best knowledge it is the first result of convergence in shape optimization. The complete proofs of some results are presented in report (Plotnikov and Sokolowski, Gradient flow for Kohn–Vogelius functional).

Robust a posteriori estimates for the stochastic Cahn-Hilliard equation

We derive a posteriori error estimates for a fully discrete finite element approximation of the stochastic Cahn-Hilliard equation. The a posteriori bound is obtained by a splitting of the equation into a linear stochastic partial differential equation and a nonlinear random partial differential equation. The resulting estimate is robust with respect to the interfacial width parameter and is computable since it involves the discrete principal eigenvalue of a linearized (stochastic) Cahn-Hilliard operator. Furthermore, the estimate is robust with respect to topological changes as well as the intensity of the stochastic noise. We provide numerical simulations to demonstrate the practicability of the proposed adaptive algorithm.

Time complexity analysis of quantum algorithms via linear representations for nonlinear ordinary and partial differential equations

We construct quantum algorithms to compute the solution and/or physical observables of nonlinear ordinary differential equations (ODEs) and nonlinear Hamilton-Jacobi equations (HJE) via linear representations or exact mappings between nonlinear ODEs/HJE and linear partial differential equations (the Liouville equation and the Koopman-von Neumann equation). The connection between the linear representations and the original nonlinear system is established through the Dirac delta function or the level set mechanism. We compare the quantum linear systems algorithms based methods and the quantum simulation methods arising from different numerical approximations, including the finite difference discretisations and the Fourier spectral discretisations for the two different linear representations, with the result showing that the quantum simulation methods usually give the best performance in time complexity. We also propose the Schrödinger framework to solve the Liouville equation for the HJE with the Hamiltonian formulation of classical mechanics, since it can be recast as the semiclassical limit of the Wigner transform of the Schrödinger equation. Comparison between the Schrödinger and the Liouville framework will also be made.

A Framework for Physics-Informed Deep Learning Over Freeform Domains

Deep learning is a popular approach for approximating the solutions to partial differential equations (PDEs) over different material parameters and boundary conditions. However, no work has yet been reported on learning PDE solutions over changing shapes of the underlying domain.We present a framework to train neural networks (NN) and physics-informed neural networks (PINNs) to learn the solutions to PDEs defined over varying freeform domains. This is made possible through our adoption of a parametric non-uniform rational B-Spline (NURBS) representation of the underlying physical shape. Distinct physical domains are mapped to a common parametric space via NURBS parameterization. In our approach, we formulate NNs and PINNs that learn the solutions to PDEs as a function of the shape of the domain itself through shape parameters.Under this formulation, the loss function is based on an unchanging parametric domain that maps to a variable physical domain. Residual computation in PINNs is made possible through the Jacobian of the mapping.Numerical results show that our networks can be trained to predict the solutions to a PDE defined over an entire set of shapes. We focus on the linear elasticity PDE and show how we can build a surrogate model that is able to predict displacements and stresses over a variety of freeform domains. To assess the efficacy of all networks in this work, data efficiency, network accuracy, and the capacity of networks to extrapolate are considered and compared between NNs and PINNs. The comparison includes cases where little training data is available. Transfer learning and applications to shape optimization are analyzed as well. A selection of the used codes and datasets is provided at https://github.com/fmezzadri/shape_parameterized.git.

Implementation of variational iteration method for various types of linear and nonlinear partial differential equations

There are various linear and nonlinear one-dimensional partial differential equations that are the focus of this research. There are a large number of these equations that cannot be solved analytically or precisely. The evaluation of nonlinear partial differential equations, even if analytical solutions exist, may be problematic. Therefore, it may be necessary to use approximate analytical methodologies to solve these issues. As a result, a more effective and accurate approach must be investigated and analyzed. It is shown in this study that the Lagrange multiplier may be used to get an ideal value for parameters in a functional form and then used to construct an iterative series solution. Linear and nonlinear partial differential equations may both be solved using the variational iteration method (VIM) method, thanks to its high computing power and high efficiency. Decoding and analyzing possible Korteweg-De-Vries, Benjamin, and Airy equations demonstrates the method’s ability. With just a few iterations, the produced findings are very effective, precise, and convergent to the exact answer. As a result, solving nonlinear equations using VIM is regarded as a viable option.

On a Whitham-Broer-Kaup-like system arising in the oceanic shallow water

Presently, taking care of the oceanic shallow water, people have investigated some Kaup-type systems. For example, a Whitham-Broer-Kaup-like system, which we aim to study hereby, models the dispersive long waves in the oceanic shallow water. As for the horizontal velocity of the water wave and height of the deviation from the equilibrium position of the water, using symbolic computation, we build up (1) a scaling transformation, (2) two hetero-Bäcklund transformations, each of which is from that system to a known linear partial differential equation, and (3) a set of the similarity reductions, from that system to a known ordinary differential equation with some sample solutions also presented. Our results rely on different dispersion/diffusion powers for that system, with respect to the oceanic shallow water.

Approximating the Stationary Bellman Equation by Hierarchical Tensor Products

We treat infinite horizon optimal control problems by solving the associated stationary Hamilton-Jacobi-Bellman (HJB) equation numerically to compute the value function and an optimal feedback law. The dynamical systems under consideration are spatial discretizations of non linear parabolic partial differential equations (PDE), which means that the HJB is non linear and suffers from the curse of dimensionality. Its non linearity is handled by the Policy Iteration algorithm, where the problem is reduced to a sequence of linear, hyperbolic PDEs. These equations remain the computational bottleneck due to their high dimensions. By the method of characteristics these linearized HJB equations can be reformulated via the Koopman operator in the spirit of dynamic programming. The resulting operator equations are solved using a minimal residual method. To overcome numerical infeasability we use low rank hierarchical tensor product approximation/tree-based tensor formats, in particular tensor trains (TT tensors), and multi-polynomials, together with high dimensional quadrature, e.g. Monte-Carlo. By controlling a destabilized version of viscous Burgers and a diffusion equation with unstable reaction term numerical evidences are given.

On the hyperbolic type differential equation with time involution

In the present paper, the initial value problem for the hyperbolic type involutory in t second order linear partial differential equation is studied. The initial value problem for the fourth order partial differential equations equivalent to this problem is obtained. The stability estimates for the solution and its first and second order derivatives of this problem are established.

On the closed-loop stochastic dynamics of two-state nonlinear exothermic CSTRs with PI temperature control

Fokker-Planck (FP) partial differential equation (PDE) theory is applied to characterize the stochastic dynamics of a class of open-loop (OL) 2-state nonlinear exothermic continuous reactors with: (i) zero and time-varying mean noise disturbances, and (ii) linear proportional-integral (PI) temperature control. The characterization includes: (i) the stochastic on deterministic dynamics dependency, (ii) gain condition for robust probability density function (PDF) stability over deterministic-diffusion time biscale with stationary monomodality at prescribed most probable (MP) state, (iii) evolutions of along nearly deterministic time scale of MP state and control and their variabilities, (iv) attainment of random motion in-probability (IP) stability over deterministic-diffusion time biscale, and (v) identification of the compromise between MP state regulation speed, robustness, and control effort. The methodological developments and findings are illustrated with three indicative examples with OL complex (bimodal and vulcanoid) stationary state PDFs, including analytic assessment as well as state PDF and random motion numerical simulation.

BSDEs generated by fractional space-time noise and related SPDEs

This paper is concerned with the backward stochastic differential equations whose generator is a weighted fractional Brownian field: Yt=ξ+∫tTYsW(ds,Bs)−∫tTZsdBs, 0≤t≤T, where W is a (d+1)-parameter weighted fractional Brownian field of Hurst parameter H=(H0,H1,⋯,Hd), which provide probabilistic interpretations (Feynman-Kac formulas) for certain linear stochastic partial differential equations with colored space-time noise. Conditions on the Hurst parameter H and on the decay rate of the weight are given to ensure the existence and uniqueness of the solution pair. Moreover, the explicit expression for both components Y and Z of the solution pair is given.

A localized reduced basis approach for unfitted domain methods on parameterized geometries

This work introduces a reduced order modeling (ROM) framework for the solution of parameterized second-order linear elliptic partial differential equations formulated on unfitted geometries. The goal is to construct efficient projection-based ROMs, which rely on techniques such as the reduced basis method and discrete empirical interpolation. The presence of geometrical parameters in unfitted domain discretizations entails challenges for the application of standard ROMs. Therefore, in this work we propose a methodology based on (i) extension of snapshots on the background mesh and (ii) localization strategies to decrease the number of reduced basis functions. The method we obtain is computationally efficient and accurate, while it is agnostic with respect to the underlying discretization choice. We test the applicability of the proposed framework with numerical experiments on two model problems, namely the Poisson and linear elasticity problems. In particular, we study several benchmarks formulated on two-dimensional, trimmed domains discretized with splines and we observe a significant reduction of the online computational cost compared to standard ROMs for the same level of accuracy. Moreover, we show the applicability of our methodology to a three-dimensional geometry of a linear elastic problem.

Nonlinear H∞ Control Based on Successive Gaussian Process Regression

Nonlinear <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$H_{\infty }$ </tex-math></inline-formula> controllers can be designed based on the solution of the celebrated Hamilton-Jacobi-Isaacs (HJI) equation, which is a nonlinear partial differential equation (PDE) and generally hard to solve. In this brief, a concise but effective Gaussian process regression based method is proposed, and the solution of the HJI equation is obtained without explicitly discretizing the PDE. Based on the successive approximation technique, the nonlinear HJI PDE is transformed into a sequence of linear PDEs, and it is shown in this brief that solution to the linear PDE can be obtained by dealing with easily solvable initial value problems. Then a successive Gaussian process regression based method is designed, and the solution of the HJI equation is calculated by successively solving initial value problems and constructing regression models. Simulations are conducted and numerical results demonstrate its superior performance in accuracy and efficiency.

PDE-constrained models with neural network terms: Optimization and global convergence

Recent research has used deep learning to develop partial differential equation (PDE) models in science and engineering. The functional form of the PDE is determined by a neural network, and the neural network parameters are calibrated to available data. Calibration of the embedded neural network can be performed by optimizing over the PDE. Motivated by these applications, we rigorously study the optimization of a class of linear elliptic PDEs with neural network terms. The neural network parameters in the PDE are optimized using gradient descent, where the gradient is evaluated using an adjoint PDE. As the number of parameters become large, the PDE and adjoint PDE converge to a non-local PDE system. Using this limit PDE system, we are able to prove convergence of the neural network-PDE to a global minimum during the optimization. Finally, we use this adjoint method to train a neural network model for an application in fluid mechanics, in which the neural network functions as a closure model for the Reynolds-averaged Navier–Stokes (RANS) equations. The RANS neural network model is trained on several datasets for turbulent channel flow and is evaluated out-of-sample at different Reynolds numbers.

A local domain boundary element method for solving 2D incompressible fluid flow problems

The Boundary Element Method (BEM) is a well-established, accurate numerical method for solving engineering problems described by linear partial differential equations. However, the application of BEM to non-linear problems leads to the appearance of volume integrals in the integral formulation, and combined with the non-local character of the method the required computational effort is dramatically increased. In this work, a Local Domain BEM (LD-BEM) is proposed for solving incompressible fluid flow problems. The domain of interest is fragmented into small subdomains, in each of which the integral representation of the solution is considered separately. Eliminating the fluxes at all subdomain interfaces, the proposed LD-BEM leads to sparse linear system coefficient matrices and reduced computational complexity, overcoming some of the well-known disadvantages of the conventional BEM. The proposed method is applied to the solution of the well-known two-dimensional (2D) lid-driven cavity problem, for Reynolds numbers up to Re=125 × 103, with a discretization involving more than 10 million unknowns.