arXiv subjects

Julius Fergy T. Rabago

Publications and source records attributed to Julius Fergy T. Rabago.

11 recordsLinked to original sources

arXiv2023

Numerical solution to a free boundary problem for the Stokes equation using the coupled complex boundary method in shape optimization settings

A new reformulation of a free boundary problem for the Stokes equations governing a viscous flow with overdetermined condition on the free boundary is proposed. The idea of the method is to transform the governing equations to a boundary value problem with a complex Robin boundary condition coupling the two boundary conditions on the free boundary. The proposed formulation give rise to a new cost functional that apparently has not been exploited yet in the literature, specifically, and at least, in the context of free surface problems. The shape derivatives of the cost function constructed by the imaginary part of the solution in the whole domain in order to identify the free boundary is explicitly determined. Using the computed shape gradient information, a domain variation method from a preconditioned steepest descent algorithm is applied to solve the shape optimization problem. Numerical results illustrating the applicability of the method is then provided both in two and three spatial dimensions. For validation and evaluation of the method, the numerical results are compared with the ones obtained via the classical tracking Dirichlet data.

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arXiv2022

On the new coupled complex boundary method in shape optimization framework for solving stationary free boundary problems

We expose here a novel application of the so-called coupled complex boundary method -- first put forward by Cheng et al. (2014) to deal with inverse source problems -- in the framework of shape optimization for solving the exterior Bernoulli problem, a prototypical model of stationary free boundary problems. The idea of the method is to transform the overdetermined problem to a complex boundary value problem with a complex Robin boundary condition coupling the Dirichlet and Neumann boundary conditions on the free boundary. Then, we optimize the cost function constructed by the imaginary part of the solution in the whole domain in order to identify the free boundary. We also prove the existence of the shape derivative of the complex state with respect to the domain. Afterwards, we compute the shape gradient of the cost functional, and characterize its shape Hessian at the optimal domain under a strong, and then a mild regularity assumption on the domain. We then examine the instability of the proposed method by proving the compactness of the latter expression. Also, we devise an iterative algorithm based on a Sobolev gradient scheme via finite element method to solve the minimization problem. Finally, we illustrate the applicability of the method through several numerical examples, both in two and three spatial dimensions.

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arXiv2017

Forbidden Set of the Rational Difference Equation $x_{n+1} = x_n x_{n-k}/(ax_{n-k+1} +x_n x_{n-k+1} x_{n-k})$

This short note aims to answer one of the open problems raised by F. Balibrea and A. Cascales in \cite{bc}. In particular, the forbidden set of the nonlinear difference equation $x_{n+1} = x_n x_{n-k}/(ax_{n-k+1} +x_n x_{n-k+1} x_{n-k})$, where $k$ is a positive integer and $a$ is a positive constant, is found by first computing the closed form solution of the given equation. Additional results regarding the limiting properties and periodicity of its solutions are also discussed. Numerical examples are also provided to illustrate the exhibited results. Lastly, a possible generalization of this present work is offered as an open problem.

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arXiv2016

On the solutions of a second-order difference equations in terms of generalized Padovan sequences

This paper deals with the solution, stability character and asymptotic behavior of the rational difference equation \begin{equation*} x_{n+1}=\frac{αx_{n-1}+β}{ γx_{n}x_{n-1}},\qquad n \in \mathbb{N}_{0}, \end{equation*} where $\mathbb{N}_{0}=\mathbb{N}\cup \left\{0\right\}$, $α,β,γ\in\mathbb{R}^{+}$, and the initial conditions $x_{-1}$ and $x_{0}$ are non zero real numbers such that their solutions are associated to generalized Padovan numbers. Also, we investigate the two-dimensional case of the this equation given by \begin{equation*} x_{n+1} = \frac{αx_{n-1} + β}{γy_n x_{n-1}}, \qquad y_{n+1} = \frac{αy_{n-1} +β}{γx_n y_{n-1}} ,\qquad n\in \mathbb{N}_0, \end{equation*} and this generalizes the results presented in \cite{yazlik}

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arXiv2016

On the Closed-Form Solution of a Nonlinear Difference Equation and Another Proof to Sroysang's Conjecture

The purpose of this paper is twofold. First, we derive theoretically, using appropriate transformation on $x_n$, the closed-form solution of the nonlinear difference equation \[ x_{n+1} = \frac{1}{\pm 1 + x_n},\qquad n\in \mathbb{N}_0. \] We mention that the solution form of this equation was already obtained by Tollu et al. in 2013, but through induction principle, and one of our purpose is to clearly explain how was the formula appeared in such structure. After that, with the solution form of the above equation at hand, we prove a case of Sroysang's conjecture (2013); i.e., given a fixed positive integer $k$, we verify the validity of the following claim: \[ \lim_{x \rightarrow \infty}\left\{ \frac{f(x+k)}{f(x)}\right\}= ϕ, \] where $ϕ=(1+\sqrt{5})/2$ denotes the well-known golden ratio and the real valued function $f$ on $\mathbb{R}$ satisfies the functional equation $f(x+2k)=f(x+k) + f(x)$ for every $x\in \mathbb{R}$. We complete the proof of the conjecture by giving out an entirely different approach for the other case.

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arXiv2016

Solution Form of a Higher Order System of Difference Equation and Dynamical Behavior of Its Special Case

The solution form of the system of nonlinear difference equations \begin{equation*} x_{n+1} = \frac{x_{n-k+1}^{p}y_{n}}{a y_{n-k}^{p}+b y_{n}},\ y_{n+1} = \frac{y_{n-k+1}^{p}x_{n}}{αx_{n-k}^{p}+βx_{n}}, \quad n, p \in \mathbb{N}_{0},\ k\in \mathbb{N}, \end{equation*} where the coefficients $a, b, α, β$ and the initial values $x_{-i},y_{-i},i\in\{0,1,\ldots,k\}$ are real numbers, is obtained. Furthermore, the behavior of solutions of the above system when $p=1$ is examined. Numerical examples are presented to illustrate the results exhibited in the paper.

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arXiv2016

Olver's Method for Approximating Roots of p-Adic Polynomials Equations

Let $\mathbb{Z}_p[x]$ be the set of all functions whose coefficients are in the field of $p$-adic integers $\mathbb{Z}_p$. This work considers a problem of finding a root of a polynomial equation $P(x)=0$ where $P(x)\in\mathbb{Z}_p[x]$. The solution is approached through an analogue of Olver's method for finding roots of polynomial equations $P(x)=0$ in $\mathbb{Z}_p$.

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arXiv2015

On Two Nonlinear Difference Equations

The behavior of solutions of the following nonlinear difference equations \[ x_{n+1}=\displaystyle\frac{q}{p+x_n^ν} \quad \text{and} \quad y_{n+1}=\displaystyle\frac{q}{-p+y_n^ν}, \] where $p, q \in\mathbb{R}^+$ and $ν\in \mathbb{N}$ are studied. The solution form of these two equations when $ν=1$ are expressed in terms of Horadam numbers. Furthermore, the behavior of their solutions are investigated for all integer $ν> 0$ and several numerical examples are presented to illustrate the results exhibited. The present work generalizes those seen in [{\it Adv. Differ. Equ.}, {\bf 2013}:174 (2013), 7 pages].

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arXiv2015

Effective Methods on Determining the Periodicity and Form of Solutions of Some Systems of Nonlinear Difference Equations

Recently, various systems of nonlinear difference equations, of different forms, were studied. In this existing work, two earlier published papers, due respectively to Bayram and Das. [Appl. Math. Sci. (Ruse), 4(7) (2010) pp. 817-821] and Elsayed [Fasciculi Mathematici, 40 (2008), pp. 5-13], are revisited. The results exhibited in these previous investigations are re-examined through a new approach, more theoretical and explanative compared to the ones offered in these aforementioned works. Furthermore, the qualitative behavior of solutions of a system of nonlinear difference equations of higher-order is investigated through analytical methods. The system, which is considered here, generalizes those that are first presented in [Fasciculi Mathematici, 40 (2008), pp. 5-13] but are treated differently from this pre-existing work. The results delivered here are important, not only for the reason that they provide theoretical explanations on several earlier results, but also because the system being studied can be use to model real-life phenomena exhibiting periodic behaviors.

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arXiv2015

On Linear Recursive Sequences with Coefficients in Arithmetic-Geometric Progressions

We present a certain generalization of a recent result of M. I. Cirnu on linear recurrence relations with coefficient in progressions [2]. We provide some interesting examples related to some well-known integer sequences, such as Fibonacci sequence, Pell sequence, Jacobsthal sequence, and the Balancing sequence of numbers. The paper also provides several approaches in solving the linear recurrence relation under consideration. We end the paper by giving out an open problem.

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arXiv2015

On Generalized Fibonacci Numbers

We provide a formula for the $n^{th}$ term of the $k$-generalized Fibonacci-like number sequence using the $k$-generalized Fibonacci number or $k$-nacci number, and by utilizing the newly derived formula, we show that the limit of the ratio of successive terms of the sequence tends to a root of the equation $x + x^{-k} = 2$. We then extend our results to $k$-generalized Horadam ($k$GH) and $k$-generalized Horadam-like ($k$GHL) numbers. In dealing with the limit of the ratio of successive terms of $k$GH and $k$GHL, a lemma due to Z. Wu and H. Zhang [8] shall be employed. Finally, we remark that an analogue result for $k$-periodic $k$-nary Fibonacci sequence can also be derived.

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