Numerical Solution of linear Second Order Partial Differential Equation

Finite Difference Method: A Brief Study

The scope of this book is to present well known simple and advanced numerical methods for solving partial differential equations (PDEs) and how to implement these methods using the programming environment of the software package Diffpack. A basic background in PDEs and numerical methods is required by the potential reader. Further, a basic knowledge of the finite element method and its implementation in one and two space dimensions is required. The authors claim that no prior knowledge of the package Diffpack is required, which is true, but the reader should be at least familiar with an object oriented programming language like C++ in order to better comprehend the programming environment of Diffpack. Certainly, a prior knowledge or usage of Diffpack would be a great advantage to the reader.

This article is concerned with a class of singularly perturbed semilinear parabolic convection‐diffusion partial differential equations (PDEs) with discontinuous source function. Solutions of these PDEs usually exhibit a weak interior layer at one side of the point of discontinuity along with a boundary layer at one side of the spatial domain. We begin our study by proving existence of the analytical solution of the considered nonlinear PDE by means of the upper and lower solutions approach; and the ‐uniform stability of the analytical solution is established by using the comparison principle for the continuous nonlinear operator. In order to realize the asymptotic behavior of the analytical solution, we derive a priori bounds of the solution derivatives via decomposition of the solution into the smooth and the layer components. For an efficient numerical solution of the nonlinear PDE, the time‐derivative is approximated by the Crank–Nicolson method on an equidistant mesh, and we approximate the spatial derivative by a finite difference scheme on a suitable layer‐adapted mesh. We establish the comparison principle for the nonlinear difference operator to prove the ‐uniform stability of the discrete solution and further construct a suitable decomposition of the discrete solution for pursuing the convergence analysis. The computational method is proven to be parameter‐robust with second‐order time accuracy in the discrete supremum norm. The theoretical estimate is finally verified by the numerical experiments.

In the present study, Modified Taylor wavelets Galerkin method-based approximation scheme is used to get out appropriate numerical solution of certain partial differential equations. The Modified Taylor wavelets are used with weight functions and these wavelets are supposed basis elements which permit the derivision of numerical solutions of partial differential equations. Some of the counter problems are given to show the numerical results extractive by proposed schemes which are compared with already established numerical methods, i.e., Coifman wavelet method, Finite Difference Method (FDM), Hermite Wavelet Galarkin Method (HWGM), and exact solution to set out the relevancy and efficiency of the introduced method.

In this paper we will discuss Euler’s theorem for homogenous functions to solve different order partial differential equations. We will see that how we can predict the solution of partial differential Equation using different approaches of this theorem. In fact we also consider the case when more than two independent variables will be involved in the partial differential equation whenever dependent functions will be homogenous functions. We will throw a light on one method called Ajayous rules to predict the solution of homogenous partial differential equation.

Partial differential equations (PDEs) usually apply for modeling complex physical phenomena in the real world, and the corresponding solution is the key to interpreting these problems. Generally, traditional solving methods suffer from inefficiency and time consumption. At the same time, the current rise in machine learning algorithms, represented by the Deep Operator Network (DeepONet), could compensate for these shortcomings and effectively predict the solutions of PDEs by learning the operators from the data. The current deep learning-based methods focus on solving one-dimensional PDEs, but the research on higher-dimensional problems is still in development. Therefore, this paper proposes an efficient scheme to predict the solution of two-dimensional PDEs with improved DeepONet. In order to construct the data needed for training, the functions are sampled from a classical function space and produce the corresponding two-dimensional data. The difference method is used to obtain the numerical solutions of the PDEs and form a point-value data set. For training the network, the matrix representing two-dimensional functions is processed to form vectors and adapt the DeepONet model perfectly. In addition, we theoretically prove that the discrete point division of the data ensures that the model loss is guaranteed to be in a small range. This method is verified for predicting the two-dimensional Poisson equation and heat conduction equation solutions through experiments. Compared with other methods, the proposed scheme is simple and effective.

The Rate of Convergence of Some Difference Schemes

In the past few years deep artificial neural networks (DNNs) have been successfully employed in a large number of computational problems including, e.g., language processing, image recognition, fraud detection, and computational advertisement. Recently, it has also been proposed in the scientific literature to reformulate high-dimensional partial differential equations (PDEs) as stochastic learning problems and to employ DNNs together with stochastic gradient descent methods to approximate the solutions of such high-dimensional PDEs. There are also a few mathematical convergence results in the scientific literature which show that DNNs can approximate solutions of certain PDEs without the curse of dimensionality in the sense that the number of real parameters employed to describe the DNN grows at most polynomially both in the PDE dimension d in {mathbb {N}} and the reciprocal of the prescribed approximation accuracy varepsilon > 0. One key argument in most of these results is, first, to employ a Monte Carlo approximation scheme which can approximate the solution of the PDE under consideration at a fixed space-time point without the curse of dimensionality and, thereafter, to prove then that DNNs are flexible enough to mimic the behaviour of the employed approximation scheme. Having this in mind, one could aim for a general abstract result which shows under suitable assumptions that if a certain function can be approximated by any kind of (Monte Carlo) approximation scheme without the curse of dimensionality, then the function can also be approximated with DNNs without the curse of dimensionality. It is a subject of this article to make a first step towards this direction. In particular, the main result of this paper, roughly speaking, shows that if a function can be approximated by means of some suitable discrete approximation scheme without the curse of dimensionality and if there exist DNNs which satisfy certain regularity properties and which approximate this discrete approximation scheme without the curse of dimensionality, then the function itself can also be approximated with DNNs without the curse of dimensionality. Moreover, for the number of real parameters used to describe such approximating DNNs we provide an explicit upper bound for the optimal exponent of the dimension d in {mathbb {N}} of the function under consideration as well as an explicit lower bound for the optimal exponent of the prescribed approximation accuracy varepsilon >0. As an application of this result we derive that solutions of suitable Kolmogorov PDEs can be approximated with DNNs without the curse of dimensionality.

Caustics and non-degenerate Hamiltonians

The theory of analytic systems of partial differential equations was first systematically investigated by Riquier and Elie Cartan around 1900. The existence of local solutions involves an algebraic problem, finding formal power series solutions, and an analytic problem, proving the convergence of formal power series solutions. Cartan defined the notion of an involutive system of partial differential equations and, using his theory of exterior differential systems, was able to show the existence of formal power series solutions for involutive partial differential equations of first order and to prove the convergence by the Cauchy-Kowalewski theorem. His result was extended by Kihler to systems of partial differential equations of higher order and is known today as the Cartan-Kihler theorem. Adjoining to a system of partial differential equations of order k the equations obtained by differentiating the original equations gives rise to a system of partial differential equations of order k +1, the prolongation of the system, which has the same solutions as the original equations. Cartan conjectured that, by prolonging a system a sufficient number of times, one would obtain an involutive system; this result was proved by Kuranishi in 1957 within the framework of Cartan's theory of exterior differential systems, and is now referred to as the Cartan-Kuranishi prolongation theorem. In 1961, Spencer introduced, in his fundamental paper [6] on the deformation of pseudogroup structures, certain cohomology groups Hkj associated to a partial differential equation (see ? 3), which are dual to homology groups of a Koszul complex; and so the cohomology groups Hki vanish for all sufficiently large k (see Lemma 3.1). The vanishing of these cohomology groups was shown by Serre to be equivalent to Cartan's notion of involutiveness (see V. W. Guillemin and S. Sternberg [3]). It then became possible to analyse the role played by involutiveness in Cartan's theory of partial differential equations. In this paper, we prove the Cartan-Kahler theorem for systems of linear partial differential equations formulated in terms of Ehresmann's theory of

OSCILLATORY SOLUTIONS OF PARTIAL DIFFERENTIAL AND DIFFERENCE EQUATIONS

Partial differential equations (PDEs) are fundamental in describing various physical phenomena, such as fluid dynamics, heat conduction, and wave propagation. However, analytical solutions to these equations are often difficult or impossible to obtain due to their complexity and the boundary conditions involved. Numerical methods provide an effective alternative by approximating solutions through discretization techniques. This paper explores various numerical methods for solving PDEs, including finite difference, finite element, and finite volume methods. We discuss their theoretical foundations, implementation strategies, and advantages in handling different types of PDEs, such as elliptic, parabolic, and hyperbolic equations. Moreover, the paper addresses key challenges such as stability, convergence, and computational efficiency, and reviews the use of high-performance computing in tackling large-scale problems. The applications of these methods in scientific computing and engineering are highlighted, demonstrating their versatility and importance in solving real-world problems.The numerical solution of partial differential equations (PDEs) plays a crucial role in solving real-world problems across various fields, including physics, engineering, and finance. Exact analytical solutions to PDEs are often not feasible due to their complexity and the nature of boundary conditions. As a result, numerical methods such as the finite difference, finite element, and finite volume methods are widely employed to approximate solutions. This paper provides an overview of these methods, emphasizing their formulation, implementation, and application to different types of PDEs, including elliptic, parabolic, and hyperbolic equations. Key considerations such as stability, convergence, and accuracy are discussed, along with strategies for improving computational efficiency. The paper also highlights the use of advanced computational techniques and parallel computing in addressing large-scale and complex PDE systems. Overall, numerical methods offer powerful tools for solving PDEs and are essential for simulating and analyzing complex phenomena in science and engineering.

Sylvester Equations and the numerical solution of partial fractional differential equations

The Radial Basis Function (RBF) method has been considered an important meshfree tool for numerical solutions of Partial Differential Equations (PDEs). For various situations, RBF with infinitely differentiable functions can provide accurate results and more flexibility in the geometry of computation domains than traditional methods such as finite difference and finite element methods. However, RBF does not suit large scale problems, and, therefore, a combination of RBF and the finite difference (RBF-FD) method was proposed because of its own strengths not only on feasibility and computational cost, but also on solution accuracy. In this study, we try the RBF-FD method on elliptic PDEs and study the effect of it on such equations with different shape parameters. Most importantly, we study the solution accuracy after additional ghost node strategy, preconditioning strategy, regularization strategy, and floating point arithmetic strategy. We have found more satisfactory accurate solutions in most situations than those from global RBF, except in the preconditioning and regularization strategies.

Partial differential equations are paramount in mathematical modelling with applications in engineering and science. The book starts with a crash course on partial differential equations in order to familiarize the reader with fundamental properties such as existence, uniqueness and possibly existing maximum principles. The main topic of the book entails the description of classical numerical methods that are used to approximate the solution of partial differential equations. The focus is on discretization methods such as the finite difference, finite volume and finite element method. The manuscript also makes a short excursion to the solution of large sets of (non)linear algebraic equations that result after application of discretization method to partial differential equations. The book treats the construction of such discretization methods, as well as some error analysis, where it is noted that the error analysis for the finite element method is merely descriptive, rather than rigorous from a mathematical point of view. The last chapters focus on time integration issues for classical time-dependent partial differential equations. After reading the book, the reader should be able to derive finite element methods, to implement the methods and to judge whether the obtained approximations are consistent with the solution to the partial differential equations. The reader will also obtain these skills for the other classical discretization methods. Acquiring such fundamental knowledge will allow the reader to continue studying more advanced methods like meshfree methods, discontinuous Galerkin methods and spectral methods for the approximation of solutions to partial differential equations.