Numerical solution of differential-algebraic equations with Hessenberg index-3 is considered by variational iteration method. We applied this method to two examples, and solutions have been compared with those obtained by exact solutions.
By constructing a Lyapunov function, a new instability result is established, which guarantees that the trivial solution of a certain nonlinear vector differential equation of the fifth order is unstable. An example is also given to illustrate the importance of the result obtained. By this way, our findings improve an instability result related to a scalar differential equation in the literature to instability of the trivial solution to the afore-mentioned differential equation.
In this article, the generalized Rosenau-KdV equation is split into two subequations such that one is linear and the other is nonlinear. The resulting subequations with the prescribed initial and boundary conditions are numerically solved by the first order Lie-Trotter and the second-order Strang time-splitting techniques combined with the quintic B-spline collocation by the help of the fourth order Runge-Kutta (RK-4) method. To show the accuracy and reliability of the proposed techniques, two test problems having exact solutions are considered. The computed error norms L 2 and L ∞ with the conservative properties of the discrete mass Q(t) and energy E(t) are compared with those available in the literature. The convergence orders of both techniques have also been calculated. Moreover, the stability analyses of the numerical schemes are investigated.
Numerical computations for natural systems and acquiring travelling wave
solutions of nonlinear wave equations in relation to sciences such as
optics, fluid mechanics, solid state physics, plasma physics, kinetics,
and geology have become very important in the field of mathematical
modeling recently. For this, many methods have been suggested. The
strategy applied for this article is to obtain more perfect numerical
solutions of Modified Equal Width equation (MEW), which is one of the
equations used to model the nonlinear phenomena mentioned. For this
purpose, the Lie-Trotter splitting technique is applied to the MEW
equation. Firstly, the problem is split into two sub-problems, one
linear and the other nonlinear, containing derivative with respect to
time. Secondly, each subproblem is reduced to the algebraic equation
system by using collocation finite element method (FEM) based on the
quintic B-spline approximate functions for spatial discretization and
the convenient classical finite difference approaches for temporal
discretization. Then, the obtained systems are solved with the Lie
Trotter splitting algorithm. Explanatory test problems are considered,
showing that the newly proposed algorithm has superior accuracy than
previous methods, and the numerical results produced by the proposed
algorithm are shown in tables and graphs. In addition, the stability
analysis of the new approach is examined. Therefore, it is appropriate
to state that this new technique can be easily applied to partial
differential equations used in other disciplines in terms of the results
obtained and the cost of Matlab calculation software.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.