Similarity Solution of (2+1)-Dimensional Calogero-Bogoyavlenskii-Schiff Equation Lax Pair
Shaimaa Salem,
Magda Kassem,
Samah Mohamed Mabrouk
Issue:
Volume 7, Issue 5, October 2019
Pages:
137-144
Received:
10 September 2019
Accepted:
23 September 2019
Published:
14 October 2019
Abstract: In this paper, we discussed and studied the solutions of the (2+1)-dimensional Calogero-Bogoyavlenskii-Schiff (CBS) equation. The Calogero-Bogoyavlenskii-Schiff equation describes the propagation of Riemann waves along the y-axis, with long wave propagating along the x-axis. Lax pair and Bäcklund transformation of the Calogero-Bogoyavlenskii-Schiff equation are derived by using the singular manifold method (SMM). The optimal Lie infinitesimals of the Lax pair are obtained. The detected Lie infinitesimals contain eight unknown functions. These functions are optimized through the commutator table. The eight unknown functions are evaluated through the solution of a set of linear differential equations, in which solutions lead to optimal Lie vectors. The CBS Lax pair is reduced by using the optimal Lie vectors to a system of ordinary differential equations (ODEs). The solitary wave solutions of Calogero-Bogoyavlenskii-Schiff equation Lax pair’s show soliton and kink waves. The obtained similarity solutions are plotted for different arbitrary functions and compared with previous analytical solutions. The comparison shows that we derive new solutions of Calogero-Bogoyavlenskii-Schiff equation by using the combination of two methods, which is different from the previous findings.
Abstract: In this paper, we discussed and studied the solutions of the (2+1)-dimensional Calogero-Bogoyavlenskii-Schiff (CBS) equation. The Calogero-Bogoyavlenskii-Schiff equation describes the propagation of Riemann waves along the y-axis, with long wave propagating along the x-axis. Lax pair and Bäcklund transformation of the Calogero-Bogoyavlenskii-Schiff...
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A Mathematical Model for SIS Cholera Epidemic with Quarantine Effect
Deepti Mokati,
Viqar Hussain Badshah,
Nirmala Gupta
Issue:
Volume 7, Issue 5, October 2019
Pages:
145-151
Received:
10 July 2019
Accepted:
28 August 2019
Published:
4 November 2019
Abstract: Cholera was prevalent in the U.S. in the 1800s, before modern water and sewage treatment systems eliminated its spread by contaminated water. Cholera is an acute intestinal infectious disease caused by the bacterium vibrio cholerae. We propose and analyse a mathematical model for cholera considering quarantine. Quarantine plays an important role to control the disease. Our goal is to control the disease through the quarantine even if infected population again becomes suscepted. Determine two equilibrium points of the model: disease-free and endemic. Also basic reproduction number Rq is obtained. Reproduction number plays as a key role for analyzing stability for disease-free and endemic equilibrium points. Stability has been discussed for both equilibrium points using Ruth-Hurwitz criterian. We concluded that the disease-free and endemic equilibria are locally asymptotically stable if Rq<1 and Rq>1 respectively. Also, Numerical simulations are carried out for the model. From the graphically representation it is more clearly seen that when the disease becomes dies out and when it persistence.
Abstract: Cholera was prevalent in the U.S. in the 1800s, before modern water and sewage treatment systems eliminated its spread by contaminated water. Cholera is an acute intestinal infectious disease caused by the bacterium vibrio cholerae. We propose and analyse a mathematical model for cholera considering quarantine. Quarantine plays an important role to...
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