This paper will solve one of the fractional mathematical physics models, a one-dimensional time-fractional differential equation, by utilizing the second-order quarter-sweep finite-difference scheme and the preconditioned accelerated over-relaxation method. The proposed numerical method offers an efficient solution to the time-fractional differential equation by applying the computational complexity reduction approach by the quarter-sweep technique. The finite-difference approximation equation will be formulated based on the Caputo’s time-fractional derivative and quarter-sweep central difference in space. The developed approximation equation generates a linear system on a large scale and has sparse coefficients. With the quarter-sweep technique and the preconditioned iterative method, computing the time-fractional differential equation solutions can be more efficient in terms of the number of iterations and computation time. The quarter-sweep computes a quarter of the total mesh points using the preconditioned iterative method while maintaining the solutions’ accuracy. A numerical example will demonstrate the efficiency of the proposed quarter-sweep preconditioned accelerated over-relaxation method against the half-sweep preconditioned accelerated over-relaxation, and the full-sweep preconditioned accelerated over-relaxation methods. The numerical finding showed that the quarter-sweep finite difference scheme and preconditioned accelerated over-relaxation method can serve as an efficient numerical method to solve fractional differential equations.

The purpose of this paper is to see the impact of social media on the idea of radicalism. In writing this paper using literature study method. And from the description in this paper can be concluded that social media have a positive and negative impact. The negative impact of the free use of social media is the ease of accepting the Indonesian public receiving information including radicalism from all fields without finding out the truth of this information.

Abstrak: Penelitian ini bertujuan untuk mengembangkan modul pembelajaran IPA berbasis etnosains yang memiliki kelayakan untuk dipakai setelah divalidasi, dan melihat kepraktisan modul dalam melatih literasi sains siswa. Metode penelitian menggunakan model pengembangan Borg dan Gall yang terdiri dari 8 tahap yaitu potensi dan masalah, pengumpulan data, mengembangkan desain produk, validasi desain produk, revisi desain produk, uji coba produk kelompok kecil, revisi produk, dan produk akhir. Data hasil analisis angket dengan menghitung persentase pencapaian pada setiap komponen persentase pencapaian kriteria kelayakan modul pembelajaran IPA berbasis etnosains yang dikembangkan layak untuk digunakan dengan persentase sebesar 72% (layak), 96,25% (sangat layak), dan 93% (sangat layak). Kepraktisan modul pembelajaran IPA berdasarkan data hasil sebanyak 15 orang respon siswa dan 1 orang respon guru IPA yang dilakukan pada uji coba skala kecil/terbatas menunjukkan bahwa modul pembelajaran IPA berbasis etnosains materi pemanasan global termasuk dalam kriteria sangat praktis dengan persentase secara berurutan sebesar 87% dan 88%. Kesimpulan penelitian ini adalah modul yang dikembangkan telah layak, dan praktis digunakan sebagai bahan ajar dalam menunjang proses belajar mengajar pada materi Pemanasan Global dalam melatih kemampuan literasi sains siswa SMP untuk kelas VII. Abstract:  This study aims to develop an ethnoscience-based science learning module that has the feasibility to be used after being validated, and to see the practicality of the module in training students' scientific literacy. The research method uses the Borg and Gall development model which consists of 8 stages, namely potential and problems, data collection, product design development, product design validation, product design revision, small group product trials, product revisions, and final products. The data from the questionnaire analysis by calculating the percentage of achievement in each component of the percentage of achievement of the eligibility criteria of the ethnoscience-based learning module that was developed was feasible to use with a percentage of 72%, 96.25%, and 93% with (category eligible, very feasible, and very feasible). ). The practicality of the science learning module based on data from student responses and the response of science teachers conducted on a small/limited scale trial showed that the ethnoscience-based science learning module on global warming material was included in very practical criteria with percentages of 87% and 88% respectively. The conclusion of this study is that the module developed has been feasible, and is practically used as teaching material in supporting the teaching and learning process on Global Warming material in training the science literacy skills of junior high school students for class VII.

In this study, the numerical solution of a space-fractional parabolic partial differential equation was considered. The investigation of the solution was made by focusing on the space-fractional diffusion equation (SFDE) problem. Note that the symmetry of an efficient approximation to the SFDE based on a numerical method is related to the compatibility of a discretization scheme and a linear system solver. The application of the one-dimensional, linear, unconditionally stable, and implicit finite difference approximation to SFDE was studied. The general differential equation of the SFDE was discretized using the space-fractional derivative of Caputo with a half-sweep finite difference scheme. The implicit approximation to the SFDE was formulated, and the formation of a linear system with a coefficient matrix, which was large and sparse, is shown. The construction of a general preconditioned system of equation is also presented. This study’s contribution is the introduction of a half-sweep preconditioned successive over relaxation (HSPSOR) method for the solution of the SFDE-based system of equation. This work extended the use of the HSPSOR as an efficient numerical method for the time-fractional diffusion equation, which has been presented in the 5th North American International Conference on industrial engineering and operations management in Detroit, Michigan, USA, 10–14 August 2020. The current work proposed several SFDE examples to validate the performance of the HSPSOR iterative method in solving the fractional diffusion equation. The outcome of the numerical investigation illustrated the competence of the HSPSOR to solve the SFDE and proved that the HSPSOR is superior to the standard approximation, which is the full-sweep preconditioned SOR (FSPSOR), in terms of computational complexity.

Research into the recent developments for solving fractional mathematical equations requires accurate and efficient numerical methods. Although many numerical methods based on Caputo’s fractional derivative have been proposed to solve fractional mathematical equations, the efficiency of obtaining solutions using these methods when dealing with a large matrix requires further study. The matrix size influences the accuracy of the solution. Therefore, this paper proposes a quarter-sweep finite difference scheme with a preconditioned relaxation-based approximation to efficiently solve a large matrix, which is based on the establishment of a linear system for a fractional mathematical equation. The paper presents the formulation of the quarter-sweep finite difference scheme that is used to approximate the selected fractional mathematical equation. Then, the derivation of a preconditioned relaxation method based on a quarter-sweep scheme is discussed. The design of a C++ algorithm of the proposed quarter-sweep preconditioned relaxation method is shown and, finally, efficiency analysis comparing the proposed method with several tested methods is presented. The contributions of this paper are the presentation of a new preconditioned matrix to restructure the developed linear system, and the derivation of an efficient preconditioned relaxation iterative method for solving a fractional mathematical equation. By simulating the solutions of time-fractional diffusion problems with the proposed numerical method, the study found that computing solutions using the quarter-sweep preconditioned relaxation method is more efficient than using the tested methods. The proposed numerical method is able to solve the selected problems with fewer iterations and a faster execution time than the tested existing methods. The efficiency of the methods was evaluated using different matrix sizes. Thus, the combination of a quarter-sweep finite difference method, Caputo’s time-fractional derivative, and the preconditioned successive over-relaxation method showed good potential for solving different types of fractional mathematical equations, and provides a future direction for this field of research.