scholarly journals Quotients of Euler Equations on Space Curves

Symmetry ◽  
2021 ◽  
Vol 13 (2) ◽  
pp. 186
Author(s):  
Anna Duyunova ◽  
Valentin Lychagin ◽  
Sergey Tychkov
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Euler Equations ◽  
Gas Flow ◽  
Euler System ◽  
Partial Differential ◽  
One Dimensional ◽  
Space Curves ◽  
Ode System

Quotients of partial differential equations are discussed. The quotient equation for the Euler system describing a one-dimensional gas flow on a space curve is found. An example of using the quotient to solve the Euler system is given. Using virial expansion of the Planck potential, we reduce the quotient equation to a series of systems of ordinary differential equations (ODEs). Possible solutions of the ODE system are discussed.

scholarly journals Existence of weak solutions to SPDEs with fractional Laplacian and non-Lipschitz coefficients

2021 ◽  
Author(s):  
Shohei Nakajima
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Weak Solutions ◽  
Stochastic Partial Differential Equations ◽  
Fractional Laplacian ◽  
Existence Of Solutions ◽  
Partial Differential ◽  
Continuity Theorem ◽  
Existence Of Weak Solutions ◽  
One Dimensional

AbstractWe prove existence of solutions and its properties for a one-dimensional stochastic partial differential equations with fractional Laplacian and non-Lipschitz coefficients. The method of proof is eatablished by Kolmogorov’s continuity theorem and tightness arguments.

scholarly journals Approximation of linear one dimensional partial differential equations including fractional derivative with non-singular kernel

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Raheel Kamal ◽  
Kamran ◽  
Gul Rahmat ◽  
Ali Ahmadian ◽  
Noreen Izza Arshad ◽  
...  
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Laplace Transform ◽  
Fractional Derivative ◽  
Meshless Method ◽  
Inverse Laplace Transform ◽  
Contour Integral ◽  
Partial Differential ◽  
One Dimensional ◽  
The Laplace Transform

AbstractIn this article we propose a hybrid method based on a local meshless method and the Laplace transform for approximating the solution of linear one dimensional partial differential equations in the sense of the Caputo–Fabrizio fractional derivative. In our numerical scheme the Laplace transform is used to avoid the time stepping procedure, and the local meshless method is used to produce sparse differentiation matrices and avoid the ill conditioning issues resulting in global meshless methods. Our numerical method comprises three steps. In the first step we transform the given equation to an equivalent time independent equation. Secondly the reduced equation is solved via a local meshless method. Finally, the solution of the original equation is obtained via the inverse Laplace transform by representing it as a contour integral in the complex left half plane. The contour integral is then approximated using the trapezoidal rule. The stability and convergence of the method are discussed. The efficiency, efficacy, and accuracy of the proposed method are assessed using four different problems. Numerical approximations of these problems are obtained and validated against exact solutions. The obtained results show that the proposed method can solve such types of problems efficiently.

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.

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 GROUP PROPERTIES OF A ONE-DIMENSIONAL NONLINEAR POROELASTICITY EQUATION

2019 ◽  
Vol 4 (1) ◽  
pp. 149-155
Author(s):  
Kholmatzhon Imomnazarov ◽  
Ravshanbek Yusupov ◽  
Ilham Iskandarov
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Lie Algebra ◽  
Three Dimensional ◽  
Group Classification ◽  
Second Order ◽  
Partial Differential ◽  
One Dimensional ◽  
Preliminary Group

This paper studies a class of partial differential equations of second order , with arbitrary functions and , with the help of the group classification. The main Lie algebra of infinitely infinitesimal symmetries is three-dimensional. We use the method of preliminary group classification for obtaining the classifications of these equations for a one-dimensional extension of the main Lie algebra.

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.

Some Invariant Solutions of Two-Dimensional Elastodynamics in Linear Homogeneous Isotropic Materials

2013 ◽  
Vol 5 (2) ◽  
pp. 212-221
Author(s):  
Houguo Li ◽  
Kefu Huang
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Theoretical Method ◽  
Two Dimensional ◽  
Invariant Solutions ◽  
Partial Differential ◽  
Isotropic Materials ◽  
Group Theoretical Method ◽  
Infinitesimal Operators

AbstractInvariant solutions of two-dimensional elastodynamics in linear homogeneous isotropic materials are considered via the group theoretical method. The second order partial differential equations of elastodynamics are reduced to ordinary differential equations under the infinitesimal operators. Three invariant solutions are constructed. Their graphical figures are presented and physical meanings are elucidated in some cases.

Lie Group Analysis for a Mixed Convective Flow and Heat Mass Transfer Over a Permeable Stretching Surface with Soret and Dufour Effects

2013 ◽  
Vol 30 (1) ◽  
pp. 67-75 ◽  
Author(s):  
Reda G. Abdel-Rahman ◽  
Ahmed M. Megahed
Keyword(s):  
Mass Transfer ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Lie Group ◽  
Group Analysis ◽  
Group Method ◽  
Stretching Surface ◽  
Partial Differential ◽  
Soret And Dufour Effects

ABSTRACTThe Lie group transformation method is applied for solving the problem of mixed convection flow with mass transfer over a permeable stretching surface with Soret and Dufour effects. The application of Lie group method reduces the number of independent variables by one and consequently the system of governing partial differential equations reduces to a system of ordinary differential equations with appropriate boundary conditions. Further, the reduced non-linear ordinary differential equations are solved numerically by using the shooting method. The effects of various parameters governing the flow and heat transfer are shown through graphs and discussed. Our aim is to detect new similarity variables which transform our system of partial differential equations to a system of ordinary differential equations. In this work a special attention is given to investigate the effect of the Soret and Dufour numbers on the velocity, temperature and concentration fields above the sheet.

Sign in / Sign up

Export Citation Format

Share Document