Invariant Imbedding and Partial Differential Equations

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

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

The problem of parallel solution of partial differential equations with the help of the method of lines that ensures the reduction of the initial problem to the Cauchy problem described by a system of ordinary differential equations is considered. As a basic method, collocated multi-step block difference schemes are proposed. Obtaining a numerical solution in this case becomes possible only with the use of high-performance computing, usually with parallel architecture. This problem becomes especially important in the implementation of mathematical models based on systems of partial differential equations when it comes to the need for discretization of the solution search area, which can be significantly complicated by the geometric configuration of the boundaries. Solution of systems of this order cannot be obtained without involving multiprocessor computers. But by simply increasing the processing power, this problem cannot be solved. Only by combining the advantages of supercomputers and modern numerical simulation methods we can expect a significant improvement in the numerical solution of partial differential equations (PDEs).

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 Green’s function is widely used in solving boundary value problems for differential equations, to which many mathematical and physical problems are reduced. In particular, solutions of partial differential equations by the Fourier method are reduced to boundary value problems for ordinary differential equations. Using the Green's function for a homogeneous problem, one can calculate the solution of an inhomogeneous differential equation. Knowing the Green's function makes it possible to solve a whole class of problems of finding eigenvalues in quantum field theory. The developed method for constructing the Green’s function of boundary value problems for ordinary linear differential equations is described. An algorithm and program for calculating the Green's function of boundary value problems for differential equations of the second and third orders in an explicit analytical form are presented. Examples of computing the Green's function for specific boundary value problems are given. The fundamental system of solutions of ordinary differential equations with singular points needed to construct the Green's function is calculated in the form of generalized power series with the help of the developed programs in the Maple environment. An algorithm is developed for constructing the Green's function in the form of power series for second-order and third-order differential equations with given boundary conditions. Compiled work programs in the Maple environment for calculating the Green functions of arbitrary boundary value problems for differential equations of the second and third orders. Calculations of the Green's function for specific third-order boundary value problems using the developed program are presented. The obtained approximate Green’s function is compared with the known expressions of the exact Green’s function and very good agreement is found

A method for predicting the performance of packed columns that control gaseous air pollutants has been developed that exploits the advances in both computer software and hardware commonly used by practicing engineers. The solution of the simultaneous partial differential equations that describe the absorption process in packed columns that occurs in the presence of chemical reaction is obtained by converting the partial differential equations to systems of ordinary differential equations. These systems of ordinary differential equations are then solved using the method of lines along with a variable step, variable order numerical method. The method is applicable to systems in which there are multiple reactions within the liquid phase. The reactions can be of any order and can be reversible. The programming is simple and the machine running time is minimal. The method is illustrated here with an example.

Chaos and coherent structures in partial differential equations

A new Trapezoidal-type scheme is proposed for the direct numerical integration of time-dependent partial differential equations. The evolving system of ordinary differential equations after discretization is usually stiff, so it is desirable for the use of a numerical method in solving it to have good properties concerning stability. The method proposed in this article is L-stable and at least of algebraic order two. It is increasingly common to encounter partial differential equations (PDEs) posed on manifolds, and standard numerical methods are not available for such novel situations. Here, an L-Stable implicit Trapezoidal-like numerical integrator was developed for solving partial differential equations on manifolds. This approach allows the immediate use of familiar finite difference methods for the discretization and numerical solution. Presented here are the motivation and details of the method, illustration of its numerical convergence and stability properties for a general case. Numerical experiments illustrate the performance of the new method on different stiff systems of ODEs after discretizing in the space variable of some PDE problems. Results show accuracy of maximum error for various space and time steps. Key words: Trapezoidal-type method, partial differential equations, L-stability, implicit surfaces, manifolds, stiff systems.

In 1967 Roger Penrose introduced what is now called twistor geometry (Penrose 1967). This geometry is an amplification of the geometry of lines in projective space studied extensively by Felix Klein and others in the late 19th century. Penrose made the fundamental observation that this classical geometry could be used in conjunction with a Radon transform style integral geometry to provide rather complete sets of solutions of numerous partial differential equations, both linear and nonlinear, which arise naturally in theoretical physics. The Radon transform, which involves integration over straight lines in R 3, was used by F. John to generate solutions of hyperbolic partial differential equations in 1942. Penrose integrates over complex projective lines, real 2-spheres, and the data of the integration is complex-analytic functions of 3 complex variables. This is analogous to (and indeed related to) Weierstrass' representation of solutions of the minimal surface equation in terms of triples of holomorphic functions of a single complex variable which satisfy an algebraic relation. In this paper we discuss the origins of the differential equations which are amiable to twistor-geometric study from the realms of theoretical physics. Twistor geometry is most suitable for physical theories which are relativistically invariant, including Maxwell's description of electromagnetic phenomena from the 19th century (Maxwell's equations), Einstein's theory of gravitation (the equations of general relativity), and the contemporary descriptions of elementary particles (quantum field theory). Twistor geometry interacts in a nontrivial manner with the mathematical models involved in all of the above physical theories. In particular, it has led to new solutions of some of the nonlinear partial differential equations involved (Einstein's equations, Yang-Mills equations, equations of a magnetic monopole). We will describe some recent progress on twistorgeometric representations of solutions of classical field equations. There are two aspects to this. First, there is the problem of translating a given system of differential equations into twistor-geometric language. This has been done successfully in many different cases, and will be described in Section 3. Once a given problem has been transferred to a twistor-geometric context, it becomes, in general, a specific problem involving algebraic geometry, several complex variables, and algebraic topology involving holomorphic data on complex manifolds. The only partial differential equations remaining in the twistor context are the CauchyRiemann equations characterizing holomorphic functions. In effect, the partial differential equation has been transformed into a geometric problem. This is analogous to the Fourier transform transforming a partial differential equation with constant coefficients into an algebraic problem, where differentiation is replaced by the (algebraic) symbol of the differential operator in question. The second aspect of the recent progress on the twistor-geometric representation of the solutions of the classical field equations concerns itself with solving the geometric problem in question and transforming back to space-time (or Euclidean space), generating (sometimes new) solutions of the classical field equations.

In this chapter we discuss the prerequisites that are necessary to understand how problems in the natural sciences are formulated using (differential) equations and how to solve them numerically. For this, we discuss the basics of ordinary and partial differential equations in Sect. 4.1. We learn about the Lipschitz condition for ordinary differential equations and about initial and boundary conditions with partial differential equations, where we distinguish between three basic types of equations. A discussion of plane and spherical wave solutions of partial differential equations leads us to the study of numerical procedures in Sect. 4.2. Here, we discuss mesh-based and mesh-free methods, finite difference, finite volume, and finite element methods. We also get to know the most important algorithm used for numerical integration in molecular dynamics methods – the velocity Verlet algorithm. Finally, in Sect. 4.3 some tricks of the trade and important principles in software design and engineering which are not found in other books on computational science are presented. The chapter ends with a couple of problems for the interested reader.

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.

Fast and scalable computation of shape-morphing nonlinear solutions with application to evolutional neural networks

Exact solutions of a novel integrable partial differential equation

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.

Finite-dimensional approximations are developed for retarded delay differential equations (DDEs). The DDE system is equivalently posed as an initial-boundary value problem consisting of hyperbolic partial differential equations (PDEs). By exploiting the equivalence of partial derivatives in space and time, we develop a new PDE representation for the DDEs that is devoid of boundary conditions. The resulting boundary condition-free PDEs are discretized using the Galerkin method with Legendre polynomials as the basis functions, whereupon we obtain a system of ordinary differential equations (ODEs) that is a finite-dimensional approximation of the original DDE system. We present several numerical examples comparing the solution obtained using the approximate ODEs to the direct numerical simulation of the original non-linear DDEs. Stability charts developed using our method are compared to existing results for linear DDEs. The presented results clearly demonstrate that the equivalent boundary condition-free PDE formulation accurately captures the dynamic behaviour of the original DDE system and facilitates the application of control theory developed for systems governed by ODEs.