scholarly journals Conservation Laws for a Variable Coefficient Variant Boussinesq System

2014 ◽  
Vol 2014 ◽  
pp. 1-5 ◽  
Author(s):  
Ben Muatjetjeja ◽  
Chaudry Masood Khalique
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Conservation Laws ◽  
Fourth Order ◽  
Variable Coefficient ◽  
Boussinesq System ◽  
Order System ◽  
Conserved Quantities ◽  
Third Order ◽  
Partial Differential

We construct the conservation laws for a variable coefficient variant Boussinesq system, which is a third-order system of two partial differential equations. This system does not have a Lagrangian and so we transform it to a system of fourth-order, which admits a Lagrangian. Noether’s approach is then utilized to obtain the conservation laws. Lastly, the conservation laws are presented in terms of the original variables. Infinite numbers of both local and nonlocal conserved quantities are derived for the underlying system.

scholarly journals Conservation Laws for a Generalized Coupled Korteweg-de Vries System

2013 ◽  
Vol 2013 ◽  
pp. 1-5
Author(s):  
Daniel Mpho Nkwanazana ◽  
Ben Muatjetjeja ◽  
Chaudry Masood Khalique
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Conservation Laws ◽  
Fourth Order ◽  
Nonlinear Partial Differential Equations ◽  
Order System ◽  
Conserved Quantities ◽  
Third Order ◽  
The Third ◽  
New System

We construct conservation laws for a generalized coupled KdV system, which is a third-order system of nonlinear partial differential equations. We employ Noether's approach to derive the conservation laws. Since the system does not have a Lagrangian, we make use of the transformationu=Ux,v=Vxand convert the system to a fourth-order system inU,V. This new system has a Lagrangian, and so the Noether approach can now be used to obtain conservation laws. Finally, the conservation laws are expressed in theu,vvariables, and they constitute the conservation laws for the third-order generalized coupled KdV system. Some local and infinitely many nonlocal conserved quantities are found.

scholarly journals Method of functional parametrization for solving a semi-periodic initial problem for fourth-order partial differential equations

2020 ◽  
Vol 100 (4) ◽  
pp. 5-16
Author(s):  
A.T. Assanova ◽  
◽  
Zh.S. Tokmurzin ◽  
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Fourth Order ◽  
Periodic Problem ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Hyperbolic Type ◽  
Order System ◽  
Partial Differential ◽  
Initial Boundary Value

A semi-periodic initial boundary-value problem for a fourth-order system of partial differential equations is considered. Using the method of functional parametrization, an additional parameter is carried out and the studied problem is reduced to the equivalent semi-periodic problem for a system of integro-differential equations of hyperbolic type second order with functional parameters and integral relations. An interrelation between the semi-periodic problem for the system of integro-differential equations of hyperbolic type and a family of Cauchy problems for a system of ordinary differential equations is established. Algorithms for finding of solutions to an equivalent problem are constructed and their convergence is proved. Sufficient conditions of a unique solvability to the semi-periodic initial boundary value problem for the fourth-order system of partial differential equations are obtained.

Conservation laws and conserved quantities for the two-dimensional laminar wake

2016 ◽  
Vol 30 (28n29) ◽  
pp. 1640014 ◽  
Author(s):  
L. N. Kokela ◽  
D. P. Mason ◽  
A. J. Hutchinson
Keyword(s):  
Conservation Laws ◽  
Physical Significance ◽  
Order System ◽  
Conserved Quantities ◽  
Order Partial Differential Equation ◽  
Two Dimensional ◽  
Third Order ◽  
Partial Differential ◽  
Laminar Wake ◽  
Unified Method

A systematic and unified method is presented for the derivation of the conserved quantities for the laminar classical wake and the wake of a self-propelled body. The multiplier method for the derivation of conservation laws is applied to the far downstream wake equations expressed in terms of the velocity components which gives rise to a second-order system of two partial differential equations, and in terms of the stream function which results in one third-order partial differential equation. Once the corresponding conservation laws are obtained, they are integrated across the wake and upon imposing the boundary conditions and the condition that the drag on a self-propelled body is zero, the conserved quantities for the classical wake and the wake of a self-propelled body are derived. In addition, this method results in the discovery of a new laminar wake which may have physical significance.

scholarly journals Symmetry Analysis, Exact Solutions and Conservation Laws of a Benjamin–Bona–Mahony–Burgers Equation in 2+1-Dimensions

Symmetry ◽  
2021 ◽  
Vol 13 (11) ◽  
pp. 2083
Author(s):  
María S. Bruzón ◽  
Tamara M. Garrido-Letrán ◽  
Rafael de la Rosa
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Conservation Laws ◽  
Exact Solutions ◽  
Soliton Solutions ◽  
Third Order ◽  
Partial Differential ◽  
Symmetry Reductions ◽  
Multipliers Method ◽  
The Family

The Benjamin–Bona–Mahony equation describes the unidirectional propagation of small-amplitude long waves on the surface of water in a channel. In this paper, we consider a family of generalized Benjamin–Bona–Mahony–Burgers equations depending on three arbitrary constants and an arbitrary function G(u). We study this family from the standpoint of the theory of symmetry reductions of partial differential equations. Firstly, we obtain the Lie point symmetries admitted by the considered family. Moreover, taking into account the admitted point symmetries, we perform symmetry reductions. In particular, for G′(u)≠0, we construct an optimal system of one-dimensional subalgebras for each maximal Lie algebra and deduce the corresponding (1+1)-dimensional nonlinear third-order partial differential equations. Then, we apply Kudryashov’s method to look for exact solutions of the nonlinear differential equation. We also determine line soliton solutions of the family of equations in a particular case. Lastly, through the multipliers method, we have constructed low-order conservation laws admitted by the family of equations.

scholarly journals WELL-POSEDNESS OF A NONLOCAL PROBLEM WITH INTEGRAL CONDITIONS FOR THIRD ORDER SYSTEM OF THE PARTIAL DIFFERENTIAL EQUATIONS

2018 ◽  
pp. 33-41
Author(s):  
Assanova Anar Turmaganbetkyzy ◽  
Alikhanova Botakoz Zhorakhanovna ◽  
Nazarova Kulzina Zharkimbekovna
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Hyperbolic Equations ◽  
Nonlocal Problem ◽  
Second Order ◽  
Order System ◽  
Well Posedness ◽  
Integral Conditions ◽  
Third Order ◽  
Partial Differential

The nonlocal problem with integral conditions for the system of partial differential equations third-order is considered. The existence and uniqueness of classical solution to nonlocal problem with integral conditions for third-order system of partial differential equations are studied and the method for constructing their approximate solutions is proposed. Conditions of an unique solvability to nonlocal problem with integral conditions for third order system of partial differential equations are established. By introduction of new unknown functions, we have reduced the considered problem to an equivalent problem consisting of a nonlocal problem with integral conditions and parameters for a system of hyperbolic equations of second order and a integral relations. We have offered the algorithm for finding approximate solution to investigated problem and have proved its convergence. Sufficient conditions for the existence of unique solution to the equivalent problem with parameters are obtained. Well-posedness of the nonlocal problem with integral conditions for third order system of partial differential equations are established in the terms of well-posedness to nonlocal problem with integral conditions for system of hyperbolic equations second order. Key Words: third order partial differential equations, nonlocal problem, integral condition, system of hyperbolic equations second order, solvability, algorithm.

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