A Finite Difference Method for the Variational p-Laplacian

We propose a new monotone finite difference discretization for the variational p-Laplace operator, $$\Delta _pu=\text{ div }(|\nabla u|^{p-2}\nabla u),$$ Δ p u = div ( | ∇ u | p - 2 ∇ u ) , and present a convergent numerical scheme for related Dirichlet problems. The resulting nonlinear system is solved using two different methods: one based on Newton-Raphson and one explicit method. Finally, we exhibit some numerical simulations supporting our theoretical results. To the best of our knowledge, this is the first monotone finite difference discretization of the variational p-Laplacian and also the first time that nonhomogeneous problems for this operator can be treated numerically with a finite difference scheme.

An explicit method to calculate implicit spatial finite differences

The finite-difference solution of the second-order acoustic wave equation is a fundamental algorithm in seismic exploration for seismic forward modeling, imaging, and inversion. Unlike the standard explicit finite difference (EFD) methods that usually suffer from the so-called "saturation effect", the implicit FD methods can obtain much higher accuracy with relatively short operator length. Unfortunately, these implicit methods are not widely used because band matrices need to be solved implicitly, which is not suitable for most high-performance computer architectures. We introduce an explicit method to overcome this limitation by applying explicit causal and anti-causal integrations. We can prove that the explicit solution is equivalent to the traditional implicit LU decomposition method in analytical and numerical ways. In addition, we also compare the accuracy of the new methods with the traditional EFD methods up to 32nd order, and numerical results indicate that the new method is more accurate. In terms of the computational cost, the newly proposed method is standard 8th order EFD plus two causal and anti-causal integrations, which can be applied recursively, and no extra memory is needed. In summary, compared to the standard EFD methods, the new method has a spectral-like accuracy; compared to the traditional LU-decomposition implicit methods, the new method is explicit. It is more suitable for high-performance computing without losing any accuracy.

Comparison of Explicit and Implicit Methods of Cross-Cultural Learning in an International Classroom

The paper addresses a gap in the literature concerning the difference between enhanced and not enhanced cross-cultural learning in an international classroom. The objective of the described research was to clarify if the environment of international classrooms could enhance cross-cultural competences significantly enough or if additional focus on cross-cultural learning as an explicit objective of learning activities would add substantially to the experience. The research question was defined as “how can a specific exercise focused on cross-cultural learning enhance the cross-cultural skills of university students in an international classroom?”. Surveys were conducted among international students in three leading Central-European Universities in Lithuania, Poland and Hungary to measure the increase of their cross-cultural competences. The Lithuanian and Polish classes were composed of international students and concentrated on International Management/Business topics (explicit method). The Hungarian survey was done in a general business class that just happened to be international in its composition (implicit method). Overall, our findings prove that the implicit method resulted in comparable, somewhat even stronger effectiveness than the explicit method. The study method included the analyses of students’ individual increases in each study dimension and construction of a compound measure to note the overall results. Our findings confirm the power of the international classroom as a stimulating environment for latent cross-cultural learning even without specific exercises focused on cross-cultural learning itself. However, the specific exercise did induce additional learning, especially related to cross-cultural awareness and communication with representatives of other cultures, even though the extent of that learning may be interpreted as underwhelming. The main conclusion from the study is that the diversity of the students engaged in a project provided an environment that supported cross-cultural learning, even without specific culture-focused reflections or exercises.

Numerical Simulation of Partial Differential Equations Using Finite Difference Methods

This paper explores the use of the software package, Matlab and Excel in the implementation of the finite difference method to solve partial differential equations (PDE’s.). It aimed to examine the strength of the forward explicit method and backward implicit method in solving PDE’s. A comparison was made between the forward explicit method and the backward implicit method for their stability. The FDM method was used to solve partial differential equations of heat. Numerical examples were also created and analysed to show the strengths of each method. The results shows that the forward explicit method is conditionally stable because the stability it requires a small step size of time ‘t’ compared to space ‘x’ for stability. The backward implicit method is unconditionally stable because it depends on the local truncation error considerations.

ANALISIS INSTAGRAM SEBAGAI SOCIAL MEDIA MARKETING DI INDUSTRI PERHOTELAN

Social media marketing in Instagram provides an opportunity for consumers to interact better with the hotel industry, it also has greater cost effectiveness, in addition to the application, more and more unique features are emerging, making it easier for marketers to be as creative as possible. The emergence of Instagram with unique features that can be accessed around the world encourages the hotel industry to use it to market products and provides opportunities for consumers to interact better with companies so that they have higher effectiveness. The purpose of this study is to explore the reasons for the hospitality industry to use Instagram as a marketing medium as well as topics that are often researched in previous journals regarding the application of Instagram as social media marketing in the hospitality industry. The method used is qualitative with a systematic literature review approach to find relevant theories on this topic and then identified, assessed, and synthesized according to a predetermined explicit method so that the SLR approach provides clarity, transparency and impartiality as well as inclusive coverage in certain areas. The result of this research is that hotels use Instagram social media as their marketing because Instagram has many features such as hashtags and stories that can reach a broad audience. In addition, posting in image format also has a big impact on customer decisions. However, this is not the full power of Instagram marketing. On the other hand, earlier research mostly discuss aouut the application of social media marketing in hotels.

Numerical method for fractional diffusion-wave equations with functional delay

For a fractional diffusion-wave equation with a nonlinear effect of functional delay, an implicit numerical method is constructed. The scheme is based on the L2-method of approximation of the fractional derivative of the order from 1 to 2, interpolation and extrapolation with the given properties of discrete prehistory and an analogue of the Crank-Nicolson method. The order of convergence of the method is investigated using the ideas of the general theory of difference schemes with heredity. The order of convergence of the method is more significant than in previously known methods, depending on the order of the starting values. The main point of the proof is the use of the stability of the L2-method. The results of comparing numerical experiments with other schemes are presented: a purely implicit method and a purely explicit method, these results showed, in general, the advantages of the proposed scheme.