Kinds Of Partial Differential Equations Research Articles

An adaptive energy-based sequential method for training PINNs to solve gradient flow equations

Recent developments in numerous scientific computing fields have been made possible by physics-informed neural networks (PINNs), a class of machine learning techniques that incorporate physical knowledge into neural networks. However, PINNs still encounter difficulties in simulating a kind of partial differential equations (PDEs), i.e. gradient flow equations of complex dynamic systems. Through analysis from the neural tangent kernel's (NTK) point of view, we find that directly training PINNs on the large temporal window may lead to failure. To tackle this problem, we utilize one of the gradient flow equations' significant physical properties, the energy dissipation law, to propose an adaptive energy-based sequential training method for PINNs. Specifically, the gradient of free energy is employed to determine the adaptive temporal steps of the training window, which also enhances the training efficiency. Numerical experiments are demonstrated among gradient flow equations including Cahn-Hilliard, heat conduction, and Allen-Cahn equations. Our method not only outperforms PINNs but also preserves the energy dissipation law, which shows the ability to solve complex long-time simulation problems of gradient flow systems.

Applications of Partial Differential Equations in Fluid Physics

Partial differential equations, or PDEs, assume a critical part in grasping and outlining different fluid physics peculiarities. They have an expansive scope of utilizations, from expecting weather patterns to consolidating ocean streams, fire cycles, and fluid streams into system plan. These equations oversee the way of behaving of fluid amounts like as speed, stress, temperature, and consistency. They portray complex collaborations like changes in precipitation, scattering, and fluid-solid associations. Partial differential equations are utilized to apply the developing methodology. The arrangement is equivalent to for the recently concentrated on examples of typical differential equations. There are two kinds of partial differential equations: nonlinear and straight. Some certifiable equations, for example, those in electrostatics, heat conduction, transmission lines, quantum mechanics, and wave hypothesis, feature the significance of partial differential equations (PDEs). To make sense of something other than one, two, or three pieces of the partial differential equations, we will check out at the speculative piece of those applications that utilization PDEs in this examination. In all parts of science and development, partial differential equations, or PDEs, are generally utilized. Partial differential equations handle most of genuine frameworks. A condition communicating a connection between a piece of no less than two free factors and the partial helpers of this cutoff concerning these free factors is known as a partial differential condition, or PDE.

A numerical method for solving uncertain wave equation

Partial differential equations are usually used to deal with natural or physical problems, and wave equations are an important branch of them. Considering the fluctuation phenomena with uncertain noise in real life, this paper will focus on the study of the uncertain wave equation. The uncertain wave equation is a special kind of partial differential equation excited by the Liu process, which is widely used to describe wave propagation phenomena. However, not all uncertain wave equations can be solved analytically, so numerical solutions are of particular importance. The main objective of this paper is to study the numerical solution of uncertain wave equations. We first introduce the concept of α-path and, based on it, prove an important formula that reveals the connection between the uncertain wave equation and the classical wave equation. Based on this finding, we design a new numerical solution method to solve the uncertain wave equation. To verify the feasibility and accuracy of the method, we conduct numerical experiments. In addition, as an application, we obtain the uncertainty distribution, the inverse uncertainty distribution and the expected value of the solution of the equation based on the important formula. The originality and novelty of the conclusions of this work are demonstrated by comparing the results with those of other researchers.

Exact and Approximate Solutions for Linear and Nonlinear Partial Differential Equations via Laplace Residual Power Series Method

The Laplace residual power series method was introduced as an effective technique for finding exact and approximate series solutions to various kinds of differential equations. In this context, we utilize the Laplace residual power series method to generate analytic solutions to various kinds of partial differential equations. Then, by resorting to the above-mentioned technique, we derive certain solutions to different types of linear and nonlinear partial differential equations, including wave equations, nonhomogeneous space telegraph equations, water wave partial differential equations, Klein–Gordon partial differential equations, Fisher equations, and a few others. Moreover, we numerically examine several results by investing some graphs and tables and comparing our results with the exact solutions of some nominated differential equations to display the new approach’s reliability, capability, and efficiency.

Solving the partial differential equations by using SEJI integral transform

The integral transformations have become very important in solving differential equations, especially those that have an applied aspect in the physical, chemical, economic and other scientific fields. In this article, the suggested transform which named by SEJI integral transform is developed for solving the partial derivatives and applied to solve several partial differential equations. This paper introduced the definition of the suggested transform with its properties that have the importance in this field. This new transform has an ability to applying in {PDEs) and finding the exact solution for these equations. Kinds of partial differential equations have solved by using this transform.

Physics and equality constrained artificial neural networks: Application to forward and inverse problems with multi-fidelity data fusion

Physics-informed neural networks (PINNs) have been proposed to learn the solution of partial differential equations (PDE). In PINNs, the residual form of the PDE of interest and its boundary conditions are lumped into a composite objective function as soft penalties. Here, we show that this specific way of formulating the objective function is the source of severe limitations in the PINN approach when applied to different kinds of PDEs. To address these limitations, we propose a versatile framework based on a constrained optimization problem formulation, where we use the augmented Lagrangian method (ALM) to constrain the solution of a PDE with its boundary conditions and any high-fidelity data that may be available. Our approach is adept at forward and inverse problems with multi-fidelity data fusion. We demonstrate the efficacy and versatility of our physics- and equality-constrained deep-learning framework by applying it to several forward and inverse problems involving multi-dimensional PDEs. Our framework achieves orders of magnitude improvements in accuracy levels in comparison with state-of-the-art physics-informed neural networks.

A novel kind of equations linking the quantum dynamics and the classical wave motions based on the catastrophe theory

Considering that the catastrophe theory could describe quantitatively any phase transition process, we adopt the folding and cusp catastrophe types as the potential functions in the Schrödinger equation to attempt to link the quantum dynamics and the classical wave motions. Thus, through the dimensionless analysis a novel kind of partial differential equations is derived out. When the scaling parameter of the novel equation is equal to the Planck's constant, this equation becomes a detailed time-independent Schrödinger equation, from which Bohr correspondence principle can be found. On the other hand, when the scaling parameter tends to zero, this equation could degenerate to the classical Helmholtz equation. Therefore, this novel kind of equations could describe quantitatively the variation process of the wave functions from the macroscopic level to the quantum size.

Traveling Wave Solutions of Partial Differential Equations Via Neural Networks

This paper focuses on how to approximate traveling wave solutions for various kinds of partial differential equations via artificial neural networks. A traveling wave solution is hard to obtain with traditional numerical methods when the corresponding wave speed is unknown in advance. We propose a novel method to approximate both the traveling wave solution and the unknown wave speed via a neural network and an additional free parameter. We proved that under a mild assumption, the neural network solution converges to the analytic solution and the free parameter accurately approximates the wave speed as the corresponding loss tends to zero for the Keller–Segel equation. We also demonstrate in the experiments that reducing loss through training assures an accurate approximation of the traveling wave solution and the wave speed for the Keller–Segel equation, the Allen–Cahn model with relaxation, and the Lotka–Volterra competition model.

Source recovery with a posteriori error estimates in linear partial differential equations

Abstract We consider inverse problems of recovering a source term in initial boundary value problems for linear multidimensional partial differential equations (PDEs) of a general form. A universal stable method suitable for solving such inverse problems is proposed. The method allows one to obtain in the same way approximations to exact sources in different kinds of PDEs using various types of linear supplementary conditions specified with an error. The method is suitable for both spacewise dependent and time-dependent sources. The method consists in preliminary calculation of a special matrix introduced in the article, the matrix of the source inverse problem, and then inverting it using Tikhonov regularization. The matrix can be obtained by solving a number of initial boundary value problems in question with sources in the form of basis functions. Having spent some time for preliminary finding the matrix (for example, by finite element method with a sufficiently detailed grid), we can then use this matrix to quickly solve the inverse problem with various data. The same technique can be applied to solve inverse source problems in linear steady-state PDEs. We also propose an a posteriori error estimation method for the obtained approximate solution and give a numerical algorithm for such estimation. In addition, a relationship is established between the posterior estimate and the lower estimate for the optimal accuracy of solving the inverse problem. The proposed method of solving inverse source problems is illustrated by the numerical solution of model examples for one-dimensional and two-dimensional PDEs of different kinds with a posteriori error estimates.

On the fractional version of Leibniz rule

AbstractThis manuscript is dedicated to prove a new inequality that involves an important case of Leibniz rule regarding Riemann–Liouville and Caputo fractional derivatives of order . In the context of partial differential equations, the aforesaid inequality allows us to address the Faedo–Galerkin method to study several kinds of partial differential equations with fractional derivative in the time variable; particularly, we apply these ideas to prove the existence and uniqueness of solution to the fractional version of the 2D unsteady Stokes equations in bounded domains.

Discontinuous Legendre Wavelet Element Method for Reaction–Diffusion Equation from Mathematical Chemistry

This paper presents discontinuous Legendre wavelet element (DLWE) approach for solving nonlinear reaction–diffusion equation (RDE) arising in mathematical chemistry. Firstly, weak formulation of the RDE and corresponding numerical fluxes are devised by utilizing the advantages of both Legendre wavelet and discontinuous Galerkin (DG) approach. Secondly, stability and error estimates of the proposed method have been addressed. Finally, numerical experiments demonstrate the validity and utility of the DLWE method, which is also applicable to solving some other kinds of partial differential equations.

Pseudospectral Meshless Radial Point Hermit Interpolation Versus Pseudospectral Meshless Radial Point Interpolation

This paper develops pseudospectral meshless radial point Hermit interpolation (PSMRPHI) and pseudospectral meshless radial point interpolation (PSMRPI) in order to apply to the elliptic partial differential equations (PDEs) held on irregular domains subject to impedance (convective) boundary conditions. Elliptic PDEs in simplest form, i.e., Laplace equation or Poisson equation, play key role in almost all kinds of PDEs. On the other hand, impedance boundary conditions, from their application in electromagnetic problems, or convective boundary conditions, from their application in heat transfer problems, are nearly more complicated forms of the boundary conditions in boundary value problems (BVPs). Based on this problem, we aim also to compare PSMRPHI and PSMRPI which belong to more influence type of meshless methods. PSMRPI method is based on a combination of meshless methods and spectral collocation techniques. The point interpolation method with the help of radial basis functions is used to construct shape functions which act as basis functions in the frame of PSMRPI and PSMRPHI methods. While the latter one has been rarely used in applications, we observe that is more accurate and reliable than PSMRPI method.

Moving least squares particle hydrodynamics method for Burgers’ equation

In this paper, a meshless method named Moving Least Squares Particle Hydrodynamics (MLSPH) method is applied to obtain numerical solutions of the Burgers’ equation with a set of initial and boundary conditions. The influence of kernel function and smoothing length on the accuracy of MLSPH method is investigated. Several various forms of one- and two- dimensional Burgers’ equations are successfully calculated. It is concluded that the proposed method can obtain second-order accuracy. The way of non-uniform distribution is applied to present the desired performance near the shocks. The numerical results obtained by the proposed method are compared with the analytical solutions and the results obtained by some other methods. The satisfactorily small errors indicate the high precision of the proposed method to solve this kind of partial differential equation.

Traveling and Shock Wave Simulations in A Viscous Burgers’ Equation with Periodic Boundary Conditions

A fourth order numerical method based on cubic B-spline functions has been proposed to solve the periodic Burgers’ equation. The method is capable of capturing the physical behavior of the Burgers’ equation very efficiently. Obtained solutions confirm that the Burgers’ equation sets a balance between nonlinearity and diffusion. The term $$u u_{x}$$ produces a shocking up effect that causes waves to break while the diffusion term $$u_{xx}$$ helps to smooth out shocks. As the diffusion term decreases, the smooth viscous solutions tend to become discontinuous shock waves. Approximate solutions are in good agreement with the earlier studies. Four well known problems of the periodic Burgers’ equation have been considered to test the efficiency of the method. With small modifications, method can be employed to solve different kind of partial differential equations arising in various areas of science and engineering.

An approach for robust PDE-constrained optimization with application to shape optimization of electrical engines and of dynamic elastic structures under uncertainty

We present a robust optimization framework that is applicable to general nonlinear programs (NLP) with uncertain parameters. We focus on design problems with partial differential equations (PDE), which involve high computational cost. Our framework addresses the uncertainty with a deterministic worst-case approach. Since the resulting min–max problem is computationally intractable, we propose an approximate robust formulation that employs quadratic models of the involved functions that can be handled efficiently with standard NLP solvers. We outline numerical methods to build the quadratic models, compute their derivatives, and deal with high-dimensional uncertainties. We apply the presented approach to the parametrized shape optimization of systems that are governed by different kinds of PDE and present numerical results.

An extrapolation full multigrid algorithm combined with fourth-order compact scheme for convection\u2013diffusion equations

In this paper, we propose an extrapolation full multigrid (EXFMG) algorithm to solve the large linear system arising from a fourth-order compact difference discretization of two-dimensional (2D) convection diffusion equations. A bi-quartic Lagrange interpolation for the solution on previous coarser grid is used to construct a good initial guess on the next finer grid for V- or W-cycles multigrid solver, which greatly reduces the number of relaxation sweeps. Instead of performing a fixed number of multigrid cycles as used in classical full multigrid methods, a series of grid level dependent relative residual tolerances is introduced to control the number of the multigrid cycles. Once the fourth-order accurate numerical solutions are obtained, a simple method based on the midpoint extrapolation is employed for the fourth-order difference solutions on two-level grids to construct a sixth-order accurate solution on the entire fine grid cheaply and directly. Numerical experiments are conducted to verify that the proposed method has much better efficiency compared to classical multigrid methods. The proposed EXFMG method can also be extended to solve other kinds of partial differential equations.

Implicit numerical techniques for Fisher equation

A very common known equations which is having higher level of significance in biology, chemistry, ecology and heat transfer is Fisher equation. Fisher equation is described as a KPP equation, Kolmogorov Petrovsky-Piscounov or Fisher KPP equation in mathematics. Fisher equation is a kind of partial differential equation which is nonlinear in nature of second order with nonlinear source term and this PDE have combined effects of interaction in between reactions and diffusions involved in physical phenomenon. Two implicit methods have been suggested in the paper presented to get the solution of Fisher equation. This methods are based on a semi implicit finite difference scheme. First scheme is modified CN method and second one is modified Keller Box and both the designed schemes having accuracy of second order in both time and space. The one major advantage of the designed schemes is that both are unconditionally stable. Numerical results have been calculated for the mentioned example at different values of coefficient of diffusion at various time and space steps which gives satisfactory results with exact solution.

The Hunter‐Saxton Equation: A Numerical Approach Using Collocation Method

In this study, we are going to present an overview on the Hunter‐Saxton equation which is a famous equation modelling waves in a massive director field of a nematic liquid crystal. The collocation finite element method is based on quintic B‐spline basis for obtaining numerical solutions of the equation. Using this method, after discretization, solution of the equation expressed as linear combination of shape functions and B‐spline basis. So, Hunter‐Saxton equation converted to nonlinear ordinary differential equation system. With the aid of the error norms and , some comparisons are presented between numeric and exact solutions for different step sizes. As a result, the authors observed that the method is a powerful, suitable and reliable numerical method for solving various kind of partial differential equations.

The BDF orthogonal spline collocation method for the two-dimensional evolution equation with memory

ABSTRACTIn this paper, a second-order backward differentiation formula (BDF) orthogonal spline collocation (OSC) method with the truncation error of order in time and in space is presented for a kind of partial differential equation (PDE) with memory. The stability and convergence of the BDF OSC method in a new norm are proved. Besides, we provide three numerical experiments to demonstrate the results of theoretical analysis and show the accuracy and effectiveness of the BDF OSC method. Numerical experiments also exhibit the optimal error estimates in the , and-norms.

Numerical solutions of nonlinear fractional Schrödinger equations using nonstandard discretizations

In this article, numerical study for both nonlinear space‐fractional Schrödinger equation and the coupled nonlinear space‐fractional Schrödinger system is presented. We offer here the weighted average nonstandard finite difference method (WANSFDM) as a novel numerical technique to study such kinds of partial differential equations. The space fractional derivative is described in the sense of the quantum Riesz‐Feller definition. Stability analysis of the proposed method is studied. To show that this method is reliable and computationally efficient different numerical examples are provided. We expect that the proposed schemes can be applicable to different systems of fractional partial differential equations. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 1399–1419, 2017