From symmetries of partial differential equations towards secondary (“quantized”) calculus

Essential connections between the classical symmetry and nonclassical symmetry of a partial differential equations (PDEs) are established. Through these connections, the sufficient conditions for the nonclassical symmetry of PDEs can be derived directly from the inconsistent conditions of the system determining equations of the classical symmetry of the PDE. Based on the connections, a new algorithm for determining the nonclassical symmetry of a PDEs is proposed. The algorithm make the determination of the nonclassical symmetry easier by adding compatibility extra equations obtained from system of determining equations of the classical symmetry to the system of determining equations of the nonclassical symmetry of the PDE. The findings of this study not only give an alternative method to determine the nonclassical symmetry of a PDE, but also can help for better understanding of the essential connections between classical and nonclassical symmetries of a PDE. Concurrently, the results obtained here enhance the efficiency of the existing algorithms for determining the nonclassical symmetry of a PDE. As applications of the given algorithm, a nonclassical symmetry classification of a class of generalized Burgers equations and the nonclassical symmetries of a KdV-type equations are given within a relatively easier way and some new nonclassical symmetries have been found for the Burgers equations.

Type-II hidden symmetries of the linear, two-dimensional and three-dimensional wave equations are analysed. These hidden symmetries are Lie point symmetries that appear in addition to the inherited point symmetries when the number of independent and dependent variables of a partial differential equation is reduced by a Lie point symmetry. The provenance of these hidden symmetries of partial differential equations is identified to be the same as found recently for some nonlinear partial differential equations. The appearance of Type-II hidden symmetries depends not only on the Lie symmetries used but on the order in which the symmetries are applied. The presence of Type-II hidden symmetries of partial differential equations complicates the prediction of symmetry reductions based on the Lie algebra associated with the original Lie point symmetries.

A MATHEMATICA package for finding Lie symmetries of partial differential equations is presented. The package is designed to create and solve the associated determining system of equations, the full set of solutions of which generates the widest permissible local Lie group of point symmetry transformations. Examples illustrating the functionality of the package's tools are given. The results of the package application to performing a full Lie group analysis of coupled nonlinear Schrödinger equations from nonlinear fiber optics are presented. Comparisons with earlier published computer algebra implementations of the Lie group method are discussed.

The subject of this section is to discuss the basic tools of Lie’s symmetry method in connection with partial differential equations. These tools will support the practical calculations. We will show how the theory of Lie becomes vital again by using computer algebra calculations.

We consider nonclassical symmetries of partial differential equations (PDEs) in dimensions. Given a th‐order ordinary differential equation in the unknown we are able to find the most general scalar PDE of a given order which can be reduced via a nonclassical symmetry to .

Geometry of finite-order contact structures and the classical theory of symmetries of partial differential equations

A description is given of a software package to compute symmetries of partial differential equations, using computer algebra. As an application, the computation of higher-order symmetries of the classical Boussinesq equation is given leading to the recursion operator for symmetries in a straightforward way. Nonlocal symmetries for the Federbush model are obtained yielding the linearization of the model.Key wordsComputer algebrasymmetriesrecursion operators.

Symmetry detection, the task of discovering the underlying symmetries of a given dataset, has been gaining popularity in the machine learning community, particularly in science and engineering applications. Most previous works focus on detecting ‘canonical’ symmetries such as translation, scaling, and rotation, and cast the task as a modeling problem involving complex inductive biases and architecture design of neural networks. We challenge these assumptions and propose that instead of constructing biases, we can learn to detect symmetries from raw data without prior knowledge. The approach presented in this paper provides a flexible way to scale up the detection procedure to non-canonical symmetries, and has the potential to detect both known and unknown symmetries alike. Concretely, we focus on predicting the generators of Lie point symmetries of partial differential equations, more specifically, evolutionary equations for ease of data generation. Our results demonstrate that well-established neural network architectures are capable of recognizing symmetry generators, even in unseen dynamical systems. These findings have the potential to make non-canonical symmetries more accessible to applications, including model selection, sparse identification, and data interpretability.

On apporte plusieurs ameliorations au programme REDUCE pour determiner les symetries des equations aux derivees partielles

We consider the symmetries of partial differential equations based on the use of differential-geometric and algebraic methods of the theory of dynamical control 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.

Mathematical Modelling of Dehydration Processes of Agricultural Crops

We present in this paper a series of operators coming from various domains of mathematical analysis (approximation theory, differential geometry, optimization and partial differential equations (PDE's)) and able to give adequate responses to low level image processing tasks like contrasting, segmenting and contouring objects of interest in bio-medical images. These operators are essentially based on powerful analytic tools, like splines, wavelets, singularities and PDE's, which will be described in a brief historical view.

The Survey and Review article in this issue is “The Narrow Escape Problem,” by D. Holcman and Z. Schuss. We are all familiar with the jagged plots formed by share prices and interest rates as they fluctuate over time. It is natural to ask questions like -when will my portfolio of Facebook shares reach $1 Million? -when will my bank's interest rate leave the 2% -- 3% range? These questions concern hitting times. Modeling the evolution as a stochastic process (a random variable that changes over time), we want to know the first occasion when the value reaches a specified level. More generally, for processes in a higher dimension, such as Robert Brown's pollen particles suspended in a fluid, we could ask when a tagged particle will first meet a specified region on the boundary of a given domain. The hitting time is itself a random variable, and we can reduce to a simpler deterministic quantity by asking for its expected value. It will then be no surprise to many readers that this deterministic value can be obtained by solving an appropriate partial differential equation. The topic of this article is a commonly arising special case which has wide practical significance and offers the added attraction of allowing mathematicians to introduce a small parameter. The narrow escape time (NET) relates to the first time that Brownian motion meets a small absorbing window on the boundary of a domain that is otherwise reflecting. Figure 1.3 illustrates the setting very clearly. A closely related problem is to determine when a particle traverses a narrow passage between almost disconnected regions, as illustrated in Figure 1.1. The article addresses the development of asymptotic methods that deal with various geometries, motivated by applications in cell biology. The underlying mathematical challenge can be viewed as solving a mixed Neumann--Dirichlet boundary value problem for the Poisson equation in a bounded domain, and the authors survey a range of developments in nonstandard asymptotics that have been brought to bear in recent years. This article will be a valuable resource for readers with interests at the interface between probability theory and partial differential equations, and those who study asymptotic analysis of singular perturbation problems and applications to physics, biochemistry, and neuroscience.

We consider the quasilinear elliptic partial differential equation Qu(x) = aij(x, u(x), Du(x))Diju(x) + a(x, u(x), Du(x)) = 0 in Ω (0.1) with the imposed nonlinear boundary condition Gu(x) = g(x, u(x), Du(x)) = 0 on ∂Ω. (0.2) A homotopy argument for the Leray-Schauder degree is presented which reduces the question of existence of solutions to the above problem to that of establishing suitable a priori bounds on solutions of an approximating problem. The above problem is re-written asaij(x,u, Du)DijU + a(x,u, Du) = 0 in Ω (0.3) u(x) = ω(x) on ∂Ω.where ω(.) is a function which satisfies the requirement that g(x,ω(x), Du(x)) = 0 on ∂Ω. (0.4)The gradient term in the boundary condition is replaced with an approximating function of greater smoothness, allowing the application of Leray-Schauder degree theory. We present simple assumptions which ensure the existence of both an upper solution and a lower solution of the differential operator Q, and impose simple conditions upon the boundary operator g(x,z,p) which allow us to use a homotopy argument to reduce the problem to two 'disconnected' halves, one of which is a simple Dirichlet problem in u with no reference to the function ω, the other an equation in ω with no reference to u or its derivatives. A reduction argument and reasonable assumptions upon g(x,z,p) then imply the existence of solutions to the approximating problems. It is then shown that an a priori bound on the solutions to the approximating equations in a Holder space of the form C1,α(Ω) where α > 0 suffices to ensure the convergence of a subsequence of these functions to a classical solution of the original problem. The result described here differ from the method of continuity in that the boundary ∂Ω, is assumed to be only C1,α, and the monotonicity of the differential operator with respect to u is significantly relaxed. The techniques described also apply in situations where degree theory cannot be applied directly.