An Optimal Fourth-Order Finite Difference Scheme for the Helmholtz Equation Based on the Technique of Matched Interface Boundary

In this paper, a new optimal fourth-order 21-point finite difference scheme is proposed to solve the 2D Helmholtz equation numerically, with the technique of matched interface boundary (MIB) utilized to treat boundary problems. For the approximation of Laplacian, two sets of fourth-order difference schemes are derived firstly based on the Taylor formula, with a total of 21 grid points involved. Then, a weighted combination of the two schemes is employed in order to reduce the numerical dispersion, and the weights are determined by minimizing the dispersion. Similarly, for the discretization of the zeroth-order derivative term, a weighted average of all the 21 points is implemented to obtain the fourth-order accuracy. The new scheme is noncompact; hence, it encounters great difficulties in dealing with the boundary conditions, which is crucial to the order of convergence. To tackle this issue, the matched interface boundary (MIB) method is employed and developed, which is originally used to accommodate free edges in the discrete singular convolution analysis. Convergence analysis and dispersion analysis are performed. Numerical examples are given for various boundary conditions, which show that new scheme delivers a fourth order of accuracy and is efficient in reducing the numerical dispersion as well.

Analysis of the General Tool Life Function of Cutting Tools by Application of the Catastrophe-Theory

Production technology planning requires information on tool life T and its relation to cutting speed v. As the Taylor formula often cannot be linearized on an lg-lg scale, a general tool life function has been developed for describing a v-T function with a convex-concave part. Using catastrophe theory, an analogy is established between the general tool life function and the cusp catastrophe, allowing topological mapping of the general v-T function. Results were verified by machinability tests in the turning of C35 and C60 conventional and specially deoxidized C-steels during steelmaking. It was found that in the convex-concave section of this function, 2–3 cutting speeds can be selected for a given tool life, which is advantageous for harmonizing tool changes in multi-operation technology.

Extended ball convergence for a seventh order derivative free class of algorithms for nonlinear equations

In the earlier work, expensive Taylor formula and conditions on derivatives up to the eighthorder have been utilized to establish the convergence of a derivative free class of seventh orderiterative algorithms. Moreover, no error distances or results on uniqueness of the solution weregiven. In this study, extended ball convergence analysis is derived for this class by imposingconditions on the first derivative. Additionally, we offer error distances and convergence radiustogether with the region of uniqueness for the solution. Therefore, we enlarge the practicalutility of these algorithms. Also, convergence regions of a specific member of this class are displayedfor solving complex polynomial equations. At the end, standard numerical applicationsare provided to illustrate the efficacy of our theoretical findings.

Solving Riccati type q-Difference Equations via Difference Transform Method

In this paper, we deal on the time scale that its delta derivative of graininess function is a nonzero positive constant. Based on the Taylor formula for this time scale, we investigate the difference transform method (DTM). This method has been applied successfully to solve Riccati type difference equations in quantum calculus. To demonstrate the ability and efficacy of this method, some examples have been provided.

Formalizing Calculus without Limit Theory in Coq

Formal verification of mathematical theory has received widespread concern and grown rapidly. The formalization of the fundamental theory will contribute to the development of large projects. In this paper, we present the formalization in Coq of calculus without limit theory. The theory aims to found a new form of calculus more easily but rigorously. This theory as an innovation differs from traditional calculus but is equivalent and more comprehensible. First, the definition of the difference-quotient control function is given intuitively from the physical facts. Further, conditions are added to it to get the derivative, and define the integral by the axiomatization. Then some important conclusions in calculus such as the Newton–Leibniz formula and the Taylor formula can be formally verified. This shows that this theory can be independent of limit theory, and any proof does not involve real number completeness. This work can help learners to study calculus and lay the foundation for many applications.

Effect of 6-DOF Oscillation of Ship Target on SAR Imaging

Ship targets are high-value military and civilian targets with broad application prospects. However, the precise focusing of ships is still a difficult issue because of their complicated six-degree-of-freedom motions on the sea surface. This paper focused on investigating the effect of ship six-degree-of-freedom oscillation on Synthetic Aperture Radar imaging. Firstly, based on the six-degree-of-freedom motions, the accurate range models for ship linear oscillation and angular oscillation were built, and the superiority was verified by comparing them with the models described in published literature. Secondly, we used the Taylor formula and Bessel function to expand the phase error introduced by ship oscillation, then their effects on imaging were further analyzed. Finally, based on the measured ship attitude data, we generated the semi-physical echoes of the oscillatory ship to validate the analysis throughout this article. Based on the proposed range model, we also made some tentative on the phase compensation method by fitting ship attitude angles with multiple sinusoidal functions.

Gambaran Estimasi Usia Biologis dengan Menggunakan Metode Blenkin-Taylor (Modifikasi Sistem Demirjian) di Kota Semarang

Unexpected incident such as natural disaster and accident often occur in many countries including Indonesia which causes many victims with unknown identity. Tooth is one of the indicators to assess and determine a person's identity. Blenkin-Taylor method is used for age estimation of an individual by using teeth. This study was aimed to obtain the estimation of biological age by using Blenkin-Taylor method in Semarang. This was a descriptive study with a cross sectional design. Samples were panoramic digital radiograph data of patients aged 5-15 years, copied in the form of a soft file. The observation and measurement were performed on seven teeth of right lower jaw by using the DICOM RadiAnt application. Data of observations and measurements of maturation scores were calculated and converted into the Blenkin-Taylor formula to determine the biological age. The results showed that the difference between biological and chronological age was ±0.32 years. This value was lower than the Blenkin-Taylor previous study result which was ±0,6 years. In conclusion, by using the Blenkin-Taylor method, there was a difference between biological age and chronological age as many as ±0,32 years in individuals aged 5-15 years old in Semarang.Keywords: biological age, the Blenkin-Taylor method, panoramic radiography Abstrak: Kejadian tidak terduga seperti bencana alam dan kecelakaan sering terjadi di berbagai negara, salah satunya di Indonesia yang menimbulkan banyak korban jiwa yang tidak diketahui identitasnya. Gigi merupakan salah satu indikator untuk menilai dan menentukan identitas seseorang. Salah satu metode dalam menentukan estimasi usia dengan menggunakan gigi ialah metode Blenkin-Taylor. Penelitian ini bertujuan untuk mendapatkan gambaran estimasi usia biologis dengan menggunakan metode Blenkin-Taylor di Kota Semarang. Jenis penelitian ialah deskriptif dengan desain potong lintang. Sampel penelitian ialah data file digital radiograf panoramik pasien berusia 5-15 tahun yang disalin ke dalam bentuk soft file, kemudian dilakukan pengamatan dan pengukuran pada 7 gigi regio kanan rahang bawah dengan menggunakan aplikasi RadiAnt DICOM. Hasil pengamatan dan pengukuran skor maturasi dihitung dan dikonversikan ke dalam rumus metode Blenkin-Taylor untuk menentukan usia biologis. Hasil penelitian menunjukkan bahwa selisih usia biologis dan usia kronologis sebesar 0,32 tahun. Hal ini lebih kecil dibandingkan penelitian Blenkin-Taylor terdahulu sebesar 0,6 tahun. Simpulan penelitian ini ialah dengan mengggunakan metode Blenkin-Taylor terdapat selisih rerata usia kronologis dan usia biologis sebesar ± 0,32 tahun pada individu usia 5-15 tahun di Kota Semarang.Kata kunci: usia biologis, metode Blenkin-Taylor, radiograf panoramik