The Bäcklund Transformations and Abundant Exact Explicit Solutions for a General Nonintegrable Nonlinear Convection-Diffusion Equation

The Bäcklund Transformations and Abundant Exact Explicit Solutions for a General Nonintegrable Nonlinear Convection-Diffusion Equation

and Applied Analysis 3 From 2.1 , we have ut f ′′ ( φ ) φxφt f ′ ( φ ) φxt u1t, 2.2 ux f ′′ ( φ ) φ2 x f ′φ ) φxx u1x, 2.3 uxx f ′′′ ( φ ) φ3 x 3f ′′φ ) φxφxx f ′ ( φ ) φxxx u1xx, 2.4 u2 ( f ′ )2( φ ) φ2 x 2f ′φxu1 x, t u1 x, t , u3 ( f ′ )3( φ ) φ3 x 3 ( f ′ )2 φ2 xu1 x, t 3f φxu1 x, t u 3 1 x, t . 2.5 Substituting 2.1 – 2.5 into the left side of 1.1 and collecting all terms with φ3 x, we obta...

Exact and numerical solutions for nonlinear differential equation of Jeffrey-Hamel flow

Finite Element Methods for Convection Diffusion Equation

This paper deals with the finite element solution of the convection diffusion equation in one and two dimensions. Two main techniques are adopted and compared. The first one includes Petrov-Galerkin based on Lagrangian tensor product elements in conjunction with streamlined upwinding. The second approach represents Bubnov/Petrov-Galerkin schemes based on a new group of exponential elements. It ...

Exact Multisoliton Solutions of General Nonlinear Schrödinger Equation with Derivative

Multisoliton solutions are derived for a general nonlinear Schrödinger equation with derivative by using Hirota's approach. The dynamics of one-soliton solution and two-soliton interactions are also illustrated. The considered equation can reduce to nonlinear Schrödinger equation with derivative as well as the solutions.

Exact Solutions of Diffusion–Convection Equations

where f = f(x), g = g(x), h = h(x), A = A(u) and B = B(u) are arbitrary smooth functions of their variables, f(x)g(x)A(u) 6=0. Our aim is not to give a physical interpretation of the solution of diffusion equations (that is too huge and cannot be reached in the scope of a short paper), but to list the already known exact solutions of equations from the class under consideration. However, in som...