Symmetries and Solvability of a Class of Higher Order Systems of Ordinary Difference Equations

We perform Lie analysis for a system of higher order difference equations with variable coefficients and derive non-trivial symmetries. We use these symmetries to find exact formulas for the solutions in terms of k. Furthermore, a detailed study for a specific value of k is presented. Our findings generalize some results in the literature.

Advanced fibonacci sequence and its sum by second order variable co-efficient difference operator

We introduce a second order difference operator with specific powers of variable co-efficient and its inverse in this study, which allows us to derive the (α1tr1, α2tr2 )-Fibonacci sequence and its summation. This series is known as the Fibonacci sequence with variable co-efficients (VCFS). On the sum of the terms of the variable co-efficient Fibonacci sequence, some theorems and intriguing findings are generated. To demonstrate our findings, appropriate instances arepresented.

Behaviors of the Solutions via Their Closed-Form Formulas for Two Rational Third-Order Difference Equations

In this work, we derive the solution formulas and study their behaviors for the difference equations x n + 1 = α x n x n − 3 / − β x n − 3 + γ x n − 2 , n ∈ ℕ 0 and x n + 1 = α x n x n − 3 / β x n − 3 − γ x n − 2 , n ∈ ℕ 0 with real initials and positive parameters. We show that there exist periodic solutions for the second equation under certain conditions when β 2 < 4 α γ . Finally, we give some illustrative examples.

Valuation of the American put option as a free boundary problem through a high-order difference scheme

Valuation of the American options encountered commonly in finance is quite difficult due to the possibility of early exercise alternatives. Since an exact solution for the American options does not exist, effective numerical methods are needed to understand the behavior of option pricing models. Therefore, in this paper, a new approach based on a high-order difference scheme is proposed to discuss the valuation of an American put option as a free boundary problem. Using a front-fixing approach that transforms the unknown free boundary (optimal stopping) into a fixed one, a sixth-order finite difference scheme (FD6) in space and a third-order strong-stability preserving Runge–Kutta (SSPRK3) in time are applied to the model converted to a nonlinear partial differential equation. The computed results revealed that the combined method is seen to attempt to pull up the capacity of the algorithm to achieve higher accuracy. It is seen that the quantitative and qualitative results produced by the method proposed with minimal computational effort are sufficiently accurate and meaningful. Therefore, this article provides some new insights about the physical characteristics of financial problems and such realistic phenomena.

Hadean/Eoarchean Tectonics and Mantle Mixing Induced by Impacts: a Three-dimensional Study

The timing of the onset of plate tectonics on Earth remains a topic of strong debate, as does the tectonic mode that preceded modern plate tectonics. Understanding possible tectonic modes and transitions between them is also important for other terrestrial planets such as Venus and rocky exoplanets. Recent two-dimensional modelling studies have demonstrated that impacts can initiate subduction during the early stages of terrestrial planet evolution - the Hadean and Eoarchean in Earth's case. Here, we perform three-dimensional simulations of the influence of ongoing multiple impacts on early Earth tectonics and its effect on the distribution of compositional heterogeneity in the mantle, including the distribution of impactor material. We compare two-dimensional and three-dimensional simulations to determine when geometry is important. Results show that impacts can induce subduction in both 2-D and 3-D and thus have a great influence on the tectonic regime. The effect is particularly strong in cases that otherwise display stagnant-lid tectonics: impacts can shift them to having a plate-like regime. In such cases, however, plate-like behaviour is temporary: as the impactor flux decreases the system returns to what it was without impacts. Impacts result in both greater production of oceanic crust and greater recycling of it, increasing the build-up of subducted crust above the core-mantle boundary and in the transition zone. Impactor material is mainly located in the upper mantle, at least at the end of the modelled 500 million year period. In 2-D simulations, in contrast to 3-D simulations, impacts are less frequent but each has a larger effect on surface mobility, making the simulations more stochastic. These stronger 2-D subduction events can mix both recycled basalt and impactor material into the lower mantle. These results thus demonstrate that impacts can make a first-order difference to the early tectonics and mantle mixing of Earth and other large terrestrial planets, and that three-dimensional simulations are important so that effects are not over- or under-predicted.