### M.Sc./ Ph.D. Applied and Computational Mathematics Program Courses

__Compulsory Core courses:__

**ACM 501: **Partial differential equations** **

**ACM 502: **Numerical analysis** **

__Elective Courses:__

**ACM 503: **Advanced numerical methods for differential equations

**ACM 504: **Applied linear algebra

**ACM 505: **Dynamical systems

**ACM 506: **Fractional differential equations I

**ACM 507: **Scientific computing

**ACM 508: **Applied functional analysis

**ACM 509: **Mathematical and computational modeling in science and engineering

**ACM 601:** Advanced ordinary differential equations

**ACM 602:** Finite element analysis

**ACM 603:** Computational methods for partial differential equations

**ACM 604:** High performance computing I

**ACM 605:** High performance computing II

**ACM 606:** Fractional differential equations II

**ACM 607**: Computational science and engineering

**ACM 608:** Selected topics in applied and computational mathematics

__Project-Based-Learning/____ Research Seminar Courses:__

**ACM 701**: Project based learning in applied and computational mathematics

**ACM 702: **Seminars on advanced topics in applied and computational mathematics I

**ACM 703:** Seminars on advanced topics in applied and computational mathematics II

**ACM 501 – Partial differential equations:** The course will start with a quick reminder on partial differential equations, first-order PDEs, and Classification of second-order linear equations (elliptic, parabolic, hyperbolic). It will cover the following core topics: Elliptic equations (Laplace and Poisson equations; finite difference, Gauss’s theorem, Green’s function). Parabolic equations (Heat equations fundamental solutions, maximum principles, finite difference and convergence, Stefan Problems). Wave equation and vibrations. Water waves and various approximations.

**ACM 502 – Numerical analysis:** This course gives an opportunity to the students to be familiar with data fitting; interpolation using Fourier transform, orthogonal polynomials and splines, numerical quadrature, relevant linear algebra, numerical methods for solving initial value problems, boundary value problems and eigenvalue problems, finite difference methods, conjugate gradient methods for PDEs (elliptic, parabolic and hyperbolic equations). stability analysis, convergence analysis and error propagation will be covered, Computer programming in MATLAB is required.

**ACM 503 – Advanced numerical methods for differential equations:** The purpose of this course is to provide students with numerical methods for solving ordinary, fractional and partial differential equations (finite difference, iterative, finite element, pseudo-spectral, and method of lines and multi-grid methods). To get a handle on these issues, this course will fuse together knowledge from a variety of relevant topics including introduction to parallel algorithms, Grid computing, Visualization tools and software development.

**ACM 504 – Applied linear algebra:** This course covers selected topics in inner product spaces, matrix norms, orthogonality, the Gram-Schmidt process, least squares approximation, sparse matrices, positive definite, symmetric matrices and Cholesky’s algorithm, quadratic forms, eigenvalues and eigenvectors. The course will also present the spectral theorem, optimization principles for eigenvalues and eigenvectors, singular value decomposition, the Schur decomposition and methods for computing eigenvalues, non-negative matrices, graphs, networks, random walks, the Perron-Frobenius theorem.

**ACM 505 – Dynamical systems:** Dynamical systems play a central role in modeling complex phenomena in science and engineering. We begin the course with existence and uniqueness for solutions of ordinary differential equations, linear systems, nonlinear systems, stability, Limit cycle, bifurcation theory and Chaos. Theory and theoretical examples are complemented by computational, model driven examples from biological, physical and engineering sciences will also be included and discussed.

**ACM 506- Fractional differential equations I:** This is an introductory course in fractional differential equations. Different types of fractional derivatives and fractional integrals, ordinary fractional differential equations are considered. The course will also examine the existence and the uniqueness of solutions. Numerical and analytical methods for solving fractional differential equations are provided.

**ACM 507- Scientific computing:** This course covers fundamental material necessary for using high performance computing in science and engineering (Introduction to C++ programming language and/ or introduction to Matlab, Mathematica, and Maple). There is a special emphasis on algorithm development, computer implementation, Numerical solutions of linear and nonlinear systems, approximation of functions, integration, differential equations and the application of these methods to specific problems in science and engineering.

**ACM 508- Applied functional analysis:** The aim of this course is to provide students with both a deep understanding of the fundamentals of the elements of functional analysis such as Normed spaces, Banach space, Escolli theory, Weierstrass theory, Banach theory, Ritz theory, Hilbert’s space and Sobolev space. Also, applications to DEs are concluded.

**ACM 509- Mathematical and computational modeling in science and engineering: **Introductory course in applied mathematics and computational modeling with emphasis on modeling of biological problems, engineering problems in terms of differential equations and stochastic dynamical systems. Students will be working in groups on several projects and will present them in class at the end of the course.

**ACM 601- Advanced ordinary differential equations:**** **This course covers selected topics in existence and uniqueness of solutions (autonomous systems, non-autonomous systems) and linear systems with constant, periodic and analytic coefficients. Singularities of autonomous systems, self-adjoint Eigenvalue problems, and expansion in terms of Eigenfunctions are included. Also, the students will be exposed to stability theory and Lyapunov functions, the stability of autonomous systems and stability of nonautonomous systems.

**ACM 602- Finite element analysis:** This course presents fundamental material necessary for dealing with engineering problems. In this class, the student will be familiar with basic concepts, direct stiffness approach, method of weighted residuals, weak formulation and variation techniques in the solution of physical and engineering problems such as (trusses, beams, plane frames, time-dependent problems of fluid and heat flow, one and two-dimensional problems, Elasticity problems).

**ACM 603- Computational methods for partial differential equations:** This course is primarily concerned with classes of problems and PDEs, types of boundary conditions, classes of numerical methods for PDEs, analysis of numerical methods for PDEs, errors, time, and memory. In this class, the student will deal with Boundary value problems: Elliptic PDEs (Finite difference methods, Finite element methods and overview of linear solvers for PDEs). Also, initial value problems: Parabolic and Hyperbolic PDEs (Parabolic PDEs, Finite difference methods, Hyperbolic PDEs, Finite difference methods, Convergence, Stability and Method of lines) are included. Advanced and parallel methods for PDEs (Parallel methods for BVPs, Domain decomposition, Schur complement method, Schwarz alternating method, Multigrid method, Fast Fourier Transform solvers), software development.

**ACM 604 – High performance computing I:** The aim of this course is to provide students with the fundamentals of parallel computing, linear systems (direct methods such as Gauss elimination, LU factorization, symmetric and symmetric positive definite matrices, banded systems, pivoting in banded systems, tridiagonal systems, banded systems, partitioning methods) and linear systems (iterative methods such as Jacobi and Gauss-Seidel methods, CG method ,preconditioning , preconditioned CG method, application to the GS, SOR and SSOR methods, Block iterative methods).

**ACM 605- High performance computing II:** The main topic of the course is the study of partial differential equations (Schur complement method, arrowhead matrix, application to the 1D BVP, the use of CG for the solution of the Schur complement system, Schur complement method, application to the 2D BVP, Schwarz alternating (splitting) method, preconditioning, Multigrid method, FFT methods, application to the 1D BVP, Tensor products of matrices).

**ACM 606- Fractional differential equations II:** This class aims to provide a basic theory of fractional differential equations, Integral transform method for explicit solutions to fractional differential equations, sequential linear differential equations of fractional order and system of fractional order differential equations. Partial differential equations of fractional order. Also, numerical methods for solving fractional differential equations arising in science and engineering are considered.

**ACM 607- Computational science and engineering:**** **Computational science and engineering has applications to a wide variety of fields, from physics to engineering and medicine. This is one of its most exciting aspects that it brings researchers from many disciplines together with a common language. The class is concerned with development and analysis of algorithms used in the solution of science and engineering problems. Numerical analysis of discretization schemes for partial differential equations. Also, a survey of finite difference, finite element, finite volume and spectral approximations for the numerical solution of the incompressible and compressible flow, Euler and Navier-Stokes equations, including shock-capturing methods are concluded.

**ACM 608- Selected topics in applied and computational mathematics:** Selected topics in applied and computational mathematics will vary according to student and instructor interest.

**ACM 701- Project-based learning in applied and computational mathematics: **This module includes students participation in Project-Based Learning activities in new advanced topics related to the field of research, suggested by the student’s supervisors. Students will be encouraged to demonstrate knowledge and skills by working for an extended period of time to investigate and respond to an engaging and complex question, problem, or challenge. Student’s evaluation will be based on the presented written materials and his/her participation in discussion sessions.

**ACM 702 and ACM 703- Seminars on advanced topics on applied and computational mathematics I and II:** Series of research seminars conducted by Ph.D. students and based on self-learning and presentations of new advanced topics in the fields of ACM, selected by professors specialized in those topics. The student’s evaluation is based on his/her understanding of the presented topics and presentations skills.

**ACM 801– M.Sc. Thesis:** For the Thesis Master’s in ACM, students will be trained, with the help of their supervisors, to perform a literature review, identify important issues in a specific field and understand the scientific approach to research questions in ACM, carry out a scientific study and appropriately managing the obtained data. Also, students will be trained to express themselves clearly in science (when speaking and writing). The students will be trained and encouraged to prepare their work for publication in high impact scientific journals. The student will be guided to submit a research thesis not exceeding 60,000 words, including tables, figures, and footnotes, and present an appropriate defense in an oral examination.

**ACM 802 – Ph.D. Thesis:** For the Thesis Ph.D.’s in ACM, students will be trained, with the help of their supervisors, to perform a literature review, identify important issues in a specific field and understand the scientific approach to research questions in ACM, carry out a scientific study and appropriately managing the obtained data. Importantly, the Ph.D. graduate students will be provided with a complete and thorough opportunity to become a research scientist, to be exposed to the highest quality research methods and techniques in the field of ACM. Also, the student will be supported by an environment that fosters critical thinking. In addition, they will be provided with an appreciation for the value of multidisciplinary collaborations. The students will be trained and encouraged to prepare their work for publication in high impact scientific journals. The student will be guided to submit a research thesis not exceeding 100,000 words, including tables, figures, and footnotes, and present an appropriate defense in an oral examination.