Asymptotic Methods for Partial Differential Equations: The Reduced Wave Equation and Maxwell’s Equations

1995 ◽  
pp. 1-82 ◽  
Author(s):  
Joseph B. Keller ◽  
Robert M. Lewis
Keyword(s):  
Wave Equation ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Maxwell’S Equations ◽  
Maxwell's Equations ◽  
Asymptotic Methods ◽  
Partial Differential

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.

scholarly journals Ordinary and partial difference equations

1986 ◽  
Vol 27 (4) ◽  
pp. 488-501 ◽  
Author(s):  
Renfrey B. Potts
Keyword(s):  
Wave Equation ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Difference Equations ◽  
Hyperbolic Functions ◽  
Partial Differential ◽  
Partial Difference Equations ◽  
Partial Difference

AbstractOrdinary difference equations (OΔE's), mostly of order two and three, are derived for the trigonometric, Jacobian elliptic, and hyperbolic functions. The results are used to derive partial difference equations (PΔE's) for simple solutions of the wave equation and three nonlinear evolutionary partial differential equations.

Uniform progressing wave solutions of the wave equation

1971 ◽  
Vol 70 (3) ◽  
pp. 455-465
Author(s):  
Erich Zauderer
Keyword(s):  
Wave Equation ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Standard Form ◽  
Hyperbolic Partial Differential Equations ◽  
Partial Differential ◽  
Wave Solutions ◽  
Different Types ◽  
Uniform Expansions

The solution of problems involving the propagation of discontinuities and other singularities for hyperbolic partial differential equations by means of progressing wave expansions is discussed in the book by Courant(l). He also refers to the work of Hadamard, Friedlander, Ludwig and others on this subject. More recently, Ludwig (2), Lewis(3) and others have considered 'uniform' progressing wave expansions for various problems. These expansions are valid in regions where the standard expansions are not suitable and they can be re-expanded in the standard form outside these regions. Examples of such regions are given by envelopes of bicharacteristic curves or, equivalently, caustics and by shadow boundaries such as occur in diffraction problems. In each of these regions, which we term 'transition regions' different types of uniform expansions are required.

scholarly journals Development of Methods of Asymptotic Analysis of Transition Layers in Reaction–Diffusion–Advection Equations: Theory and Applications

2021 ◽  
Vol 61 (12) ◽  
pp. 2068-2087
Author(s):  
N. N. Nefedov
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Asymptotic Methods ◽  
General Scheme ◽  
Rapid Change ◽  
Reaction Diffusion ◽  
Singularly Perturbed ◽  
Parabolic Problems ◽  
Singularly Perturbed Problems ◽  
Partial Differential

Abstract This work presents a review and analysis of modern asymptotic methods for analysis of singularly perturbed problems with interior and boundary layers. The central part of the work is a review of the work of the author and his colleagues and disciples. It highlights boundary and initial-boundary value problems for nonlinear elliptic and parabolic partial differential equations, as well as periodic parabolic problems, which are widely used in applications and are called reaction–diffusion and reaction–diffusion–advection equations. These problems can be interpreted as models in chemical kinetics, synergetics, astrophysics, biology, and other fields. The solutions of these problems often have both narrow boundary regions of rapid change and inner layers of various types (contrasting structures, moving interior layers: fronts), which leads to the need to develop new asymptotic methods in order to study them both formally and rigorously. A general scheme for a rigorous study of contrast structures in singularly perturbed problems for partial differential equations, based on the use of the asymptotic method of differential inequalities, is presented and illustrated on relevant problems. The main achievements of this line of studies of partial differential equations are reflected, and the key directions of its development are indicated.

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 Sine-Function Method In The Soliton Solution Of Nonlinear Partial Differential Equations

2013 ◽  
Vol 32 ◽  
pp. 55-60 ◽  
Author(s):  
M Abdur Rab ◽  
Jasmin Akhter
Keyword(s):  
Wave Equation ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Function Method ◽  
Nonlinear Partial Differential Equations ◽  
Traveling Wave Solution ◽  
Sine Function ◽  
Partial Differential ◽  
Regularized Long Wave Equation ◽  
Different Types

In this paper we establish a traveling wave solution for nonlinear partial differential equations using sine-function method. The method is used to obtain the exact solutions for three different types of nonlinear partial differential equations like general equal width wave equation (GEWE), general regularized long wave equation (GRLW), general Korteweg-de Vries equation(GKDV) which are the important soliton equations DOI: http://dx.doi.org/10.3329/ganit.v32i0.13647 GANIT J. Bangladesh Math. Soc. (ISSN 1606-3694) 32 (2012) 55-60

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