Julius Fergy Tiongson Rabago

• PhD
• JSPS Postdoctoral Fellow at Kanazawa University

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....

This study revisits the problem of identifying the unknown interior Robin boundary of a connected domain using Cauchy data from the exterior region of a harmonic function. It investigates two shape optimization reformulations employing least-squares boundary-data-tracking cost functionals. Firstly, it rigorously addresses the existence of optimal s...

This paper explores the reconstruction of a space-dependent parameter in inverse diffusion problems, proposing a shape-optimization-based approach. The main objective is to recover the absorption coefficient from a single boundary measurement. While conventional gradient-based methods rely on the Fr\'{e}chet derivative of a cost functional with res...

This paper introduces a method for estimating the shape and location of an embedded tumor. The approach utilizes shape optimization techniques, applying the coupled complex boundary method. By rewriting the problem--characterized by a measured temperature profile and corresponding flux (e.g., from infrared thermography)--into a complex boundary val...

The alternating direction method of multipliers within a shape optimization framework is developed for solving geometric inverse problems, focusing on a cavity identification problem from the perspective of non-destructive testing and evaluation techniques. The rationale behind this method is to achieve more accurate detection of unknown inclusions...

The purpose of this study is twofold. First, we revisit a shape optimization reformulation of a prototypical shape inverse problem and briefly propose a simple yet efficient numerical approach for solving the minimization problem. Second, we examine the existence, uniqueness, and continuous dependence of a classical solution to a Hele-Shaw-like sys...

We propose a novel application of the alternating direction method of multipliers (ADMM) to shape inverse problems in a shape optimization setting. Specifically, we address the problem of identifying a perfectly conducting inclusion inside a larger bounded domain from boundary measurements.

A new reformulation of a free boundary problem for the Stokes equations, which govern a viscous flow with an overdetermined condition on the free boundary, is proposed. The idea of the method is to transform the governing equations into a boundary value problem with a complex Robin boundary condition that couples the two boundary conditions on the...

The exterior Bernoulli problem — a prototype stationary free boundary problem — is rephrased into a shape optimization setting using an energy-gap type cost functional that is subject to two auxiliary problems: a pure Dirichlet problem and a mixed Dirichlet-Robin boundary value problem. It is demonstrated here that depending on what method is used,...

The alternating direction method of multipliers within a shape optimization framework is developed for solving geometric inverse problems, focusing on a cavity identification problem from the perspective of non-destructive testing and evaluation techniques. The rationale behind this method is to achieve more accurate detection of unknown inclusions...

A Lagrangian-type numerical scheme called the “comoving mesh method” or CMM is developed for numerically solving certain classes of moving boundary problems which include, for example, the classical Hele-Shaw flow problem and the well-known mean curvature flow problem. This finite element scheme exploits the idea that the normal velocity field of t...

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 t...

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...

A Lagrangian-type numerical scheme called the "comoving mesh method" or CMM is developed for numerically solving certain classes of moving boundary problems which include, for example, the classical Hele-Shaw flow problem and the well-known mean curvature flow problem. This finite element scheme exploits the idea that the normal velocity field of t...

The exterior Bernoulli problem is rephrased into a shape optimization problem using a new type of objective function called the Dirichlet-data-gap cost function which measures the \(L^2\)-distance between the Dirichlet data of two state functions. The first-order shape derivative of the cost function is explicitly determined via the chain rule appr...

This paper provides alternative techniques on solving some systems of difference equations. These techniques are analytical and much explanatory in nature as compared to methods used in existing literatures. We applied these methods particularly to the systems studied by Touafek in his paper Touafek (Iran J Math Sci Info 9(2): 303–305, 2014, [33])....

We propose a new shape optimization formulation of the Bernoulli problem by tracking the Neumann data. The associated state problem is an equivalent formulation of the Bernoulli problem with a Robin condition. We devise an iterative procedure based on a Lagrangian-like approach to numerically solve the minimization problem. The proposed scheme invo...

We generalise a recent result of Mansour et al. (2012) and study other related systems that deal with the dynamics of a competitive population model described by a system of nonlinear difference equations. Particularly, we consider a discrete-competitive system of the form (Iqueation Persented) where k ∈ ℕ and f: ℝ \ F f → ℝ and g: ℝ \ F g → ℝ, whe...

The solution to a free boundary problem of Bernoulli type, also known as Alt-Caffarelli problem, is studied via shape optimization techniques. In particular, a novel energy-gap cost functional approach with a state constraint consisting of a Robin condition is proposed as a shape optimization reformulation of the problem. Accordingly, the shape der...

We aim to identify the geometry (i.e., the shape and location) of a cavity inside an object through the concept of thermal imaging. More precisely, we present an identification procedure to determine the geometric shape of a cavity with convective boundary condition in a heat-conducting medium using the measured temperature on a part of the surface...

The exterior Bernoulli free boundary problem is considered and reformulated into a shape optimization setting wherein the Neumann data is being tracked. The shape differentiability of the cost functional associated with the formulation is studied, and the expression for its shape derivative is established through a Lagrangian formulation coupled wi...

Intraguild (IG) predation, which is a specific case of omnivory with an inclusion of competition for a shared resource, is known as the most basic system among food webs. The simplest form consists of three species: the IG predator, the IG prey, and the shared prey/resource. This study considers a food web model where the interaction between the sp...

Abstract. We re-examine the system of difference equations given by xn−(2k−1) ε+δxn−(2k−1) yn−(k−1) yn−(2k−1) ρ+σyn−(2k+1) xn−(k−1) , where ε, δ, ρ, σ ∈ {−1, 1} and k ∈ N, with real initial values {xn }0n=−(2k−1) and {yn}0n=−(2k−1) such that δxm−(2k−1) ym−(k−1) ̸= −ε and σym−(2k+1) xm−(k−1) ̸= −ρ for all possible values of m and k. More precisely,...

In this article, a class M2 of admissible perturbations of the special expression M0 = Σrk=0cktαkDtρk in the weighted space ℓ²ω([1,∞)) will be presented. It will be shown that the operator ωM2ω-1/2, where ω belongs to the family of completely monotonic functions, is an admissible perturbation of M0 in the non-weighted space ℓ²([1,∞)), and eventuall...

This paper deals with the solution, stability character and asymptotic behavior of the rational difference equation $$\begin{array}{} \displaystyle x_{n+1}=\frac{\alpha x_{n-1}+\beta}{ \gamma x_{n}x_{n-1}},\qquad n \in \mathbb{N}_{0}, \end{array}$$ where ℕ 0 = ℕ ∪ {0}, α , β , γ ∈ ℝ ⁺ , and the initial conditions x–1 and x0 are non zero real number...

In this study, an optimal control problem is formulated to a predatorprey model with disease in the prey population. This model is an adapted Lotka-Volterra model but with an applied SI epidemic dynamics on the prey population. Two controls are then applied to the system: first, a separating control, that is intended to separate the sound prey from...

In this study, an optimal control problem is formulated to a predator-prey model with disease in the prey population. This model is an adapted Lotka-Volterra model but with an applied SI epidemic dynamics on the prey population. Two controls are then applied to the system: first, a separating control, that is intended to separate the sound prey fro...

In Yang et al. (Acta Math Univ Comenianae LXXX(1):63–70, 2011), Yang, Chen, and Shi examined the system of difference equations: xn=ayn-p,yn=byn-pxn-qyn-q,n=0,1,…,where q is a positive integer with p<q, p∤q, and p≥3 is an odd number, both a and b are nonzero real constants, and the initial values x-q+1,x-q+2,…,x0,y-q+1,y-q+2,…,y0 are nonzero real n...

The exterior Bernoulli free boundary problem is reformulated into a shape optimization setting by tracking the Dirichlet data. The shape derivative of the corresponding cost functional is established through a Lagrangian formulation coupled with the velocity method. A numerical example using the traction method or H¹ gradient method is also provide...

This note offers a simple, yet very intriguing application of telescoping sums. Particularly, the cancellation technique which is known as the method of differences is employed to establish analytically the closed-form solutions of some systems of nonlinear difference equations. The results delivered here, in addition, generalizes several results f...

We show that the Diophantine equation 2x+ 17y = z^2, has exactlyve solutions (x; y; z) in positive integers. The only solutions are (3; 1; 5), (5; 1; 7),(6; 1; 9), (7; 3; 71) and (9; 1; 23). This note, in turn, addresses an open problemproposed by Sroysang in [10]. DOI : http://dx.doi.org/10.22342/jims.22.2.422.177-182

This note deals with the solution form of the system of difference equations $$\begin{aligned} x_{n+1}=\frac{a x_{n}y_{n-1}}{y_{n}-\alpha }+\beta ,\quad y_{n+1}=\frac{b x_{n-1}y_{n}}{x_{n}-\beta }+\alpha ,\quad n \in {\mathbb {N}}_{0}, \end{aligned}$$where the parameters \(a,\,b,\,\alpha ,\,\beta \) and initial values \(x_{-i},\,y_{-i},\,i=0,1\), a...

This work supplements the paper [Closed form solutions of some systems of rational difference equations in terms of Fibonacci numbers, Dynam. Cont. Dis. Ser. A, 21(6) (2014), 473-486.]. That is, an alternative proof - short and elegant - is offered in order to explain theoretically the results presented in the paper which were established through a...

We consider the root-finding problem f (x) = 0, f ∈ Zp[x], and seek a root in Zp of this equation through a p-adic analogue of Newton-Raphson method. We show in particular that, under appropriate assumptions, the sequence of approximants generated by the iterative formula of the Newton-Raphson method converges to a unique root of f in Zp. Also, we...

Various systems of non-linear difference equations, of different forms, were studied in the last two decades or so. In this existing work, two earlier published papers, due, respectively, to Bayram and Das and Elsayed, are revisited. The results exhibited in these previous investigations are re-examined through a new approach, more theoretical and...

This note is the second of the two supplements to the paper [Closed form solutions of some systems of rational difference equations in terms of Fibonacci numbers, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal., 21(6), (2014), 473–486.]. As in the first supplemental note, an alternative proof – more informative and detailed – is presented in...

This paper deals with the solution, stability character and asymptotic behavior of the rational difference equation \begin{equation*} x_{n+1}=\frac{\alpha x_{n-1}+\beta}{ \gamma x_{n}x_{n-1}},\qquad n \in \mathbb{N}_{0}, \end{equation*} where $\mathbb{N}_{0}=\mathbb{N}\cup \left\{0\right\}$, $\alpha,\beta,\gamma\in\mathbb{R}^{+}$, and the initial c...

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}}{\alpha x_{n-k}^{p}+\beta x_{n}}, \quad n, p \in \mathbb{N}_{0},\ k\in \mathbb{N}, \end{equation*} where the coefficients $a, b, \alpha, \beta$ and the initial values $x_...

Hensel’s lemma is an important result in valuation theory which gives information on finding roots of polynomials. A classical application of this result deals with the problem of finding roots of a p-adic number a in the set of p-adic numbers ℚp. Lately, there are several investigations concerning the problem of finding roots of p-adic numbers usi...

Let Zp[x] be the set of all functions whose coe±cients are in the field of p-adic integers Zp. This work considers a problem of finding a root of a polynomial equation P(x) = 0 where P(x) in Zp [x]. The solution is approximated through an analogue of Olver’s method for finding roots of polynomial equations P(x) = 0 in Zp.

We derive the closed-form solutions of the following difference equations x n + 1 = x n − 3 ± 1 + x n − 2 x n , n = 0 , 1 , 2 , …

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 in...

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 sol...

This work oers an analogue of Halley's method for solving a rootnding problem f(x) = 0 in the p-adic setting. In particular, a root of the general polynomial equation f(x) = a0+a1x+a2x2+..+aqxq = 0, where f ∈ Zp[x] and Zpdenotes the set of p-adic integers, is computed through Halley's method.

Let be the sequence of k-Fibonacci numbers recursively defined by and m be a fixed positive integer. In this work we prove that, for almost every x (0,1), the pattern k, k,..., k (comprising of m-digits) appears in the continued fraction expansion x = [0; a1,a2,. ..] with frequency where where

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)=...

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}}{\alpha x_{n-k}^{p}+\beta x_{n}}, \quad n, p \in \mathbb{N}_{0},\ k\in \mathbb{N}, \end{equation*} where the coefficients $a, b, \alpha, \beta$ and the initial values $x_...

Let w be a real-valued function on ℝ and k be a positive integer. If for every real number x, w (x + 2k) = rw (x + k) + sw(x) for some non-negative real numbers r and s, then we call such function a second-order linear recurrent function with period k. Similarly, we call a function w: ℝ→ℝ satisfying w(x + 2k) = - rw (x + k) + sw(x) an odd second-or...

The exterior Bernoulli free boundary problem was studied via shape optimization technique. The problem was reformulated into the minimization of the so-called Kohn-Vogelius objective functional, where two state variables involved satisfy two boundary value problems, separately. The paper focused on solving the second-order shape derivative of the o...

The behavior of solutions of the following nonlinear difference equations \[ x_{n+1}=\displaystyle\frac{q}{p+x_n^{\nu}} \quad \text{and} \quad y_{n+1}=\displaystyle\frac{q}{-p+y_n^{\nu}}, \] where $p, q \in\mathbb{R}^+$ and $\nu\in \mathbb{N}$ are studied. The solution form of these two equations when $\nu =1$ are expressed in terms of Horadam numb...

We say that w(x) : R → C is a solution to a second-order linear recurrent homogeneous differential equation with period k (k є N), if it satisfies a homogeneous differential equation of the form w(2k)(x) = pw(k)(x) + qw(x), ∀x є R, where p, q є R+ and w(k)(x) is the kth derivative of w(x) with respect to x. On the other hand, w(x) is a solution to...

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 in...

The paper deals with polynomials of the form f(x)=x^m-ϵ_1 a_1 x^(m-1)-ϵ_2 a_2 x^(m-2)- ...-ϵ_(m-1) a_(m-1) x- ϵ_m a_m , where ϵ_i ∈{-1,1} for i=1,2,3,…,m. It is shown that for any positive integers a_1,a_2,…,a_m with a_1≥a_2≥⋯ ≥a_m≥1 and m∈N with m≥2, f(x) has unique real zero outside the unit disk |z|≤1. It is also presented in this paper that the...

The main purpose of this paper is to correct the result of A. Suvarnamani that was published in this journal. In particular, A. Suvarnamani showed in [6] that (p, q, x, y, z) = (3, 5, 1, 0, 2) is the “unique solution” to the Diophantine equation p^x + q^y = z^2, (1) where p is an odd prime, q−p = 2 and x, y and z are non-negative integers. The au...

We present some new elementary properties of modified Jacobsthal (Atanassov, 2011) and Jacobsthal–Lucas numbers (Shang, 2012).

We establish all solutions of the title equation for p = 3, and determine various solutions for p > 3. Our approach is different from what is presented in [4].

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 resu...

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 s...

The paper provides a further generalization of the sequences of numbers in generalized arithmetic and geometric progressions [1].

The paper provides a generalization of the arithmetic-geometric alternate sequence introduced recently by Rabago [2].

We offer another elementary approach to the solution of the title equation in positive integers for n = 1. Three other approaches have already been presented in [9], [10], and [3]. Also, we give various solutions of the title equation for n > 1.

We exhaust all solutions of the Diophantine equation 3^x + 5^y + 7^z = w^2 in non-negative integers using elementary methods.

A certain generalization of Jacobsthal numbers was proposed in the form Js,t^n = (s^n-(-t)^n)/(s+t) , where n ≥ 0 is a natural number and s ≠ -t are arbitrary real numbers (Atanassov 2011). As an analogue, a modification of Jacobsthal-Lucas numbers was formulated in the form js,tn = s^n+(-t)^n , where n is a natural number and s and t are arbitrary...

In this paper, we study some elementary problems involving surface area and volume of a certain regular solid. In particular, we find integral dimensions of a rectangular prism in which its surface area and volume are numerically equal. The problem leads us in solving a specific case of the well-known Diophantine problem 4/n = 1/x+1/y+1/z

In [1], Murthy introduced the concept of the Smarandache Cyclic Determinant Natural Sequence, the Smarandache Cyclic Arithmetic Determinant Sequence, the Smarandache Bisymmetric Determinant Natural Sequence, and the Smarandache Bisymmetric Arithmetic Determinant Sequence and in [2], Majumdar derived the n-th terms of these four sequences. In this p...

In this short note, we answer an open problem posed by B. Sroysang [1]. That is, we show that the only solutions (x, y, z) in non-negative integers to the Diophantine equation 2^x + 31^y = z^2 are (3, 0, 3) and (7, 2, 33).

In this short note we study some Diophantine equations of the form p^x + q^y = z^2, where x,y and z are non-negative integers and, p and q are both primes, p < q, with distance two.

In this short note, we answer an open problem posed by B. Sroysang [Int. J. Pure Appl. Math. 81, No. 4, 601–604 (2012; Zbl 1279.11035)]. More precisely, we find all solutions of the Diophantine equation 8 x +17 y =z 2 where x, y and z are non-negative integers.

In this note, we introduce the concept of circulant determinant sequences with binomial and derive formulas for the n-th term of the given sequences as well as the sum of the first n terms.

This paper will present a special sequence of numbers related to arithmetic and geometric sequence of numbers. The formulas for the general term an and the sum of the first n terms, denoted by Sn, are given respectively.

In this note, some relations between Jacobsthal and Jacobsthal–Lucas numbers and their respective modifications due to K. T. Atanassov [1, 2] and Y. Shang [4] are presented.

This paper presents a formal proof of the results found on [1] and [2] of Natividad.

This paper talks about two types of special sequences. The first is the arithmetic sequence of numbers with three alternate common differences; and the other, is the geometric sequence of numbers with three alternate common ratios. The formulas for the general term an and the sum of the first n terms, denoted by Sn, are given respectively.

In [1] Rabago introduced the concept of circulant determinant sequences with binomial coefficients and derived a formula for the n−th term of the given sequence. He also fomulated an expression to find the n-th partial sum of the sequence. In this note, we will study the spectral properties of circulant matrices with binomial coefficients.

The paper presents a formula for solving the missing terms of a second-order linear recurrence sequence given its first term and last term. The paper also provides a generalization of Natividad (2011).