Computation of travelling wave solutions of scalar conservation laws with a stiff source term

2003 ◽  
Vol 32 (8) ◽  
pp. 1161-1178
Author(s):  
Vicente Martı́nez ◽  
Antonio Marquina
Keyword(s):  
Conservation Laws ◽  
Source Term ◽  
Travelling Wave ◽  
Travelling Wave Solutions ◽  
Scalar Conservation Laws ◽  
Wave Solutions ◽  
Scalar Conservation

Non-classical global solutions for a class of scalar conservation laws

2004 ◽  
Vol 134 (2) ◽  
pp. 225-240
Author(s):  
Paolo Baiti
Keyword(s):  
Conservation Laws ◽  
Global Solutions ◽  
Entropy Inequality ◽  
Travelling Wave ◽  
Scalar Conservation Laws ◽  
Kinetic Relation ◽  
Bounded Variation Functions ◽  
The Cauchy Problem ◽  
Scalar Conservation ◽  
The One

We consider the Cauchy problem for a class of scalar conservation laws with flux having a single inflection point. We prove existence of global weak solutions satisfying a single entropy inequality together with a kinetic relation, in a class of bounded variation functions. The kinetic relation is obtained by the travelling-wave criterion for a regularization consisting of balanced diffusive and dispersive terms. The result is applied to the one-dimensional Buckley-Leverett equation.

scholarly journals Lagrangian Formulation, Conservation Laws, Travelling Wave Solutions: A Generalized Benney-Luke Equation

Mathematics ◽  
2021 ◽  
Vol 9 (13) ◽  
pp. 1480
Author(s):  
Sivenathi Oscar Mbusi ◽  
Ben Muatjetjeja ◽  
Abdullahi Rashid Adem
Keyword(s):  
Conservation Laws ◽  
Exact Solutions ◽  
Lagrangian Density ◽  
Travelling Wave ◽  
Travelling Wave Solutions ◽  
Noether Symmetries ◽  
Extended Tanh Method ◽  
Wave Solutions ◽  
Tanh Method ◽  
Derivatives Of

The aim of this paper is to find the Noether symmetries of a generalized Benney-Luke equation. Thereafter, we construct the associated conserved vectors. In addition, we search for exact solutions for the generalized Benney-Luke equation through the extended tanh method. A brief observation on equations arising from a Lagrangian density function with high order derivatives of the field variables, is also discussed.

scholarly journals On the conservation laws and solutions of a (2+1) dimensional KdV-mKdV equation of mathematical physics

Open Physics ◽  
2018 ◽  
Vol 16 (1) ◽  
pp. 211-214 ◽  
Author(s):  
Tanki Motsepa ◽  
Chaudry Masood Khalique
Keyword(s):  
Partial Differential Equations ◽  
Conservation Laws ◽  
Elliptic Functions ◽  
Mathematical Physics ◽  
Travelling Wave ◽  
Mkdv Equation ◽  
Jacobi Elliptic Functions ◽  
Travelling Wave Solutions ◽  
Wave Solutions ◽  
Physical Phenomena

AbstractIn this paper, we study a (2+1) dimensional KdV-mKdV equation, which models many physical phenomena of mathematical physics. This equation has two integral terms in it. By an appropriate substitution, we convert this equation into two partial differential equations, which do not have integral terms and are equivalent to the original equation. We then work with the system of two equations and obtain its exact travelling wave solutions in form of Jacobi elliptic functions. Furthermore, we employ the multiplier method to construct conservation laws for the system. Finally, we revert the results obtained into the original variables of the (2+1) dimensional KdV-mKdV equation.

scholarly journals On the Solutions and Conservation Laws of a Coupled Kadomtsev-Petviashvili Equation

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Chaudry Masood Khalique
Keyword(s):  
Conservation Laws ◽  
Exact Solutions ◽  
Travelling Wave ◽  
Equation Method ◽  
Travelling Wave Solutions ◽  
Scientific Applications ◽  
Wave Solutions ◽  
Simplest Equation Method ◽  
Petviashvili Equation

A coupled Kadomtsev-Petviashvili equation, which arises in various problems in many scientific applications, is studied. Exact solutions are obtained using the simplest equation method. The solutions obtained are travelling wave solutions. In addition, we also derive the conservation laws for the coupled Kadomtsev-Petviashvili equation.

Undercompressive shocks and Riemann problems for scalar conservation laws with non-convex fluxes

1999 ◽  
Vol 129 (4) ◽  
pp. 733-754 ◽  
Author(s):  
Brian Hayes ◽  
Michael Shearer
Keyword(s):  
Initial Data ◽  
Conservation Laws ◽  
Riemann Problem ◽  
Initial Value Problem ◽  
Inflection Point ◽  
Travelling Wave ◽  
Problem Solution ◽  
Scalar Conservation Laws ◽  
Initial Value ◽  
Scalar Conservation

The Riemann initial value problem is studied for scalar conservation laws whose fluxes have a single inflection point. For a regularization consisting of balanced diffusive and dispersive terms, the travelling wave criterion is used to select admissible shocks. In some cases, the Riemann problem solution contains an undercompressive shock. The analysis is illustrated by exploring parameter space for the Buckley–Leverett flux. The boundary of the set of parameters for which there is a physical solution of the Riemann problem for all data is computed. Within the region of acceptable parameters, the solution hasseveral different forms, depending on the initial data; the different forms are illustrated by numerical computations. Qualitatively similar behaviour is observed in Lax–Wendroff approximations of solutions of the Buckley–Leverett equation with no dissipation or dispersion.

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