Doctor of Philosophy in Mathematics - Partial Differential Equations

学历文凭
Ph.D.

专业院系
数学

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国际学生入学条件
Applicants who have conducted their post-secondary education outside of the United States must have completed an undergraduate degree program equivalent to a University of California bachelor's degree by the beginning of their graduate program at UCSB. Equivalent undergraduate degrees usually include a minimum four years of university work and above-average scholarship. The degree must be awarded by an accredited institution. To be considered for admission, you must have received a bachelor's degree or its equivalent from an accredited university prior to the quarter for which you seek admission, and have at least a B average (3.0 GPA) in your undergraduate coursework. Satisfaction of minimal standards does not, however, guarantee admission, since the number of qualified applicants far exceeds the number of spaces available. An excellent command of written and spoken English is required prior to enrollment at UCSB. Applicants whose native language is not English are required to take the Test of English as a Foreign Language (TOEFL), the International English Language Testing System (IELTS), or the Duolingo English Test (DET). TOEFL paper-based test (PBT) – 550, TOEFL internet-based test (iBT) and TOEFL iBT Home Edition – 80, IELTS Academic and IELTS Indicator - Overall Band Score of 7.
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雅思考试总分
7.0
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- 雅思总分:7
- 托福网考总分:80
- 托福笔试总分:550
- 其他语言考试:Duolingo English Test (DET) - 120
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课程简介
Our understanding of the fundamental processes of the natural word is based to a large extent on partial differential equations (PDE). For example, the Einstein equations describe the geometry of space-time and its interaction with matter. The dynamics of fluids and elastic solids are governed by partial differential equations that go back to Euler and Cauchy. Electro-magnetic waves including the propagation of light in various media are modeled by Maxwell's equations. More recently, PDE's are gaining importance in the social and life sciences. The Black-Sholes PDE underpins the theory of option pricing in financial mathematics. Reaction-diffusion models are used in neurophysics and population dynamics.<br><br>The analysis of PDE's has given rise to new mathematical ideas. For example, the study of the equation thermal diffusion lead to the discovery of Fourier series and ultimately the field of Fourier analysis. Hilbert's investigations into eigenvalue problems arising in the PDE's for vibrating strings and Schrodinger's equation in quantum mechanics evolved into to modern theory of functional analysis and operator theory.
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