Similarity Solutions for Generalized Variable Coefficients Zakharov—Kuznetsov Equation under Some Integrability Conditions

2010 ◽  
Vol 54 (4) ◽  
pp. 603-608 ◽  
Author(s):  
M.H.M Moussa ◽  
Rehab M El-Shiekh
Keyword(s):  
Variable Coefficients ◽  
Similarity Solutions ◽  
Integrability Conditions

Painlevé Test, Bäcklund Transformation and Consistent Riccati Expansion Solvability for two Generalised Cylindrical Korteweg-de Vries Equations with Variable Coefficients

2018 ◽  
Vol 73 (3) ◽  
pp. 207-213 ◽  
Author(s):  
Rehab M. El-Shiekh
Keyword(s):  
Vries Equation ◽  
Dusty Plasmas ◽  
Variable Coefficients ◽  
Bäcklund Transformations ◽  
Painlevé Test ◽  
New Exact Solutions ◽  
Backlund Transformations ◽  
Integrability Conditions ◽  
De Vries ◽  
Consistent Riccati Expansion

AbstractIn this paper, the integrability of the (2+1)-dimensional cylindrical modified Korteweg-de Vries equation and the (3+1)-dimensional cylindrical Korteweg-de Vries equation with variable coefficients arising in dusty plasmas in its generalised form was studied by two different techniques: the Painlevé test and the consistent Riccati expansion solvability. The integrability conditions and Bäcklund transformations are constructed. By using Bäcklund transformations and the solutions of the Riccati equation many new exact solutions are found for the two equations in this study. Finally, the application of the obtained solutions in dusty plasmas is investigated.

scholarly journals Approximate Symmetry Reduction Approach: Infinite Series Reductions to the KdV-Burgers Equation

2009 ◽  
Vol 64 (11) ◽  
pp. 676-684 ◽  
Author(s):  
Xiaoyu Jiao ◽  
Ruoxia Yao ◽  
Shunli Zhang ◽  
Sen Y. Lou
Keyword(s):  
Burgers Equation ◽  
Symmetry Reduction ◽  
Variable Coefficients ◽  
Similarity Solutions ◽  
Zero Order ◽  
Approximate Symmetry ◽  
Jacobi Elliptic Function ◽  
Similarity Reduction ◽  
Weak Dissipation ◽  
Reduction Approach

For weak dispersion and weak dissipation cases, the (1+1)-dimensional KdV-Burgers equation is investigated in terms of approximate symmetry reduction approach. The formal coherence of similarity reduction solutions and similarity reduction equations of different orders enables series reduction solutions. For the weak dissipation case, zero-order similarity solutions satisfy the Painlevé II, Painlevé I, and Jacobi elliptic function equations. For the weak dispersion case, zero-order similarity solutions are in the form of Kummer, Airy, and hyperbolic tangent functions. Higher-order similarity solutions can be obtained by solving linear variable coefficients ordinary differential equations.

scholarly journals Symmetry Analysis and Exact Solutions to the Space-Dependent Coefficient PDEs in Finance

2013 ◽  
Vol 2013 ◽  
pp. 1-10
Author(s):  
Hanze Liu
Keyword(s):  
Power Series ◽  
Exact Solutions ◽  
Variable Coefficients ◽  
Similarity Solutions ◽  
Lie Symmetry ◽  
Symmetry Analysis ◽  
Series Method ◽  
Lie Symmetry Analysis ◽  
Power Series Method ◽  
Generalized Power Series

The variable-coefficients partial differential equations (vc-PDEs) in finance are investigated by Lie symmetry analysis and the generalized power series method. All of the geometric vector fields of the equations are obtained; the symmetry reductions and exact solutions to the equations are presented, including the exponentiated solutions and the similarity solutions. Furthermore, the exact analytic solutions are provided by the transformation technique and generalized power series method, which has shown that the combination of Lie symmetry analysis and the generalized power series method is a feasible approach to dealing with exact solutions to the variable-coefficients PDEs.

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