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Abstrak :
This textbook develops, analyzes, and applies numerical methods for evolutionary, or time-dependent, differential problems. Both PDEs and ODEs are discussed from a unified viewpoint. The author emphasizes finite difference and finite volume methods, specifically their principled derivation, stability, accuracy, efficient implementation, and practical performance in various fields of science and engineering. Smooth and nonsmooth solutions for hyperbolic PDEs, parabolic-type PDEs, and initial value ODEs are treated, and a practical introduction to geometric integration methods is included as well. The author bridges theory and practice by developing algorithms, concepts, and analysis from basic principles while discussing efficiency and performance issues and demonstrating methods through examples and case studies from numerous application areas.

Abstrak :
This volume aims to collect some recent advances in the area in order to allow a quick overview of ongoing research. The contributors are first rate mathematicians. This collection of research papers is centred around parametrix constructions and microlocal analysis, asymptotic constructions of solutions, energy and dispersive estimates, and associated spectral transforms. Applications concerning elasticity and general relativity complement the volume. The book gives an overview of a variety of ongoing current research in the field and, therefore, allows researchers as well as students to grasp new aspects and broaden their understanding of the area. ​

Abstrak :
Evolusi tingkat keparahan suatu penyakit terjadi secara acak sehingga dapat direpresentasikan sebagai sebuah proses stokastik. Selain itu, karena transisi pada evolusi tersebut dapat terjadi kapan saja, diperlukan proses stokastik yang juga mempertimbangkan waktu tunggu yaitu proses semi Markov. Penelitian ini bertujuan untuk mengkonstruksi proses semi Markov untuk pemodelan evolusi keadaan atau tingkat keparahan pasien HIV/AIDS berdasarkan jumlah sel CD4 dalam tubuh. Konstruksi proses semi Markov dimulai dengan pemahaman konsep proses semi Markov, kemudian membangun fungsi distribusi dan densitas dari waktu tunggu antar keadaan yang diasumsikan berdistribusi Weibull. Proses semi Markov yang dikonstruksi digunakan untuk membangun persamaan evolusi sebagai dasar perhitungan probabilitas transisi interval seorang pasien HIV/AIDS dan rata-rata total biaya pengobatan yang diperlukan oleh pasien tersebut. Hasil perhitungan tersebut dianalisis dan diinterpretasikan agar kecenderungan seorang pasien HIV/AIDS membaik atau memburuk seiring waktu dapat diketahui. Berdasarkan data pemeriksaan pasien HIV/AIDS di Ethiopia dari tahun 2005 sampai 2014 didapatkan bahwa kecenderungan seorang pasien HIV/AIDS membaik atau memburuk akan meningkat selama sekitar dua tahun pertama kemudian akan menurun seiring waktu. Seorang pasien HIV/AIDS dengan jumlah sel CD4 < 350 sel~mm3 pada pemeriksaan awal, memiliki rata-rata total biaya pengobatan yang lebih besar dibandingkan pasien lain dengan jumlah sel CD4 > 350 sel~mm3. ......The evolution of disease severity occurs randomly so that it can be represented as a stochastic process. Besides, because this transition can occur at any time in this evolution or in other words, the duration of the individual being in the state (waiting time) is also random, then a stochastic process is needed, which also considers the waiting time. The semi-Markov process is a stochastic process that considers the conditions and waiting times in these circumstances. This study aims to construct and apply a semi-Markov process in modeling the evolution of HIV/AIDS patients’ state or severity based on the number of CD4 cells in the body. The semi-Markov process construction begins with understanding the semi-Markov process concept, then constructs the distribution function and the waiting time density between states, which are assumed to have a Weibull distribution. The semi-Markov process constructed was used to construct an evolutionary equation as a basis for calculating the probability of the transition interval between HIV/AIDS patients moving from one state to another and the total average cost of treatment required by these patients. The results of these calculations are analyzed and interpreted so that the tendency for HIV/AIDS patients to improve or worsen over time can be identified. Based on examination data of people living with HIV/AIDS in Ethiopia from 2005 to 2014, it was found that the tendency for people with HIV/AIDS to get better or worse would increase during the first two years and then decrease over time. An HIV/AIDS patient with a CD4 count < 350 cells~mm3 at baseline had a greater total cost of treatment than another patient with a CD4 count > 350 cells~mm3.