arXiv subjects

N. Karjanto

Publications and source records attributed to N. Karjanto.

At least 37 records · Page 2Linked to original sources

arXiv2021

Harmonic numbers as the summation of integrals

Harmonic numbers arise from the truncation of the harmonic series. The $n^\text{th}$ harmonic number is the sum of the reciprocals of each positive integer up to $n$. In addition to briefly introducing the properties of harmonic numbers, we cover harmonic numbers as the summation of integrals that involve the product of exponential and hyperbolic secant functions. The proof is relatively simple since it only comprises the Principle of Mathematical Induction and integration by parts.

math.HO↗

arXiv2021

What is the planetary boundary layer?

This short article provides brief coverage of the planetary boundary layer (PBL) and some characteristics featuring this lowest part of the atmosphere. Any readers who are interested in deeper and more extensive coverage are encouraged to consult the references listed at the end of this article.

physics.ao-ph↗

arXiv2021

Perturbed potential temperature distribution in atmospheric boundary layers

This article discusses the modeling of perturbed potential temperature in an atmospheric boundary layer. We adopt a convection-diffusion model with specified initial and boundary conditions that resulted from simplifying the linearized equation of the standard continuity equation for potential temperature field in the state of weak turbulent fluxes. By implementing the method of separation of variables to the non-steady-state perturbed potential temperature, we obtain a regular Sturm-Liouville problem for the spatial-dependent, vertical distribution component of the perturbed potential temperature. By transforming the canonical problem into the Liouville normal form, we provide asymptotic solutions for the corresponding second-order boundary value problem using the WKB theory. Furthermore, by solving the problem numerically, we observe a remarkable qualitative agreement between the asymptotic solutions and numerical simulations. As other convection-diffusion models typically perform, the perturbed potential temperature diminishes and approaches the steady-state condition over time.

physics.flu-dyn↗

arXiv2021

Not another computer algebra system: Highlighting wxMaxima in calculus

This article introduces and explains a computer algebra system (CAS) wxMaxima for Calculus teaching and learning at the tertiary level. The didactic reasoning behind this approach is the need to implement an element of technology into classrooms to enhance students' understanding of Calculus concepts. For many mathematics educators who have been using CAS, this material is of great interest, particularly for secondary teachers and university instructors who plan to introduce an alternative CAS into their classrooms. By highlighting both the strengths and limitations of the software, we hope that it will stimulate further debate not only among mathematics educators and software users but also also among symbolic computation and software developers.

math.HO↗

arXiv2021

Calculus and digital natives in rendezvous: wxMaxima impact

This article covers how a computer algebra system (CAS) wxMaxima can be explored for teaching and learning Single-Variable and Multivariable Calculus for Korean digital natives. We present several examples where \emph{wxMaxima} can handle Calculus problems easily, not straightforwardly but still successfully solved with some human intervention, and unsuccessfully. By soliciting qualitative feedback on students' experience in exploiting the CAS, we gathered a mixed reaction. Although some students commented positively, the majority seemed to be resistant in embracing a new technological tool.

math.HO↗

arXiv2021

On spatial Fourier spectrum of rogue wave breathers

In this article, we derive exact analytical expressions for the spatial Fourier spectrum of the soliton family on a constant background. Also known as breathers, these solitons are exact solutions of the nonlinear Schrödinger equation and are considered as prototypes for rogue wave models. Depending on the periodicity in the spatial-temporal domain, the characteristics in the wavenumber-temporal domain may feature either a continuous or discrete spectrum.

nlin.PS↗

arXiv2021

Optimizing students' combinatorial-thinking skill through design-based research

The aim of this study is to construct and compose an instructional design in combinatorial learning, particularly in the concept of counting. A composed design is expected to optimize students' combinatorial-thinking skill. This research was conducted at the first author's workplace, in the topics of combinatorial and graph theory consisting of 12 students. This study consists of three phases: preliminary research, development, and assessment. In the first phase, we analyze the needs and context of learning activities, literature study, and develop a framework of thinking. In the second phase, we validates the instructional design for further improvement, classroom application, and improvement based on learning activities. In the third phase, we provide evaluation questions to students to find out the result of the application of the design related to the optimization of combinatorial-thinking. As a result, an instructional design in counting that begins with review of concept of finite sets and continues with small group discussion, class discussion, self-reflection, and evaluation.

math.HO↗

arXiv2021

On physical optics approximation of viscous stratified shear flows

We study a mathematical model of a perturbed stratified shear mean flow in the presence of eddy coefficients of turbulent viscosity. We adopt the standard Boussinesq approximation in the natural convection of the buoyancy-driven flow and neglect the influence of the eddy coefficients of turbulent diffusivity. Comprising both the vertical and horizontal viscosity effects, a model for the vertical velocity perturbation corresponds to a fourth-order Taylor-Goldstein (TG) differential equation. Considering only the latter, we obtained a modified TG equation with the same order as the classical, inviscid one. Under an assumption of the slowly varying Brunt-Väisälä frequency and background horizontal velocity, we discuss the corresponding geometrical and physical optics approximations of the modified TG equation using the WKB method. We further investigate the behavior of these asymptotic solutions near singular values of a turning point and critical level.

physics.ao-ph↗

arXiv2020

An ethnoarithmetic excursion into the Javanese calendar

A perpetual calendar, a calendar designed to find out the day of the week for a given date, employs a rich arithmetical calculation using congruence. Zeller's congruence is a well-known algorithm to calculate the day of the week for any Julian or Gregorian calendar date. Another rather infamous perpetual calendar has been used for nearly four centuries among Javanese people in Indonesia. This Javanese calendar combines the Saka Hindu, lunar Islamic, and western Gregorian calendars. In addition to the regular seven-day, lunar month, and lunar year cycles, it also contains five-day pasaran, 35-day wetonan, 210-day pawukon, octo-year windu, and 120-year kurup cycles. The Javanese calendar is used for cultural and spiritual purposes, including a decision to tie the knot among couples. In this chapter, we will explore the relationship between mathematics and the culture of Javanese people and how they use their calendar and the arithmetic aspect of it in their daily lives. We also propose an unprecedented congruence formula to compute the pasaran day. We hope that this excursion provides an insightful idea that can be adopted for teaching and learning of congruence in number theory.

math.HO↗

arXiv2020

Discrete solitons dynamics in $\cal{PT}$-symmetric oligomers with complex-valued couplings

We consider an array of double oligomers in an optical waveguide device. A mathematical model for the system is the coupled discrete nonlinear Schrödinger (NLS) equations, where the gain-and-loss parameter contributes to the complex-valued linear coupling. The array caters to an optical simulation of the parity-time ($\cal{PT}$)-symmetry property between the coupled arms. The system admits fundamental bright discrete soliton solutions. We investigate their existence and spectral stability using perturbation theory analysis. These analytical findings are verified further numerically using the Newton-Raphson method and a standard eigenvalue-problem solver. Our study focuses on two natural discrete modes of the solitons: single- and double-excited-sites, also known as onsite and intersite modes, respectively. Each of these modes acquires three distinct configurations between the dimer arms, i.e., symmetric, asymmetric, and antisymmetric. Although both intersite and onsite discrete solitons are generally unstable, the latter can be stable, depending on the combined values of the propagation constant, horizontal linear coupling coefficient, and gain-loss parameter.

nlin.PS↗

arXiv2020

Peregrine soliton as a limiting behavior of the Kuznetsov-Ma and Akhmediev breathers

This article discusses a limiting behavior of breather solutions of the focusing nonlinear Schrödinger (NLS) equation. These breathers belong to the families of solitons on a non-vanishing and constant background, where the continuous-wave envelope serves as a pedestal. The rational Peregrine soliton acts as a limiting behavior of the other two breather solitons, i.e., the Kuznetsov-Ma breather and Akhmediev soliton. Albeit with a phase shift, the latter becomes a nonlinear extension of the homoclinic orbit waveform corresponding to an unstable mode in the modulational instability phenomenon. All breathers are prototypes for rogue waves in nonlinear and dispersive media. We present a rigorous proof using the $ε$-$δ$ argument and show the corresponding visualization for this limiting behavior.

nlin.PS↗

arXiv2020

Challenges in teaching Real Analysis classes at the University of PGRI, South Sumatra, Indonesia

This paper discusses our experiences and challenges in teaching advanced undergraduate Real Analysis classes for Mathematics Education students at the University of PGRI (Persatuan Guru Republik Indonesia, Indonesian Teachers Association) Palembang, South Sumatra, Indonesia. We observe that the syllabus contains topics with a high level of difficulty for the students who are specialized in education and intend to teach mathematics at the secondary level. The conventional lecturing method is mainly implemented during the class, with some possible variations of the method, including the Texas method (also known as Moore's method) and the small group guided discovery method. In particular, the latter method has been implemented successfully for a Real Analysis class at Dartmouth College, New Hampshire by Dumitraşcu in 2006. Although it is a real challenge to apply a specific teaching method that will be able to accommodate a large number of students, the existing teaching activities can still be improved and a more effective method could be implemented in the future. Furthermore, the curriculum contents should be adapted for an audience in Mathematics Education to equip them for their future career as mathematics teachers. Any constructive suggestions are welcome for the improvement of our mathematics education system at the university as well as on the national scale.

math.HO↗

arXiv2020

Wave group evolution and interaction

This thesis deals with mathematical and physical aspects of deterministic freak wave generation in a hydrodynamic laboratory. We adopt the nonlinear Schrödinger (NLS) equation as a mathematical model for the evolution of the surface gravity wave packet envelopes. As predicted theoretically and observed experimentally by Benjamin and Feir (1967), Stokes waves on deep water are unstable under a sideband modulation. The plane-wave solution of the NLS equation, which represents the fundamental component of the Stokes wave, experiences a similar modulational instability under the linear perturbation theory. A nonlinear extension for this perturbed wave is given by the family of Soliton on Finite Background (SFB), where an exact expression is available and can be derived analytically by several techniques. We discuss the characteristics of the SFB family as a prime candidate for freak wave events. The corresponding physical wave field exhibits intriguing phenomena of wavefront dislocation and phase singularity.

nlin.PS↗

arXiv2020

Properties of Chebyshev polynomials

Ordinary differential equations and boundary value problems arise in many aspects of mathematical physics. Chebyshev differential equation is one special case of the Sturm-Liouville boundary value problem. Generating function, recursive formula, orthogonality, and Parseval's identity are some important properties of Chebyshev polynomials. Compared with a Fourier series, an interpolation function using Chebyshev polynomials is more accurate in approximating polynomial functions. -------- Des équations différentielles ordinaires et des problèmes de valeurs limites se posent dans de nombreux aspects de la physique mathématique. L'équation différentielle de Chebychev est un cas particulier du problème de la valeur limite de Sturm-Liouville. La fonction génératrice, la formule récursive, l'orthogonalité et l'identité de Parseval sont quelques propriétés importantes du polynôme de Chebyshev. Par rapport à une série de Fourier, une fonction d'interpolation utilisant des polynômes de Chebyshev est plus précise dans l'approximation des fonctions polynomiales.

math.HO↗

arXiv2020

The effects of increasing speed on the structure of axle box bogies of NT-11 trains

The occurrence of cracks on the structure of axle box bogie of NT-11 economic trains operated by PT Kereta Api Indonesia is studied. These axle box bogies were designed for the maximum speed of 90 km/hour. The current market demand led to an increase in maximum speed to 110 km/hour. We investigate the effects of this speed increase in the occurrence of cracks of the axle box bogies. Based on the data obtained from PT Kereta Api Indonesia, a statistical approach is first used to see a relationship between the occurrence of cracks and some parameters, namely the distance to one-way destination, the speed, and the number of stops that the trains make to arrive at the destination. From this statistical study, stopping frequency is the most dominant factor that affects the occurrence of cracks and that the increase in speed leads to an increase in the crack's occurrence. A mathematical model is then designed to further describe the effects of this increased velocity on the structure of axle box bogies. From this model, a resonance analysis is derived and numerical experiments are carried out. It is shown that the wagon resonance is highly dependent on the velocity. An example of typical data shows that this resonance occurs when the train reaches a velocity at approximately 104 km/hour. These resonance effects are believed to create a larger distance between parts of the axle box bogie that influence the occurrence of the cracks.

math.OC↗

arXiv2020

A theoretical study on a two-dimensional flap-type wavemaker

A mathematical model for unidirectional wave generation explained in this study. The model consists of Laplace's equation in the semi-infinite two-dimensional water interior, dynamic and kinematic boundary conditions at the free surface, lateral boundary condition on the wavemaker, and fixed wall at the bottom. The model is for a flap-type wavemaker that is commonly used in a towing tank of a hydrodynamic laboratory. To simplify the problem, some assumptions are made, namely that water is an ideal fluid. Linear wavemaker theory is used and a generation of monochromatic wave (single frequency) is considered. The relation between the wavenumber, wave height, and wavemaker stroke is further derived. -------- Pemodelan secara matematis pembangkitan gelombang arah tunggal disampaikan dalam laporan tugas akhir ini. Pemodelannya mencakup persamaan Laplace dalam kolam air dengan setengah batas tak hingga, syarat batas dinamika dan kinematika, syarat batas lateral di pembangkit gelombang, dan dinding tetap di dasar kolam. Modelnya diterapkan pada pembangkit gelombang (wavemaker) tipe flap yang sering digunakan pada suatu towing tank dalam laboratorium hidrodinamika. Untuk menyederhanakan permasalahan, beberapa asumsi diterapkan, yakni bahwa air adalah fluida yang bersifat ideal. Teori pembangkit gelombang linear digunakan dan tipe gelombang monokromatik (frekuensi tunggal) diamati. Kaitan antara bilangan gelombang, ketinggian gelombang, dan simpangan wavemaker juga akan diturunkan.

physics.flu-dyn↗

arXiv2019

The nonlinear Schrödinger equation: A mathematical model with its wide-ranging applications

The nonlinear Schrödinger equation (NLSE) models the slowly varying envelope dynamics of a weakly nonlinear quasi-monochromatic wave packet in dispersive media. In the context of Bose-Einstein condensate (BEC), it is often referred to as the Gross-Pitaevskii equation (GPE). The NLSE is one example of integrable systems of a nonlinear partial differential equation (PDE) in $(1 + 1)$D and it possesses an infinite set of conservation laws. This nonlinear evolution equation arises in various physical settings and admits a wide range of applications, including but not limited to, surface gravity waves, superconductivity, nonlinear optics, and BEC. This chapter discusses not only the modeling aspect of the NLSE but also provides an overview of the applications in these four exciting research areas. The former features derivations of the NLSE heuristically and by employing the method of multiple-scale from other mathematical models as governing equations. Depending on how the variables are interpreted physically, the resulting NLSE can model a different dynamics of the wave packet. Furthermore, depending on the adopted assumptions and the chosen governing equations, each approach may provide different values for the corresponding dispersive and nonlinear coefficients.

nlin.PS↗

arXiv2019

Note on wavefront dislocation in surface water waves

At singular points of a wave field, where the amplitude vanishes, the phase may become singular and wavefront dislocation may occur. In this Letter, we investigate for wave fields in one spatial dimension the appearance of these essentially linear phenomena. We introduce the Chu-Mei quotient as it is known to appear in the 'nonlinear dispersion relation' for wave groups as a consequence of the nonlinear transformation of the complex amplitude to real phase-amplitude variables. We show that unboundedness of this quotient at a singular point, related to unboundedness of the local wavenumber and frequency, is a generic property and that it is necessary for the occurrence of phase singularity and wavefront dislocation, while these phenomena are generic too. We also show that the 'soliton on finite background', an explicit solution of the NLS equation and a model for modulational instability leading to extreme waves, possesses wavefront dislocations and unboundedness of the Chu-Mei quotient.

nlin.PS↗