Interface solutions of partial differential equations with point singularity

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.

The numerical solution of optimization problems with partial differential equation (PDE) constraints is vital to a growing number of science and engineering applications. The development of robust and efficient algorithms for the solution of these optimization problems presents many challenges that arise out of, e.g., the intricate mathematical structure of these problems, the complicated interactions between numerical methods for PDE and optimization, the large-scale of the optimization problems, and the increasing complexity of applications. To identify and overcome these challenges an integrated approach is needed that builds on a variety of mathematical sub-disciplines, such as theory of PDEs, distributed parameter systems, numerical solution of PDEs, numerical optimization, and numerical linear algebra. This international workshop has brought together some of the leading experts in the fast developing field of optimization problems with PDE constraints to present recent developments in this area as well as to identify open problems and further research needs. Among the themes of this workshop were the design and analysis of approaches for the solution of PDE constrained optimization problems with additional point-wise constraints on controls and states (the solution of the governing PDE). State constrained problems are particularly challenging because of the low regularity properties of the Lagrange multipliers associated with point-wise constraints on the states. A second theme was the development of adaptive methods for the solution of PDE constrained optimization problems and, more generally, the development of optimization level model reduction techniques for these problems. The goal here is to develop models (through, e.g., mesh adaptation or proper orthogonal model reduction) of the PDE constrained optimization problems that capture the relevant features of the optimization problems with a specified accuracy, but involve as few degrees of freedom as possible and, hence, are computationally less expensive to work with. A third theme was The efficient solution of linear systems arising in optimization algorithms for discretized PDE constrained optimization problems represented another theme. Finally, a number of talks presented advances and challenges in the solution of PDE constrained optimization problems arising in important science and industrial applications.

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 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.

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.

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.

In scientific and engineering calculation, the effective solution of partial differential equations (PDEs) has great significance. This paper presents an innovative method based on the combination of a U-Net neural network with Fourier neural operators, aiming to improve the accuracy and efficiency of solving partial differential equations. U-Net neural networks with a unique encoding–decoder structure and hopping connections can efficiently extract and integrate spatial-domain features and accurately describe the spatial structure of PDEs. The Fourier neural operator combines the advantages of neural networks and Fourier transform to learn the mapping relationship of high-dimensional data by training the linear transformation parameters in the detailed frequency domain of the data. Fourier transform is used to explore and process the frequency-domain features and capture the frequency characteristics of the solution of partial differential equations. By combining the two, this method not only retains spatial details but also makes full use of frequency-domain information, which significantly reduces the dependence of the model on large-scale data and improves the generalization ability of the model. The experimental results show that compared with the traditional method, this method performs well in various complex partial differential equation (PDE) solving tasks and achieves high accuracy.

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

Simulating the propagation of 3D ultrasonic waves in a nonlinear medium is a demanding task. It requires the solution of time-dependent and nonlinear partial differential equations (PDEs). For such nonlinear PDEs, we have to use an implicit numerical method that is very CPU-intensive. Parallel simulation is therefore essential for studying those ultrasonic waves with satisfactory accuracy. We present the parallelization of an ultrasonic wave simulator and report some numerical experiments that have been run on a 24-node Linux cluster. Our CPU-measurements indicate that Linux clusters can deliver satisfactory parallel computing power for the numerical solution of PDEs. For simulating 3D ultrasonic waves in particular; we have found that our Linux cluster; which consists of 48 Pentium-III 500 MHz processors inter-connected with a 100 Mbit/s Ethernet network, is fully comparable with an SGI Origin 2000 machine.

Similarity solutions of nonlinear partial differential equations invariant to a family of affine groups

Moving boundary and boundary value problems occur in many physical and engineering processes involving heat transfer and phase changes, they range from icing problems, for example the Stefan problem, to the melting of metals and plastics using lasers. This book reviews and compares some well known and some recently developed methods to solve these problems. Recent advances in the finite difference solution of linear and non-linear partial differential equations are presented and algorithmic manipulations that enhance the computational efficiency are incorporated into the overall schemes. In this new approach, the the finite difference substitutions are made into the solutions of the partial differential equation rather than classically into the partial differential equation itself. This approach can be exploited for many types of partial differential equation and they may therefore be of interest to researchers studying a wide range of applications in physical problems such as diffusion and heat transfer equations, optoelectronic paraxial wave equations, the quantum mechanical wave equations, time dependent pulse propagation, and similar WKB-derived problemsM

The higher order displacement discontinuity method (HODDM) utilizing special crack tip elements has been used in the solution of linear elastic fracture mechanics (LEFM) problems. The paper has selected several example problems from the fracture mechanics literature (with available analytical solutions) including center slant crack in an infinite and finite body, single and double edge cracks, cracks emanating from a circular hole. The numerical values of Mode I and Mode II SIFs for these problems using HODDM are in excellent agreement with analytical results (reaching up to 0.001% deviation from their analytical results). The HODDM is also compared with the XFEM and a modified XFEM results. The results show that the HODDM needs a considerably lower computational effort (with less than 400 nodes) than the XFEM and the modified XFEM (which needs more than 10000 nodes) to reach a much higher accuracy. The proposed HODDM offers higher accuracy and lower computation effort for a wide range of problems in LEFM.

On the numerical solution of partial differential equations with two spatial scales

Numerical convective schemes based on accurate computation of space derivatives

The problem of checking existence of infinitely differentiable solutions of linear partial differential equations with zero boundary conditions is considered. The coefficients of the equations are assumed to be polynomials over Z in independent variables. It is proved that this problem is algorithmically undecidable. This result extends results of our earlier studies of analytic solutions. The proof relies on the result obtained by Denef and Lipshitz concerning the relationship between a certain subset of nonhomogeneous differential equations of the considered form (but without boundary conditions) and Diophantine equations.