A Solution Method for Partial Differential Equations by Fusing Fourier Operators and U-Net

We apply the homotopy perturbation method to obtain the solution of partial differential equations of fractional order. This method is powerful tool to find exact and approximate solution of many linear and nonlinear partial differential equations of fractional order. Convergence of the method is proved and the convergence analysis is reliable enough to estimate the maximum absolute truncated error of the series solution. The fractional derivatives are described in the Caputo sense. Some examples are presented to verify convergence hypothesis and simplicity of the method.

Interface solutions of partial differential equations with point singularity

Physics Informed Extreme Learning Machine (PIELM)–A rapid method for the numerical solution of partial differential equations

Aiming at the heat conduction equation, the heat conduction model is a very typical example of the partial differential equation. This paper focuses on the development and preliminary knowledge of the partial differential equation, the establishment of the heat conduction model and the solution of the heat conduction model. The partial differential equation with fractional Laplace operator is a kind of typical fractional partial differential equation, which has important applications in the fields of science and engineering. Fractional Laplace operator is a kind of nonlocal quasi-differential operator, which is the infinitesimal generator of L é vy steady-state process. It is essentially different from the classical Laplace operator, which leads to the disappearance of some classical properties, which generally brings difficulties to the study of this kind of problems. It is very difficult to solve the explicit solution of partial differential equation with fractional Laplace operator. In order to improve the accuracy and speed of solving partial differential equation, a high precision method for solving partial differential equation based on deep learning network is proposed. A multilayer radial basis function neural network is constructed and its structural layers are determined. By introducing subnetworks into each layer of the multilayer radial basis function neural network, a composite multilayer radial basis function neural network is constructed to improve the real function approximation performance and operation accuracy of the multilayer radial basis function neural network. The high-precision compound multi-layer radial basis function neural network is used to solve the partial differential equation. By giving a specific example of solving the partial differential equation, the solution effect of the method is tested. The results show that the method has very high solution accuracy under the four-layer network structure, which can improve about 1.5 orders of magnitude compared with the first layer. Under different number of training samples, it has higher solution accuracy and speed, and the comprehensive performance is superior. Therefore, it is a practical but challenging work to study the existence of solutions of partial differential equations with fractional Laplace operators. This paper mainly describes the research progress and trends of the existence of solutions of several classes of partial differential equations with fractional Laplace operators, including some of the work done by the author in this field in recent years.

Several research papers experimentally validated the derived theoretical solution of the partial differential equation (PDE) of Transient Land Drainage (TLD) using laboratory sand tanks, Hele-Shaw models, resistance networks, and field data. In this research, the main objective is to use an Arduino microcontroller linked with the LabVIEW software to derive experimentally-by using an RC analog circuit connected to the Arduino microcontroller - a solution to Kirchhoff’s partial differential equation describing the unsteady electrical flow through the RC circuit. The RC circuit electrically simulates a non-dimensional virtual aquifer for which an experimental Fourier sine series solution in terms of the analog components is to be derived and used for comparison with the theoretical Fourier sine series solution of the transient partial differential equation of land drainage. The derived Fourier sine series solution of the partial differential equation governing the unsteady electrical flow through the RC analog is written in a non-dimensional form to enable its general application to the case study of this research. Isochrones of the water table drawdown based on those derived from the analog solution are similar in trend and behavior to those of the theoretical solution, indicating that both Fourier solutions are physically and mathematically identical.

This paper explores the solutions of certain fractional partial differential equations using two methods; the first method involves separation of variables, which is a common technique for solving partial differential equations. However, since many equations cannot be separated in this way, the tensor product of Banach spaces method is applied to find the atomic solutions. To solve the resulting ordinary differential equations, the reproducing Kernel Hilbert space method is used to find numerical solutions, which are then used to find the numerical solution of the partial differential equation. The residual errors indicate that this method is effective and powerful. In summary, this paper presents a study on the solutions of certain fractional partial differential equations using two methods and demonstrates the effectiveness of these methods in finding numerical solutions.

The companion volume of this book, ‘Analytic methods for partial differential equations’, is concerned with solution of partial differential equations using classical methods which result in analytic solutions. These equations result when almost any physical situation is modelled, ranging from fluid mechanics problems, through electromagnetic problems to models of the economy. Some specific and fundamental problems were highlighted in the earlier volume, namely the three major classes of linear second order partial differential equations. The heat equation, the wave equation and Laplace’s equation will form a basis for study from a numerical point of view for the same reason as they did in the analytic case. That is, the three equations are the canonical forms to which any quasi-linear second order equation may be reduced using the characteristic transformation.KeywordsPartial Differential EquationCanonical FormSpectral RadiusLinear Algebraic EquationBackground MathematicThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

A comparison of differential quadrature methods for the solution of partial differential equations

In this article, a novel mesh‐free, moving Kriging (MK) based collocation scheme for the numerical solution of partial differential equations (PDEs) is introduced. In contrast to methods that are based on a Galerkin weak form of the governing PDEs, the MK collocation (MKC) approach, which is strong form based, is truly mesh‐free in the sense that no background mesh is required for numerical integration. In fact, the presented approach does not require the evaluation of any integrals. Since the approximation function in the MK framework can be conditioned on point value‐ as well as derivative‐information, the pointwise exact imposition of essential as well as natural boundary conditions is rendered straightforward. By incorporating an explicit linear basis into the MK framework, the first‐order consistency condition is fulfilled, and thus rigid body motions are captured accurately. Moreover, Kriging functions may be conceived that comply with constraints on higher order derivatives such as the PDE at hand at certain locations. This possibility proves useful in improving the solution accuracy in the vicinity of Dirichlet boundaries. This article provides a study of the method's characteristics by means of 2D linear elasticity examples. It concludes with a suggestion on how to apply MKC to nonlinear PDEs.

The section opens with a review well written by a giant in the field of numerical solution of partial differential equations, Roland Glowinski. In this featured review, he analyzes the book Think Before You Compute: A Prelude to Computational Fluid Dynamics, by E. J. Hinch, in much detail and contributes his own thoughts on appraisal or possibilities for improvement to every chapter of the book. The reviewer's verdict on the book is: “The book is a gem.” The following three reviewed books, written in an earlier and different world, take on a whole new relevance against the background of the current pandemic situation. First, we have the book by Irina Kareva and Georgy Karev, Modeling Evolution of Heterogeneous Populations: Theory and Applications. This book is diligently reviewed by Karen Page, who rates the book as “enlightening" even if it still contains small annoying errors. The following book, Delay-Adaptive Linear Control, by Yang Zhu and Miroslav Krstic, also seems to anticipate some aspects of the current modeling and control of the COVID-19 pandemic. This book is reviewed in detail by Bozenna Pasik-Duncan and is classified as an important contribution to control in delay systems. For the classification and mining of numerous medical data, such as those being generated right now, Mehrshad Sadria and Anita Layton suggest reading Bioinformatics, edited by Andreas D. Baxevanis, Gary D. Bader, and David S. Wishart, and recommend it as an “easy to read, greatly informative, and highly timely” book. Finally, I would like to mention the book by Zhendong Luo and Goong Chen on Proper Orthogonal Decomposition Methods for Partial Differential Equations, which is reviewed by the expert Stefan Volkwein, who praises the book as providing “a really good recipe for the application of the POD method to many nonstationary PDEs,” and also the book Basic Calculus of Planetary Orbits and Interplanetary Flight: The Missions of the Voyagers, Cassini, and Juno, by Alexander J. Hahn, that allows the reader to escape the current narrow earthly confinement in mathematical thoughts and delve into interplanetary explorations.

This work considers the partial solutions of partial differential equations for initial/boundary conditions using the Adomian decomposition method. The study formally shows that the partial solutions are always identical for all styles of boundary conditions. We also prove that the partial solution in the t-direction requires less computational work if compared with other partial solutions developed in any space variable direction. In addition, several mathematical models that govern the heat distribution and the wave propagation phenomenas have been tested, and the results obtained have shown that the t-solution minimizes the size of calculations if compared with the traditional techniques.

We investigate the existence and Borel summability of formal power series in one variable with holomorphic coefficients, solutions of nonlinear partial differential equations with shrinkings having non-Kowalewski type in the complex domain. Sufficient conditions are given in terms of the shape of the functional partial differential equations and initial conditions. The existence and asymptotic behavior of local sectorial holomorphic solutions of these nonlinear functional partial differential equations are obtained as a by-product. We apply the results to the study of asymptotic the behavior of solutions of some advanced-argument partial differential equations in a neighborhood of infinity.

The fractional partial differential equation is an equation with non-integer partial derivative which is now widely used in solving various problems so that in recent years, many researchers are interested to study fractional partial differential equation. The equation which is used in this paper is the non-homogeneous fractional partial differential equation. The aim of this study is to determine general solution of the partial differential equation through three different order equations (α; 1), (β; 2) and (α; β), where 0 < α ≤ 2 and 0 < β ≤ 1 by using Homotopy Analysis Method (HAM). Furthermore, by taking a convergent sequence, it will be analyzed that the convergence of the sequence of differential equations results in the sequence of solutions of partial fractional non-homogeneous partial differential equations converging to the solution function of the fractional partial differential equation.

In this paper, the homotopy analysis method is applied to solve linear fractional problems. Based on this method, a scheme is developed to obtain approximation solution of fractional wave, Burgers, Korteweg-de Vries (KdV), KdV-Burgers, and Klein-Gordon equations with initial conditions, which are introduced by replacing some integer-order time derivatives by fractional derivatives. The fractional derivatives are described in the Caputo sense. So the homotopy analysis method for partial differential equations of integer order is directly extended to derive explicit and numerical solutions of the fractional partial differential equations. The solutions are calculated in the form of convergent series with easily computable components. The results of applying this procedure to the studied cases show the high accuracy and efficiency of the new technique.

We conducted a six-month investigation of the design, analysis, and software implementation of a class of singularity-insensitive, scalable, parallel nonlinear iterative methods for the numerical solution of nonlinear partial differential equations. The solutions of nonlinear PDEs are often nonsmooth and have local singularities, such as sharp fronts. Traditional nonlinear iterative methods, such as Newton-like methods, are capable of reducing the global smooth nonlinearities at a nearly quadratic convergence rate but may become very slow once the local singularities appear somewhere in the computational domain. Even with global strategies such as line search or trust region the methods often stagnate at local minima of {parallel}F{parallel}, especially for problems with unbalanced nonlinearities, because the methods do not have built-in machinery to deal with the unbalanced nonlinearities. To find the same solution u* of F(u) = 0, we solve, instead, an equivalent nonlinearly preconditioned system G(F(u*)) = 0 whose nonlinearities are more balanced. In this project, we proposed and studied a nonlinear additive Schwarz based parallel nonlinear preconditioner and showed numerically that the new method converges well even for some difficult problems, such as high Reynolds number flows, when a traditional inexact Newton method fails.