scholarly journals Equivalent Lagrangians: Generalization, Transformation Maps, and Applications

2012 ◽  
Vol 2012 ◽  
pp. 1-19 ◽  
Author(s):  
N. Wilson ◽  
A. H. Kara
Keyword(s):  
Differential Equation ◽  
Wave Equation ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Conservation Laws ◽  
Ordinary Differential Equations ◽  
Wave Equations ◽  
Symmetry Algebra ◽  
Lie Point Symmetries ◽  
Partial Differential

Equivalent Lagrangians are used to find, via transformations, solutions and conservation laws of a given differential equation by exploiting the possible existence of an isomorphic algebra of Lie point symmetries and, more particularly, an isomorphic Noether point symmetry algebra. Applications include ordinary differential equations such as theKummer equationand thecombined gravity-inertial-Rossbywave equationand certain classes of partial differential equations related to multidimensional wave equations.

scholarly journals Hidden and Not So Hidden Symmetries

2012 ◽  
Vol 2012 ◽  
pp. 1-11 ◽  
Author(s):  
P. G. L. Leach ◽  
K. S. Govinder ◽  
K. Andriopoulos
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Recent Work ◽  
Nonlocal Symmetry ◽  
Partial Differential ◽  
Hidden Symmetries ◽  
Contact Symmetry

Hidden symmetries entered the literature in the late Eighties when it was observed that there could be gain of Lie point symmetry in the reduction of order of an ordinary differential equation. Subsequently the reverse process was also observed. Such symmetries were termed “hidden”. In each case the source of the “new” symmetry was a contact symmetry or a nonlocal symmetry, that is, a symmetry with one or more of the coefficient functions containing an integral. Recent work by Abraham-Shrauner and Govinder (2006) on the reduction of partial differential equations demonstrates that it is possible for these “hidden” symmetries to have a point origin. In this paper we show that the same phenomenon can be observed in the reduction of ordinary differential equations and in a sense loosen the interpretation of hidden symmetries.

scholarly journals On classical symmetries of ordinary differential equations related to stationary integrable partial differential equations

2021 ◽  
Vol 41 (5) ◽  
pp. 685-699
Author(s):  
Ivan Tsyfra
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Theoretical Method ◽  
Partial Differential ◽  
Classical Symmetry ◽  
Kdv Equations ◽  
Classical Symmetries

We study the relationship between the solutions of stationary integrable partial and ordinary differential equations and coefficients of the second-order ordinary differential equations invariant with respect to one-parameter Lie group. The classical symmetry method is applied. We prove that if the coefficients of ordinary differential equation satisfy the stationary integrable partial differential equation with two independent variables then the ordinary differential equation is integrable by quadratures. If special solutions of integrable partial differential equations are chosen then the coefficients satisfy the stationary KdV equations. It was shown that the Ermakov equation belong to a class of these equations. In the framework of the approach we obtained the similar results for generalized Riccati equations. By using operator of invariant differentiation we describe a class of higher order ordinary differential equations for which the group-theoretical method enables us to reduce the order of ordinary differential equation.

scholarly journals Direct construction method for conservation laws of partial differential equations Part I: Examples of conservation law classifications

2002 ◽  
Vol 13 (5) ◽  
pp. 545-566 ◽  
Author(s):  
STEPHEN C. ANCO ◽  
GEORGE BLUMAN
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Conservation Laws ◽  
Conservation Law ◽  
Wave Equations ◽  
Explicit Construction ◽  
Nonlinear Wave Equations ◽  
Local Conservation ◽  
Partial Differential ◽  
Direct Construction

An effective algorithmic method is presented for finding the local conservation laws for partial differential equations with any number of independent and dependent variables. The method does not require the use or existence of a variational principle and reduces the calculation of conservation laws to solving a system of linear determining equations similar to that for finding symmetries. An explicit construction formula is derived which yields a conservation law for each solution of the determining system. In the first of two papers (Part I), examples of nonlinear wave equations are used to exhibit the method. Classification results for conservation laws of these equations are obtained. In a second paper (Part II), a general treatment of the method is given.

Numerical Reservoir Simulation Using an Ordinary-Differential-Equations Integrator

1975 ◽  
Vol 15 (03) ◽  
pp. 255-264 ◽  
Author(s):  
R.F. Sincovec
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Reservoir Simulation ◽  
Nonlinear Partial Differential Equations ◽  
Method Of Lines ◽  
Variable Order ◽  
Partial Differential

Abstract The method of lines used in conjunction with a sophisticated ordinary-differential-equations integrator is an effective approach for solving nonlinear, partial differential equations and is applicable to the equations describing fluid flow through porous media. Given the initial values, the integrator is self-starting. Subsequently, it automatically and reliably selects the time step and order, solves the nonlinear equations (checking for convergence, etc.), and maintains a user-specified time-integration accuracy, while attempting to complete the problems in a minimal amount of computer time. The advantages of this approach, such as stability, accuracy, reliability, and flexibility, are discussed. The method is applied to reservoir simulation, including high-rate and gas-percolation problems, and appears to be readily applicable to problems, and appears to be readily applicable to compositional models. Introduction The numerical solution of nonlinear, partial differential equations is usually a complicated and lengthy problem-dependent process. Generally, the solution of slightly different types of partial differential equations requires an entirely different computer program. This situation for partial differential equations is in direct contrast to that for ordinary differential equations. Recently, sophisticated and highly reliable computer programs for automatically solving complicated systems of ordinary differential equations have become available. These computer programs feature variable-order methods and automatic time-step and error control, and are capable of solving broad classes of ordinary differential equations. This paper discusses how these sophisticated ordinary-differential-equation integrators may be used to solve systems of nonlinear partial differential equations. partial differential equations.The basis for the technique is the method of lines. Given a system of time-dependent partial differential equations, the spatial variable(s) are discretized in some manner. This procedure yields an approximating system of ordinary differential equations that can be numerically integrated with one of the recently developed, robust ordinary-differential-equation integrators to obtain numerical approximations to the solution of the original partial differential equations. This approach is not new, but the advent of robust ordinary-differential-equation integrators has made the numerical method of lines a practical and efficient method of solving many difficult systems of partial differential equations. The approach can be viewed as a variable order in time, fixed order in space technique. Certain aspects of this approach are discussed and advantages over more conventional methods are indicated. Use of ordinary-differential-equation integrators for simplifying the heretofore rather complicated procedures for accurate numerical integration of systems of nonlinear, partial differential equations is described. This approach is capable of eliminating much of the duplicate programming effort usually associated with changing equations, boundary conditions, or discretization techniques. The approach can be used for reservoir simulation, and it appears that a compositional reservoir simulator can be developed with relative ease using this approach. In particular, it should be possible to add components to or delete components possible to add components to or delete components from the compositional code with only minor modifications. SPEJ P. 255

scholarly journals Lie Point Symmetries, Traveling Wave Solutions and Conservation Laws of a Non-linear Viscoelastic Wave Equation

Mathematics ◽  
2021 ◽  
Vol 9 (17) ◽  
pp. 2131
Author(s):  
Almudena P. Márquez ◽  
María S. Bruzón
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Wave Equation ◽  
Differential Equations ◽  
Conservation Laws ◽  
Lie Point Symmetries ◽  
Viscoelastic Wave ◽  
Non Linear ◽  
Linear Viscoelastic ◽  
Damping And Source Terms

This paper studies a non-linear viscoelastic wave equation, with non-linear damping and source terms, from the point of view of the Lie groups theory. Firstly, we apply Lie’s symmetries method to the partial differential equation to classify the Lie point symmetries. Afterwards, we reduce the partial differential equation to some ordinary differential equations, by using the symmetries. Therefore, new analytical solutions are found from the ordinary differential equations. Finally, we derive low-order conservation laws, depending on the form of the damping and source terms, and discuss their physical meaning.

Practical Applications of Optimal-Control Theory to History-Matching Multiphase Simulator Models

1975 ◽  
Vol 15 (04) ◽  
pp. 347-355 ◽  
Author(s):  
M.L. Wasserman ◽  
A.S. Emanuel ◽  
J.H. Seinfeld
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Optimal Control ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
History Matching ◽  
Single Phase ◽  
Continuous Functions ◽  
Partial Differential

Abstract This paper applies material presented by Chen et al. and by Chavent et al to practical reservoir problems. The pressure history-matching algorithm used is initially based on a discretized single-phase reservoir model. Multiphase effects are approximately treated in the single-phase model by multiplying the transmissibility and storage terms by saturation-dependent terms that are obtained from a multiphase simulator run. Thus, all the history matching is performed by a "pseduo" single-phase model. The multiplicative factors for transmissibility and storage are updated when necessary. The matching technique can change any model permeability thickness or porosity thickness value. Three field examples are given. Introduction History matching using optimal-control theory was introduced by two sets of authors. Their contributions were a major breakthrough in attacking the long-standing goal of automatic history matching. This paper extends the work presented by Chen et al. and Chavent et al. Specifically, we focus on three areas.We derive the optimal-control algorithm using a discrete formulation. Our reservoir simulator, which is a set of ordinary differential equations, is adjoined to the function to be minimized. The first variation is taken to yield equations for computing Lagrange multipliers. These Lagrange multipliers are then used for computing a gradient vector. The discrete formulation keeps the adjoint equations consistent with the reservoir simulator.We include the effects of saturation change in history-matching pressures. We do this in a fashion that circumvents the need for developing a full multiphase optimal-control code.We show detailed results of the application of the optimal-control algorithm to three field examples. DERIVATION OF ADJOINT EQUATIONS Most implicit-pressure/explicit-saturation-type, finite-difference reservoir simulators perform two calculation stages for each time step. The first stage involves solving an "expansivity equation" for pressure. The expansivity equation is obtained by summing the material-balance equations for oil, gas, and water flow. Once the pressures are implicitly obtained from the expansivity equation, the phase saturations can be updated using their respective balance equations. A typical expansivity equation is shown in Appendix B, Eq. B-1. When we write the reservoir simulation equations as partial differential equations, we assume that the parameters to be estimated are continuous functions of position. The partial-differential-equation formulation is partial-differential-equation formulation is generally termed a distributed-parameter system. However, upon solving these partial differential equations, the model is discretized so that the partial differential equations are replaced by partial differential equations are replaced by sets of ordinary differential equations, and the parameters that were continuous functions of parameters that were continuous functions of position become specific values. Eq. B-1 is a position become specific values. Eq. B-1 is a set of ordinary differential equations that reflects lumping of parameters. Each cell has three associated parameters: a right-side permeability thickness, a bottom permeability thickness, and a pore volume. pore volume.Once the discretized model is written and we have one or more ordinary differential equations per cell, we can then adjoin these differential equations to the integral to be minimized by using one Lagrange multiplier per differential equation. The ordinary differential equations for the Lagrange multipliers are now derived as part of the necessary conditions for stationariness of the augmented objective function. These ordinary differential equations are termed the adjoint system of equations. A detailed example of the procedure discussed in this paragraph is given in Appendix A. The ordinary-differential-equation formulation of the optimal-control algorithm is more appropriate for use with reservoir simulators than the partial-differential-equation derivation found in partial-differential-equation derivation found in Refs. 1 and 2. SPEJ P. 347

5. What’s the frequency, Kenneth? Waves, quakes, and solitons

2018 ◽  
pp. 70-84
Author(s):  
Alain Goriely
Keyword(s):  
Wave Propagation ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Conservation Laws ◽  
Ordinary Differential Equations ◽  
Mathematical Tool ◽  
Partial Differential ◽  
Conservation Of Mass ◽  
Linear Waves ◽  
Multiple Variables

Since we live in a world where interesting things happen both in time and space, the description of many natural phenomena requires the study of functions that depend on multiple variables. These laws can be expressed as partial differential equations. Just like ordinary differential equations, these equations are the natural mathematical tool for modelling because they express, locally, basic conservation laws such as the conservation of mass and energy. ‘What’s the frequency, Kenneth? Waves, quakes, and solitons’ introduces the wonder of partial differential equations by considering two generic spatiotemporal behaviours: wave propagation through linear waves on a string and earthquakes, and non-linear waves and the discovery of solitons, which behave like discrete particles.

scholarly journals A REVIEW ON NUMERICAL METHODS FOR NON-LINEAR PARTIAL DIFFERENTIAL EQUATIONS

2021 ◽  
Vol 23 (07) ◽  
pp. 1342-1352
Author(s):  
◽  
Dr. Jatinder kaur ◽  
Keyword(s):  
Differential Equation ◽  
Partial Differential Equations ◽  
Numerical Methods ◽  
Differential Equations ◽  
Wave Equations ◽  
Numerical Models ◽  
Nonlinear Partial Differential Equations ◽  
Partial Differential ◽  
Non Linear ◽  
Linear Differential

Progression in innovation and engineering presents us with numerous difficulties, comparably to conquer such engineering difficulties with the assistance of various numerical models, equations are taken. Since in the first place Mathematicians, Designers and Engineers make progress toward accuracy what’s more, exactness while addressing equations Differential equations, specifically, hold an enormous application in engineering and numerous different areas. One such sort of Differential equation is known as partial differential equation. The range of application of partial differential equations comprises of recreation, calculation age, and investigation of higher request PDE and wave equations. Adjusting diverse numerical methods prompts an assortment of answers and contrast among them, subsequently the determination of the method of addressing is one of the urgent boundaries to produce exact outcomes. Our work centres’ around the survey of various numerical methods to settle Non-linear differential equations based on exactness and effectiveness, in order to diminish the emphases. These would orchestrate rules to existing numerical methods of nonlinear partial differential equations.[1]

A Novel General and Robust Method Based on NAOP for Solving Nonlinear Ordinary Differential Equations and Partial Differential Equations by Cellular Neural Networks

2013 ◽  
Vol 135 (3) ◽  
Author(s):  
Jean Chamberlain Chedjou ◽  
Kyandoghere Kyamakya
Keyword(s):  
Neural Networks ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Cellular Neural Networks ◽  
Nonlinear Ordinary Differential Equations ◽  
Van Der Pol ◽  
Partial Differential ◽  
Nonlinear Odes ◽  
First Order

This paper develops and validates through a series of presentable examples, a comprehensive high-precision, and ultrafast computing concept for solving nonlinear ordinary differential equations (ODEs) and partial differential equations (PDEs) with cellular neural networks (CNN). The core of this concept is a straightforward scheme that we call "nonlinear adaptive optimization (NAOP),” which is used for a precise template calculation for solving nonlinear ODEs and PDEs through CNN processors. One of the key contributions of this work is to demonstrate the possibility of transforming different types of nonlinearities displayed by various classical and well-known nonlinear equations (e.g., van der Pol-, Rayleigh-, Duffing-, Rössler-, Lorenz-, and Jerk-equations, just to name a few) unto first-order CNN elementary cells, and thereby enabling the easy derivation of corresponding CNN templates. Furthermore, in the case of PDE solving, the same concept also allows a mapping unto first-order CNN cells while considering one or even more nonlinear terms of the Taylor's series expansion generally used in the transformation of a PDE in a set of coupled nonlinear ODEs. Therefore, the concept of this paper does significantly contribute to the consolidation of CNN as a universal and ultrafast solver of nonlinear ODEs and/or PDEs. This clearly enables a CNN-based, real-time, ultraprecise, and low-cost computational engineering. As proof of concept, two examples of well-known ODEs are considered namely a second-order linear ODE and a second order nonlinear ODE of the van der Pol type. For each of these ODEs, the corresponding precise CNN templates are derived and are used to deduce the expected solutions. An implementation of the concept developed is possible even on embedded digital platforms (e.g., field programmable gate array (FPGA), digital signal processor (DSP), graphics processing unit (GPU), etc.). This opens a broad range of applications. Ongoing works (as outlook) are using NAOP for deriving precise templates for a selected set of practically interesting ODEs and PDEs equation models such as Lorenz-, Rössler-, Navier Stokes-, Schrödinger-, Maxwell-, etc.

A method for constructing solutions of homogeneous partial differential equations: localized waves

1992 ◽  
Vol 437 (1901) ◽  
pp. 673-692 ◽  
Keyword(s):  
Wave Equation ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Gordon Equation ◽  
Partial Differential ◽  
Wave Solutions ◽  
Propagation Axis ◽  
Potential Applications ◽  
Klein Gordon ◽  
Localized Waves

We introduce a method for constructing solutions of homogeneous partial differential equations. This method can be used to construct the usual, well-known, separable solutions of the wave equation, but it also easily gives the non-separable localized wave solutions. These solutions exhibit a degree of focusing about the propagation axis that is dependent on a free parameter, and have many important potential applications. The method is based on constructing the space-time Fourier transform of a function so that it satisfies the transformed partial differential equation. We also apply the method to construct localized wave solutions of the wave equation in a lossy infinite medium, and of the Klein-Gordon equation. The localized wave solutions of these three equations differ somewhat, and we discuss these differences. A discussion of the properties of the localized waves, and of experiments to launch them, is included in the Appendix.

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