Solutions of the variable coefficient Radhakrishnan-Kundu-Lakshmanan equation using the method of similarity reduction

Optik ◽  
2021 ◽  
pp. 167254
Author(s):  
A. Ghose-Choudhury ◽  
Sudip Garai
Keyword(s):  
Variable Coefficient ◽  
Similarity Reduction

Direct Similarity Reduction and New Exact Solutions for the Variable-Coefficient Kadomtsev–Petviashvili Equation

2015 ◽  
Vol 70 (6) ◽  
pp. 445-450 ◽  
Author(s):  
Rehab M. El-Shiekh
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Variable Coefficient ◽  
Nonlinear Partial Differential Equation ◽  
Direct Methods ◽  
Periodic Wave ◽  
Variable Coefficients ◽  
Nonlinear Phenomena ◽  
Similarity Reduction ◽  
Petviashvili Equation

AbstractIn this paper, the generalized (3+1)-dimensional variable-coefficient Kadomtsev–Petviashvili equation (VCKPE), which can describe nonlinear phenomena in fluids or plasmas, is studied by using two different Clarkson and Kruskal (CK) direct methods, namely, the classical CK and the modified enlarged CK method. A similarity reduction to a (2+1)-dimensional nonlinear partial differential equation and a direct similarity reduction to a nonlinear ordinary differential equation are obtained, respectively. By solving the reduced ordinary differential equation, new solitary, periodic, and singular solutions for the VCKPE are obtained. Some figures for the soliton and periodic wave solutions are given to reflect the effect of the variable coefficients on the solution propagation. Finally, the comparison between the two different CK techniques indicates that the modified enlarged CK technique is clearly more powerful and simple than the classical CK technique.

Eigenvalue Problem for the Second Order Differential Equation with Nonlocal

2006 ◽  
Vol 11 (1) ◽  
pp. 13-32 ◽  
Author(s):  
B. Bandyrskii ◽  
I. Lazurchak ◽  
V. Makarov ◽  
M. Sapagovas
Keyword(s):  
Differential Equation ◽  
Eigenvalue Problem ◽  
Variable Coefficient ◽  
Order Differential Equation ◽  
Integral Condition ◽  
Second Order ◽  
Computational Results ◽  
Discrete Method ◽  
Complex Eigenvalues ◽  
Nonlocal Integral Condition

The paper deals with numerical methods for eigenvalue problem for the second order ordinary differential operator with variable coefficient subject to nonlocal integral condition. FD-method (functional-discrete method) is derived and analyzed for calculating of eigenvalues, particulary complex eigenvalues. The convergence of FD-method is proved. Finally numerical procedures are suggested and computational results are schown.

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