An accurate moving grid Eulerian Lagrangian localized adjoint method for solving the one-dimensional variable-coefficient ADE

2004 ◽  
Vol 45 (2) ◽  
pp. 157-178 ◽  
Author(s):  
Anis Younes
Keyword(s):  
Variable Coefficient ◽  
Adjoint Method ◽  
Moving Grid ◽  
One Dimensional ◽  
The One ◽  
Dimensional Variable

Solitons interaction and integrability for a (2+1)-dimensional variable-coefficient Broer–Kaup system in water waves

2018 ◽  
Vol 32 (08) ◽  
pp. 1750268 ◽  
Author(s):  
Xue-Hui Zhao ◽  
Bo Tian ◽  
Yong-Jiang Guo ◽  
Hui-Min Li
Keyword(s):  
Water Waves ◽  
Variable Coefficient ◽  
Soliton Solutions ◽  
Resonance Phenomenon ◽  
Bell Polynomials ◽  
Bilinear Forms ◽  
Linear Superposition ◽  
Elastic Interactions ◽  
The One ◽  
Dimensional Variable

Under investigation in this paper is a (2+1)-dimensional variable-coefficient Broer–Kaup system in water waves. Via the symbolic computation, Bell polynomials and Hirota method, the Bäcklund transformation, Lax pair, bilinear forms, one- and two-soliton solutions are derived. Propagation and interaction for the solitons are illustrated: Amplitudes and shapes of the one soliton keep invariant during the propagation, which implies that the transport of the energy is stable for the (2+1)-dimensional water waves; and inelastic interactions between the two solitons are discussed. Elastic interactions between the two parabolic-, cubic- and periodic-type solitons are displayed, where the solitonic amplitudes and shapes remain unchanged except for certain phase shifts. However, inelastically, amplitudes of the two solitons have a linear superposition after each interaction which is called as a soliton resonance phenomenon.

On Standard Eigenvalues of Variable-Coefficient Heat and Rod Equations

1989 ◽  
Vol 56 (1) ◽  
pp. 146-148 ◽  
Author(s):  
H. P. W. Gottlieb
Keyword(s):  
Heat Equation ◽  
Power Law ◽  
Heat Conductivity ◽  
Variable Coefficient ◽  
Continuous Case ◽  
Coefficient Function ◽  
Heat Conductivity Coefficient ◽  
One Dimensional ◽  
Variable Heat ◽  
The One

Forms of the variable-heat-conductivity coefficient function in the one-dimensional heat equation are determined which yield a standard harmonic eigenvalue sequence as in the case of homogeneity. The continuous case is found to correspond to a four-thirds power law dependence on coordinate. For the stepped case, the condition on the ratio of segmental heat conductivities in terms of the junction location is presented.

Explicit Analytical Solutions of Linear and Nonlinear Interior Heat and Mass Transfer Equation Sets for Drying Process

2003 ◽  
Vol 125 (1) ◽  
pp. 175-178 ◽  
Author(s):  
Ruixian Cai ◽  
Na Zhang
Keyword(s):  
Analytical Solutions ◽  
Variable Coefficient ◽  
Infinite Plate ◽  
Explicit Solutions ◽  
Drying Process ◽  
Mass Transfer Equation ◽  
Mathematical Skills ◽  
One Dimensional ◽  
Linear And Nonlinear ◽  
Dimensional Variable

Some algebraic explicit analytical solutions of the unsteady one-dimensional variable coefficient partial differential equation set describing drying process of an infinite plate are derived and given. Such explicit solutions have not yet been published before. Besides their irreplaceable theoretical value, analytical solutions can also serve as benchmark to check the results of recently rapidly developing numerical calculation and study various computational methods. In addition, some mathematical skills used in this paper are special and deserve further attention.

ANALYTIC ANALYSIS ON A GENERALIZED (2+1)-DIMENSIONAL VARIABLE-COEFFICIENT KORTEWEG–DE VRIES EQUATION USING SYMBOLIC COMPUTATION

2010 ◽  
Vol 24 (27) ◽  
pp. 5359-5370 ◽  
Author(s):  
CHENG ZHANG ◽  
BO TIAN ◽  
LI-LI LI ◽  
TAO XU
Keyword(s):  
Symbolic Computation ◽  
Variable Coefficient ◽  
Vries Equation ◽  
Nonlinear Superposition ◽  
Hirota Bilinear Form ◽  
De Vries ◽  
The One ◽  
Dimensional Variable ◽  
Hirota Bilinear ◽  
Nonlinear Superposition Formula

With the help of symbolic computation, a generalized (2+1)-dimensional variable-coefficient Korteweg–de Vries equation is studied for its Painlevé integrability. Then, Hirota bilinear form is derived, from which the one- and two-solitary-wave solutions with the corresponding graphic illustration are presented. Furthermore, a bilinear auto-Bäcklund transformation is constructed and the nonlinear superposition formula and Lax pair are also obtained. Finally, the analytic solution in the Wronskian form is constructed and proved by direct substitution into the bilinear equation.

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