20 research outputs found
Lie symmetry and μ-symmetry methods for nonlinear generalized Camassa–Holm equation
Abstract
In this paper, a Lie symmetry method is used for the nonlinear generalized Camassa–Holm equation and as a result reduction of the order and computing the conservation laws are presented. Furthermore, μ-symmetry and μ-conservation laws of the generalized Camassa–Holm equation are obtained
On similarity and pseudo-similarity solutions of Falkner-Skan boundary layers
The present work deals with the two-dimensional incompressible,laminar,
steady-state boundary layer equations. First, we determinea family of velocity
distributions outside the boundary layer suchthat these problems may have
similarity solutions. Then, we examenin detail new exact solutions, called
Pseudo--similarity, where the external velocity varies inversely-linear with
the distance along the surface $ (U_e(x) = U_\infty x^{-1}). The present work
deals with the two-dimensional incompressible, laminar, steady-state boundary
layer equations. First, we determine a family of velocity distributions outside
the boundary layer such that these problems may have similarity solutions.
Then, we examenin detail new exact solutions. The analysis shows that solutions
exist only for a lateral suction. For specified conditions, we establish the
existence of an infinite number of solutions, including monotonic solutions and
solutions which oscillate an infinite number of times and tend to a certain
limit. The properties of solutions depend onthe suction parameter. Furthermore,
making use of the fourth--order Runge--Kutta scheme together with the shooting
method, numerical solutions are obtained.Comment: 15 page
Numerical treatment of first order delay differential equations using extended block backward differentiation formulae
In this research, we developed and implemented extended backward differentiation
methods (formulae) in block forms for step numbers k = 2, 3 and 4 to evaluate numerical solutions
for certain first-order differential equations of delay type, generally referred to as delay differential
equations (DDEs), without the use of interpolation methods for estimating the delay term. The
matrix inversion approach was applied to formulate the continuous composition of these block
methods through linear multistep collocation method. The discrete schemes were established
through the continuous composition for each step number, which evaluated the error constants,
order, consistency, convergent and area of absolute equilibrium of these discrete schemes. The
study of the absolute error results revealed that, as opposed to the exact solutions, the lower step
number implemented with super futures points work better than the higher step numbers
implemented with super future points