Solution Of PDE Research Articles

Asymptotic properties of PDEs in compact spaces

In this article we combine the study of solutions of PDEs with the study of asymptotic properties of the solutions via compactification of the domain. We define new spaces of functions on which study the equations, prove a version of Ascoli–Arzelà Theorem, develop the fixed point index results necessary to prove existence and multiplicity of solutions in these spaces and also illustrate the applicability of the theory with an example.

Extreme learning machine collocation for the numerical solution of elliptic PDEs with sharp gradients

We address a new numerical method based on machine learning and in particular based on the concept of the so-called Extreme Learning Machines, to approximate the solution of linear elliptic partial differential equations with collocation. We show that a feedforward neural network with a single hidden layer and sigmoidal transfer functions and fixed, random, internal weights and biases can be used to compute accurately enough a collocated solution for such problems. We discuss how one can set the range of values for both the weights between the input and hidden layer and the biases of the hidden layer in order to obtain a good underlying approximating subspace, and we explore the required number of collocation points. We demonstrate the efficiency of the proposed method with several one-dimensional diffusion–advection–reaction benchmark problems that exhibit steep behaviors, such as boundary layers. We point out that there is no need of iterative training of the network, as the proposed numerical approach results to a linear problem that can be easily solved using least-squares and regularization. Numerical results show that the proposed machine learning method achieves a good numerical accuracy, outperforming central Finite Differences, thus bypassing the time-consuming training phase of other machine learning approaches.

Spectral solutions of PDEs on networks

To solve linear PDEs on metric graphs with standard coupling conditions (continuity and Kirchhoff's law), we develop and compare a spectral, a second-order finite difference, and a discontinuous Galerkin method. The spectral method yields eigenvalues and eigenvectors of arbitrary order with machine precision and converges exponentially. These eigenvectors provide a Fourier-like basis on which to expand the solution; however, more complex coupling conditions require additional research. The discontinuous Galerkin method provides approximations of arbitrary polynomial order; however computing high-order eigenvalues accurately requires the respective eigenvector to be well-resolved. The method allows arbitrary non-Kirchhoff flux conditions and requires special penalty terms at the vertices to enforce continuity of the solutions. For the finite difference method, the standard one-sided second-order finite difference stencil reduces the accuracy of the vertex solution to O(h3/2). To preserve overall second-order accuracy, we used ghost cells for each edge. For all three methods we provide the implementation details, their validation, and examples illustrating their performance for the eigenproblem, Poisson equation, and the wave equation.

Temporal smoothing - A step forward for time-spectral methods

Time-spectral methods may feature substantial advantages over time-stepping solvers for solution of initial-value ODEs and PDEs, but their efficiency depends on the smoothness of the solution. We present two methods to overcome this problem. The first involves transforming the differential equation to an equation for a new variable, related to the time-integrated solution, before applying the solution algorithm. In the second method, a procedure for transformation to exact differential equations of a running average is outlined. Examples of solution of stiff problems and problems with multiple time scales are presented, employing the time-spectral Generalized Weighted Residual Method (GWRM). It is found that the smoothing algorithms have a significant positive effect on convergence.

Solution of Parabolic Partial Differential Equations by Non-Polynomial Cubic Spline Technique

Parabolic partial differential equation having a great impact on our scientific, engineering and technology. Enormous research have been conducted for the solution of parabolic PDEs. . In this research work, we introduced a novel technique for the numerical solution of fourth order PDEs. The novel technique is based upon the polynomial cubic cutting method (PCSM) was used with Adomian breakdown technique (ADM).The constraint for the alternative variables was decomposed by Edomian decomposition for the successive approximation. A numerical test problem of parabolic PDEs solved by purposed technique

Superconvergent HDG methods for Maxwell’s equations via the [formula omitted]-decomposition

The concept of the M-decomposition was introduced by Cockburn et al. (2017) to provide criteria to guarantee optimal convergence rates for the Hybridizable Discontinuous Galerkin (HDG) method for coercive elliptic problems. In that paper they systematically constructed superconvergent hybridizable discontinuous Galerkin (HDG) methods to approximate the solutions of elliptic PDEs on unstructured meshes. In this paper, we use the M-decomposition to construct HDG methods for the Maxwell’s equations on unstructured meshes in two dimension. In particular, we show the any choice of spaces having an M-decomposition, together with sufficiently rich auxiliary spaces, has an optimal error estimate and superconvergence even though the problem is not in general coercive. Motivated by the elliptic case, we obtain a superconvergent rate for the curl and flux of the solution, and this is confirmed by our numerical experiments.

Solutions of Hyperbolic Stochastic PDEs on Bounded and Unbounded Domains

We treat several classes of hyperbolic stochastic partial differential equations in the framework of white noise analysis, combined with Wiener–Itô chaos expansions and Fourier integral operator methods. The input data, boundary conditions and coefficients of the operators are assumed to be generalized stochastic processes that have both temporal and spatial dependence. We prove that the equations under consideration have unique solutions in the appropriate Sobolev–Kondratiev or weighted-Sobolev–Kondratiev spaces. Moreover, an explicit chaos form of the solutions is obtained, useful for calculating expectations, variances and higher order moments of the solution.

Remarks on Regularized Stokeslets in Slender Body Theory

We remark on the use of regularized Stokeslets in the slender body theory (SBT) approximation of Stokes flow about a thin fiber of radius ϵ>0. Denoting the regularization parameter by δ, we consider regularized SBT based on the most common regularized Stokeslet plus a regularized doublet correction. Given sufficiently smooth force data along the filament, we derive L∞ bounds for the difference between regularized SBT and its classical counterpart in terms of δ, ϵ, and the force data. We show that the regularized and classical expressions for the velocity of the filament itself differ by a term proportional to log(δ/ϵ); in particular, δ=ϵ is necessary to avoid an O(1) discrepancy between the theories. However, the flow at the surface of the fiber differs by an expression proportional to log(1+δ2/ϵ2), and any choice of δ∝ϵ will result in an O(1) discrepancy as ϵ→0. Consequently, the flow around a slender fiber due to regularized SBT does not converge to the solution of the well-posed slender body PDE which classical SBT is known to approximate. Numerics verify this O(1) discrepancy but also indicate that the difference may have little impact in practice.

Simulations of nonlinear parabolic PDEs with forcing function without linearization

Abstract This paper aims at producing numerical solutions of nonlinear parabolic PDEs with forcing term without any linearization. Since the linearization of nonlinear term leads to lose real features, without doing linearization, this paper focuses on capturing natural behaviour of the mechanism. Therefore we concentrate on analysis of the physical processes without losing their properties. To carry out this study, a backward differentiation formula in time and a spline method in space have been combined in leading to the discretized equation. This method leads to a very reliable alternative in solving the problem by conserving the physical properties of the nature. The efficiency of the present method are proved theoretically and illustrated by various numerical tests.

Approximate solutions of vector fields and an application to Denjoy–Carleman regularity of solutions of a nonlinear PDE

AbstractIn this paper we study microlocal regularity of a ‐solution u of the equation where is ultradifferentiable in the variables and holomorphic in the variables . We proved that if is a regular Denjoy–Carleman class (including the quasianalytic case) then: where is the Denjoy–Carleman wave‐front set of u and is the characteristic set of the linearized operator :

The local meshless collocation method for numerical simulation of shallow water waves based on generalized equal width (GEW) equation

The current paper concerns to develop an efficient and robust numerical technique to solve the shallow water equation based on the generalized equal width (GEW) model. The considered model i.e. the generalized equal width (GEW) equation is a PDE that it can be classified in the category of hyperbolic PDEs. The solution of hyperbolic PDEs is similar to a fixed or moving wave. Thus, for solving these problems, a suitable numerical procedure that its basis functions are similar to a flat or shape wave should be selected. For this aim, the local collocation method via two different basis functions is utilized. First, the space derivative is approximated by the local collocation procedure that this manner yields a system of nonlinear ODEs depends on the time variable. Furthermore, the constructed system of ODEs is solved by a fourth-order algorithm to get high-numerical results. The mentioned process is applied on several test problems to verify the efficiency of the numerical formulation.

Unbiased Deep Solvers for Linear Parametric PDEs

We develop several deep learning algorithms for approximating families of parametric PDE solutions. The proposed algorithms approximate solutions together with their gradients, which in the context of mathematical finance means that the derivative prices and hedging strategies are computed simultaneously. Having approximated the gradient of the solution, one can combine it with a Monte Carlo simulation to remove the bias in the deep network approximation of the PDE solution (derivative price). This is achieved by leveraging the Martingale Representation Theorem and combining the Monte Carlo simulation with the neural network. The resulting algorithm is robust with respect to the quality of the neural network approximation and consequently can be used as a black box in case only limited a-priori information about the underlying problem is available. We believe this is important as neural network-based algorithms often require fair amount of tuning to produce satisfactory results. The methods are empirically shown to work for high-dimensional problems (e.g., 100 dimensions). We provide diagnostics that shed light on appropriate network architectures.

Singular integrals in quantum Euclidean spaces

We shall establish the core of singular integral theory and pseudodifferential calculus over the archetypal algebras of noncommutative geometry: quantum forms of Euclidean spaces and tori. Our results go beyond Connes’ pseudodifferential calculus for rotation algebras, thanks to a new form of Calderón-Zygmund theory over these spaces which crucially incorporates nonconvolution kernels. We deduceLpL_p-boundedness and Sobolevpp-estimates for regular, exotic and forbidden symbols in the expected ranks. In theL2L_2level both Calderón-Vaillancourt and Bourdaud theorems for exotic and forbidden symbols are also generalized to the quantum setting. As a basic application of our methods, we proveLpL_p-regularity of solutions for elliptic PDEs.

SOLUTION OF NEWEL-WHITEHEAD-SEGEL EQUATION USING CONFORMABLE FRACTIONAL SUMUDU DECOMPOSITION METHOD

In the present paper, a numerical method is proposed to solve the time fractional Newel-Whitead-Segel equation subject to initial condition. This method is based d on the unification of conformable Sumudu transform (CST) and Adomian decomposition method (ADM), and then it is used to find the analytical solutions of linear-nonlinear fractional PDE’s. The test examples are given for illustration

Rank-Adaptive Tensor Methods for High-Dimensional Nonlinear PDEs

We present a new rank-adaptive tensor method to compute the numerical solution of high-dimensional nonlinear PDEs. The method combines functional tensor train (FTT) series expansions, operator splitting time integration, and a new rank-adaptive algorithm based on a thresholding criterion that limits the component of the PDE velocity vector normal to the FTT tensor manifold. This yields a scheme that can add or remove tensor modes adaptively from the PDE solution as time integration proceeds. The new method is designed to improve computational efficiency, accuracy and robustness in numerical integration of high-dimensional problems. In particular, it overcomes well-known computational challenges associated with dynamic tensor integration, including low-rank modeling errors and the need to invert covariance matrices of tensor cores at each time step. Numerical applications are presented and discussed for linear and nonlinear advection problems in two dimensions, and for a four-dimensional Fokker–Planck equation.

Physics Informed Neural Networks (PINNs) For Approximating Nonlinear Dispersive PDEs

We propose a novel algorithm, based on physics-informed neural networks (PINNs) to efficiently approximate solutions of nonlinear dispersive PDEs such as the KdV-Kawahara, Camassa-Holm andBenjamin-Ono equations. The stability of solutions of these dispersive PDEs is leveraged to prove rigorous bounds on the resulting error. We present several numerical experiments to demonstrate thatPINNs can approximate solutions of these dispersive PDEs very accurately

A hybrid MGA-MSGD ANN training approach for approximate solution of linear elliptic PDEs

We introduce a hybrid “Modified Genetic Algorithm-Multilevel Stochastic Gradient Descent” (MGA-MSGD) training algorithm that considerably improves accuracy and efficiency of solving 3D mechanical problems described, in strong-form, by PDEs via ANNs (Artificial Neural Networks). This presented approach allows the selection of a number of locations of interest at which the state variables are expected to fulfil the governing equations associated with a physical problem. Unlike classical PDE approximation methods such as finite differences or the finite element method, there is no need to establish and reconstruct the physical field quantity throughout the computational domain in order to predict the mechanical response at specific locations of interest. The basic idea of MGA-MSGD is the manipulation of the learnable parameters’ components responsible for the error explosion so that we can train the network with relatively larger learning rates which avoids trapping in local minima. The proposed training approach is less sensitive to the learning rate value, training points density and distribution, and the random initial parameters. The distance function to minimise is where we introduce the PDEs including any physical laws and conditions (so-called, Physics Informed ANN). The Genetic algorithm is modified to be suitable for this type of ANN in which a Coarse-level Stochastic Gradient Descent (CSGD) is exploited to make the decision of the offspring qualification. Employing the presented approach, a considerable improvement in both accuracy and efficiency, compared with standard training algorithms such as classical SGD and Adam optimiser, is observed. The local displacement accuracy is studied and ensured by introducing the results of Finite Element Method (FEM) at sufficiently fine mesh as the reference displacements. A slightly more complex problem is solved ensuring the feasibility of the methodology.

Integration of neural networks with numerical solution of PDEs for closure models development

Modeling often requires closures to account for the multiscale/multiphysics nature of certain phenomena. Recently, there has been interest in the application of machine learning (ML) for their development. Most of the applications are purely data-driven; however, incorporation of the knowledgebase is an opportunity to enhance flexibility and predictive capability of ML models. This paper presents a PDE-integrated ML framework. PDEs are solved using convolutional operators and integrated with neural networks (NNs). Such integration allows one to train the NNs directly on observed field variables. To demonstrate the framework's viability, NNs are integrated with heat conduction, Navier-Stokes, and RANS equations.

Construction of new traveling and solitary wave solutions of a nonlinear PDE characterizing the nonlinear low-pass electrical transmission lines

In this study, we intend to analyze the traveling and several other solitary wave solutions in the nonlinear low-pass electrical transmission line model using the new mapping method, the new extended auxiliary equation method, and the extended Kudryashov method. A type of traveling and solitary wave solutions emerge, consisting of hyperbolic function, trigonometric, rational, periodic, and doubly periodic solutions that reflect kink, anti-kink wave solitons, bright-dark optical solitons, singular solitons, and other traveling waves. The three integration techniques applied are efficient, effective, and versatile for the creation of new bright, dark, singular, and non-singular periodic and solitary wave propagation solutions in nonlinear low-pass electrical transmission lines. To see the extant physical significance of the considered equation, we present some 2D and 3D figures for some solutions. We compare the obtained results with those obtained in the literature. We investigate and demonstrate the stability of the soliton solutions.

Comments on the paper: “On the solution of a parabolic PDE involving a gas flow through a semi-infinite porous medium” by Daniel N. Pop et al. [Results Physics 22 (2021) 103884

The aim of this comment paper is to show that the paper (Pop et al., 2021) contains many serious errors.