Rapidly decaying solutions of an ordinary differential equation with applications to semilinear elliptic and parabolic partial differential equations

1986 ◽  
Vol 91 (3) ◽  
pp. 247-266 ◽  
Author(s):  
Fred B. Weissler
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Parabolic Partial Differential Equations ◽  
Partial Differential ◽  
Semilinear Elliptic

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.

The Reduction of a Type of Singular Partial Differential Equations

2014 ◽  
Vol 602-605 ◽  
pp. 3616-3619
Author(s):  
Ping Li Li ◽  
Tian Tian Sun
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Complex Domain ◽  
Partial Differential ◽  
Singular Ordinary Differential Equation

Using the theory with respect to the existence of holomorphic solutions to a singular ordinary differential equation in the complex domain, this paper shows how to constructs biholomorphic trans-formations in different cases to reduce partial differential equations with singularity at the origin to more simple forms.

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

Nonlinear Numerical Analysis of Extruding Cantilever Laminated Composite Plates

2012 ◽  
Author(s):  
Wei Zhang ◽  
Shufeng Lu
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Numerical Analysis ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Laminated Composite ◽  
Composite Plates ◽  
Nonlinear Ordinary Differential Equation ◽  
Laminated Composite Plates ◽  
Partial Differential

This paper focus on the nonlinear numerical analysis for an extruding cantilever laminated composite plates subjected to transversal and in-plane excitation. Based on the Reddy’s shear deformable plate theory, the nonlinear partial differential equations of motion were established by using the Hamilton Principal. And then, after choosing suitable vibration mode-shape functions, the Galerkin method was used to reduce the governing partial differential equations to a two-degree-of-freedom nonlinear ordinary differential equation. Finally, we numerical solved the nonlinear ordinary differential equation, and analyzed the influences of varying extruding speeds and thickness of plates on the stability of the plates.

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