### Required Core Courses

#### MATH 221: Partial-Differential Equations l

Partial differential equations (PDEs) of applied mathematics. Topics include modeling physical phenomena, linear and nonlinear first-order PDEs, D'Alembert's solution, second-order linear PDEs, characteristics, initial and boundary value problems, separation of variables, Sturm-Liouville problem, Fourier series, Duhamel's Principle, linear and nonlinear stability.

#### MATH 222: Partial-Differential Equations II

Continuation of Math 221. Topics include integral transforms, asymptotic methods for integrals, integral equations, weak solutions, point sources and fundamental solutions, conservation laws, Green's functions, generalized functions, variational properties of eigenvalues and eigenvectors, Euler-Lagrange equations, Maximum principles.

#### MATH 223: Asymptotics and Perturbation Methods

Asymptotic evaluation of integrals, matched asymptotic expansions, multiple scales, WKB, and homogenization. Applications are made to ODEs, PDEs, difference equations, and integral equations to study boundary and shock layers, nonlinear wave propagation, bifurcation and stability, and resonance.

#### MATH 231: Numerical Solution of Differential Equations I

Examines fundamental methods typically required in the numerical solution of differential equations. Topics include direct and indirect methods for linear systems, nonlinear systems, interpolation and approximation, eigenvalue problems, ordinary-differential equations (IVPs and BVPs), and finite differences for elliptic partial-differential equations. A significant amount of programming will be required.

#### MATH 232: Numerical Solution of Differential Equations II

Fundamental methods presented in Math 231 are used as a base for discussing modern methods for solving partial-differential equations. Numerical methods include variational, finite element, collocation, spectral, and FFT. Error estimates and implementation issues will be discussed. A significant amount of programming will be required.

### Other Courses

#### MATH 233: Scientific Computing

Theoretical and practical introduction to parallel scientific computing. Survey of hardware and software environments, and selected algorithms and applications. Topics will include linear systems, N-body problems, FFTs, and methods for solving PDEs. Practical implementation and performance analysis are emphasized in the context of demonstrative applications in science and engineering.

#### MATH 224: Advanced Methods of Applied Mathematics

Students who complete this course will achieve broad familiarity with advanced analytical structures, results, and methods in applied mathematics. The course will expose students to theoretical material from real and functional analysis that is useful for research in applied mathematics.

#### MATH 291: Applied Mathematics Seminar

The Applied Mathematics Seminar Series runs in the Spring and Fall semesters, with talks held nearly each week. Talks cover a broad spectrum of mathematical problems and novel applications. All students are required to enroll in MATH 291 Applied Mathematics Seminar for at least two semesters, where attendance in the Seminar Series is required. Regardless of enrollment in MATH 291, students are expected to attend all seminars in the series whenever possible. (seminar web page)

#### MATH 201: Teaching and Learning in the Sciences

Students will be introduced to 'scientific teaching' - an approach to teaching science that uses many of the same skills applied in research. Topics will include how people learn, active learning, designing, organizing and facilitating teachable units, classroom management, diversity in the classroom and assessment design.

#### MATH 399: University Teaching

This course is centered on a student's classroom experiences as a Teaching Assistant in an undergraduate Applied Mathematics course. Provides a faculty-directed opportunity to implement teaching practices presented in the course Teaching and Learning in the Sciences. Course will involve video-taping of teaching, peer review, and weekly meetings with faculty.

### Special-Topics Courses

Special Topics courses are additional graduate courses (courses numbered 200-299 and worth at least three units each) exclusive of research (MATH 295) that are appropriate to the student's research area. Doctoral and MS Plan II students mustÂ successfully complete (grade of B or better) at least two such courses. MS Plan I students must successfully complete (grade of B or better) at least one such course. There are many Special Topics courses offered both in the Applied Math Program and other graduate programs at UC Merced. Some of those that have been offered in Applied Math (as MATH 292) are linear and nonlinear wave propagation, integral equations, dynamical systems, waves in random media, and fluid dynamics, and stochastic processes. Other graduate-level courses appropriate to the student's specific field of research, including Directed Independent Study (MATH 299) may be used to meet the requirement with consent of the student's faculty committee. Courses numbered 100-199 offered outside MATH will be considered by petition. Normally, Special Topics courses should be taken during the second year of graduate study. Requirements for formal course work beyond the minimum are flexible and are determined by the individual student's background and research topic in consultation with the student's graduate research advisor.