Some oscillation criteria for solutions of a general ordinary differential equation of second order of the form (r(t)ψ(x(t))ẋ(t))+h(t)ẋ(t)+q(t)φ(g(x(t)),r(t)ψ(x(t))ẋ(t)) = H(t,x(t), ẋ(t)) with alternating coefficients are discussed. Our results improve and extend some existing results in the literature. Some illustrative examples are given with its numerical solutions which are computed using R...

The piecewise variational iteration method (VIM) for solving Riccati differential equations (RDEs) provides a solution as a sequence of iterates. Therefore, its application to RDEs leads to the calculation of terms that are not needed and more time is consumed in repeated calculations for series solutions. In order to overcome these shortcomings, we propose an easy-to-use piecewise-truncated VI...

Long time solutions to the Frank-Kamenetskii partial differential equation modelling a thermal explosion in a vessel are obtained using matrix exponentiation. Spatial derivatives are approximated by high-order finite difference approximations. A forward difference approximation to the time derivative leads to a Lawson-Euler scheme. Computations performed with a BDF approximation to the time der...

We introduce a novel local time-stepping technique for marching-in-time algorithms. The technique is denoted as Causal-Path Local Time-Stepping (CPLTS) and it is applied for two time integration techniques: fourth order low–storage explicit Runge–Kutta (LSERK4) and second order Leapfrog (LF2). The CPLTS method is applied to evolve Maxwell’s curl equations using a Discontinuous Galerkin (DG) sch...

Continuous Galerkin Petrov time discretization scheme is tested on some Hamiltonian systems including simple harmonic oscillator, Kepler’s problem with different eccentricities and molecular dynamics problem. In particular, we implement the fourth order Continuous Galerkin Petrov time discretization scheme and analyze numerically, the efficiency and conservation of Hamiltonian. A numerical comp...

In this paper, numerical simulations of nonlinear Fisher’s equation in oneand twodimensions have been considered. The derivatives and integrals are replaced by the necessary matrices, and the resulting algebraic system of equations was advanced by the popular fourth-order exponential time differencing Runge-Kutta (ETDRK4) schemes proposed by Cox and Matthew [Exponential time differencing for st...

Using the notion of integrating factors, Lawson developed a class of numerical methods for solving stiff systems of ordinary differential equations. However, the performance of these “Generalized Runge–Kutta processes” was demonstrably poorer when compared to the ETD schemes of Certaine and Nørsett, recently rediscovered by Cox and Matthews. The deficit is particularly pronounced when the schem...

We describe our progress in the development of a fourth-order, finite-volume discretization of a nonlinear, full-f gyrokinetic Vlasov-Poisson system in mapped coordinates. The approach treats the configuration and velocity components of phase space on an equal footing, using a semi-discretization with limited centered fluxes combined with a fourth-order Runge Kutta integration. The algorithm ha...