# Mathematical Modelling in Physics and Technology

**Coordinated by:** Mathematical Institute of Charles University

**Study branch coordinator:** prof. RNDr. Josef Málek, CSc., DSc.

Mathematical modelling is an interdisciplinary field connecting mathematical analysis, numerical mathematics, and physics. The curriculum is designed to provide excellent basic knowledge in all these disciplines and to allow a flexible widening of knowledge by studying specialized literature when the need arises. All students take obligatory courses in continuum mechanics, partial differential equations, and numerical mathematics. Students will acquire the ability to design mathematical models of natural phenomena (especially related to continuum mechanics and thermodynamics), analyse them, and conduct numerical simulations. After passing the obligatory classes, students get more closely involved with physical aspects of mathematical modelling (model design), with mathematical analysis of partial differential equations, or with methods for computing mathematical models. The grasp of all levels of mathematical modelling (model, analysis, simulations) allows the students to use modern results from all relevant fields to address problems in physics, technology, biology, and medicine that surpass the scope of the fields individually. Graduates will be able to pursue academic or commercial careers in applied mathematics, physics and technology.

The graduate will have mastered methods and results in continuum mechanics and thermodynamics, mathematical analysis of partial differential equations, and numerical mathematics, and will be ready to widen his/her knowledge by studying specialized literature. He/she will be able to formulate questions regarding the physical substance of natural phenomena, especially those related to the behaviour of fluids and solid matter in the framework of classical physics, with applications to technology, medicine, biology, geophysics, and meteorology. He/she will be able to choose appropriate mathematical models for such phenomena, carry out its mathematical analysis, and conduct numerical simulations with suitable methods. He/she will be able to critically analyse, evaluate, and tie in the whole modelling process. In simpler cases, he/she will be able to assess the errors in the modelling process and predict the agreement between numerical results and the physical process. The graduate will be ready to work in interdisciplinary teams. He/she will be able to pose interesting questions in a format ready for mathematical investigation and use abstract mathematical results in order to address applied problems.

*Assumed knowledge*

It is assumed that an incoming student of this branch has sufficient knowledge of the following topics and fields:

**–**Foundations of differential and integral calculus of one variable and several variables. Volume, surface and line integral. Measure theory. Lebesgue integral.**–**Foundations of linear algebra (vector spaces, matrices, determinants, Jordan canonical form, eigenvalues and eigenvectors, multilinear algebra, quadratic forms). Numerical solution of systems of linear algebraic equations (Schur theorem, QR decomposition, LU decomposition, singular value decomposition, least squares problem, partial eigenvalue problem, conjugate gradient method, GMRES, backward error, sensitivity and numerical stability, QR algorithm).**–**Foundations of complex analysis (Cauchy theorem, residual theorem, conformal mappings, Laplace transform).**–**Foundations of functional analysis and theory of metric spaces (Banach and Hilbert spaces, operators and functionals, Hahn-Banach theorem, dual space, bounded operators, compact operators, theory of distributions).**–**Foundations of theory of ordinary differential equations (existence of solution, maximal solution, systems of linear equations, stability) and partial differential equations (quasilinear first order equations, Laplace equation and heat equation – fundamental solution and maximum principle, wave equation – fundamental solution, finite propagation speed).**–**Foundations of classical mechanics (Newton laws, Lagrange equations, Hamilton equations, variational formulation, rigid body dynamics).

Should an incoming student not meet these entry requirements, the coordinator of the study programme may assign a method of acquiring the necessary knowledge and abilities, which may for example mean taking selected bachelor's courses, taking a reading course with an instructor, or following tutored independent study.

#### 6.1 Obligatory Courses

Code | Subject | Credits | Winter | Summer | |

NMMA401 | Functional Analysis 1 | 8 | 4/2 C+Ex | — | |

NMMA405 | Partial Differential Equations 1 | 6 | 3/1 C+Ex | — | |

NMMA406 | Partial Differential Equations 2 | 6 | — | 3/1 C+Ex | |

NMMO401 | Continuum Mechanics | 6 | 2/2 C+Ex | — | |

NMMO402 | Thermodynamics and Mechanics of Non-Newtonian Fluids | 5 | — | 2/1 C+Ex | |

NMMO403 | Computer Solutions of Continuum Physics Problems | 5 | — | 2/2 C+Ex | |

NMMO404 | Themodynamics and Mechanics of Solids | 5 | — | 2/1 C+Ex | |

NMNV405 | Finite Element Method 1 | 5 | 2/2 C+Ex | — | |

NMNV411 | Algorithms for matrix iterative methods | ^{*} | 5 | 2/2 C+Ex | — |

NOFY036 | Thermodynamics and Statistical Physics | 6 | 3/2 C+Ex | — | |

NSZZ023 | Diploma Thesis I | 6 | — | 0/4 C | |

NSZZ024 | Diploma Thesis II | 9 | 0/6 C | — | |

NSZZ025 | Diploma Thesis III | 15 | — | 0/10 C |

^{*}
The course replaces the obligatory course NMNV412 from study plans valid before 2021/2022.

#### 6.2 Elective Courses

It is required to earn at least 16 credits from elective courses.

Code | Subject | Credits | Winter | Summer | |

NMMA407 | Ordinary Differential Equations 2 | 5 | 2/2 C+Ex | — | |

NMMA531 | Partial Differential Equations 3 | 4 | 2/0 Ex | — | |

NMMO432 | Classical Problems of Continuum Mechanics | 4 | — | 2/1 C+Ex | |

NMMO463 | GENERIC — non-equilibrium thermodynamics | 4 | 2/1 C+Ex | — | |

NMMO531 | Biothermodynamics | ^{*} | 5 | 2/2 C+Ex | — |

NMMO532 | Mathematical Theory of Navier-Stokes Equations | 3 | — | 2/0 Ex | |

NMMO533 | Nonlinear Differential Equations and Inequalities 1 | 6 | 3/1 C+Ex | — | |

NMMO534 | Nonlinear Differential Equations and Inequalities 2 | 6 | — | 3/1 C+Ex | |

NMMO535 | Mathematical Methods in Mechanics of Solids | 3 | 2/0 Ex | — | |

NMMO536 | Mathematical Methods in Mechanics of Compressible Fluids | 3 | — | 2/0 Ex | |

NMMO537 | Saddle Point Problems and Their Solution | 5 | — | 2/2 C+Ex | |

NMMO539 | Mathematical Methods in Mechanics of Non-Newtonian Fluids | 3 | 2/0 Ex | — | |

NMMO541 | Theory of Mixtures | 4 | 2/1 C+Ex | — | |

NMMO543 | Modelling in biomechanics | ^{*} | 5 | 3/0 C+Ex | — |

NMMO567 | Simulation and Theory of Biological and Soft Matter Systems I - Biopolymers, Ions and Small Molecules | 3 | 2/0 Ex | — | |

NMMO568 | Simulation and Theory of Biological and Soft Matter Systems II — Interfaces, Self-Assembly and Networks | 3 | — | 2/0 Ex | |

NMMO660 | Non-equilibrium thermodynamics of electrochemistry | 4 | — | 2/1 C+Ex | |

NMNV403 | Numerical Software 1 | 5 | 2/2 C+Ex | — | |

NMNV404 | Numerical Software 2 | 5 | — | 2/2 C+Ex | |

NMNV412 | Analysis of matrix iterative methods — principles and interconnections | 6 | — | 4/0 Ex | |

NMNV501 | Solution of Nonlinear Algebraic Equations | ^{*} | 5 | 2/2 C+Ex | — |

NMNV503 | Numerical Optimization Methods 1 | 6 | 3/1 C+Ex | — | |

NMNV532 | Parallel Matrix Computations | 5 | — | 2/2 C+Ex | |

NMNV537 | Mathematical Methods in Fluid Mechanics 1 | 3 | 2/0 Ex | — | |

NMNV538 | Mathematical Methods in Fluid Mechanics 2 | 3 | — | 2/0 Ex | |

NMNV565 | High-Performance Computing for Computational Science | 5 | 2/2 C+Ex | — | |

NOFY026 | Classical Electrodynamics | 5 | — | 2/2 C+Ex | |

NTMF034 | Electromagnetic Field and Special Theory of Relativity | 5 | — | 2/1 Ex |

^{*} The course NMMO531 is not taught since in the academic year 2021/22. It is replaced with the course NMMO543.

#### 6.3 Recommended Optional Courses

Code | Subject | Credits | Winter | Summer | |

NMMA452 | Seminar on Partial Differential Equations | 3 | 0/2 C | 0/2 C | |

NMMA461 | Regularity of Navier — Stokes Equations | 3 | 0/2 C | 0/2 C | |

NMMA583 | Qualitative Properties of Weak Solutions to Partial Differential Equations | 3 | 2/0 Ex | — | |

NMMA584 | Regularity of Weak Solutions to Partial Differential Equations | 3 | — | 0/2 C | |

NMMO461 | Seminar in Continuum Mechanics | 2 | 0/2 C | 0/2 C | |

NMMO463 | GENERIC — non-equilibrium thermodynamics | 4 | 2/1 C+Ex | — | |

NMMO561 | Regularity of solutions of Navier-Stokes equations | 3 | 2/0 Ex | — | |

NMMO564 | Selected Problems in Mathematical Modelling | 3 | — | 0/2 C | |

NMMO660 | Non-equilibrium thermodynamics of electrochemistry | 4 | — | 2/1 C+Ex | |

NMNV406 | Nonlinear differential equations | 5 | — | 2/2 C+Ex | |

NMNV541 | Shape and Material Optimisation 1 | 3 | 2/0 Ex | — | |

NMNV542 | Shape and Material Optimisation 2 | 3 | — | 2/0 Ex |

#### 6.4 State Final Exam

*Requirements for taking the final exam*

**–**Earning at least 120 credits during the course of the study.**–**Completion of all obligatory courses prescribed by the study plan.**–**Earning at least 16 credits by completion of elective courses.**–**Submission of a completed master's thesis by the submission deadline.

*Oral part of the state final exam*

The oral part of the final exam consists of six subject areas: "Partial Differential Equations", "Funcional Analysis", "Finite element method", "Solution of algebraic equations", "Continuum kinematics and dynamics", and "Constitutive realtions of fluids and solids". One question is asked from each subject area.

*Requirements for the oral part of the final exam *

#### 1. Continuum mechanics and thermodynamics

Kinematics. Stress tensor. Balance equations. Constitutive relations. Models for fluids and solids.

#### 2. Functional analysis and partial differential equations

Linear operators and functionals, compact operators. Distributions. Function spaces. Weak solutions of the linear elliptic, parabolic and hyperbolic second order partial differential equations – foundations of the existence theory, elementary theory of qualitative properties of the solutions.

#### 3. Numerical methods

Numerical methods for partial differential equations. Finite element method. Iterative methods for solving systems of linear algebraic equations.

#### 6.5 Recommended Course of Study

*1st year*

Code | Subject | Credits | Winter | Summer | |

NMMA401 | Functional Analysis 1 | 8 | 4/2 C+Ex | — | |

NMMA405 | Partial Differential Equations 1 | 6 | 3/1 C+Ex | — | |

NMMO401 | Continuum Mechanics | 6 | 2/2 C+Ex | — | |

NOFY036 | Thermodynamics and Statistical Physics | 6 | 3/2 C+Ex | — | |

NMNV405 | Finite Element Method 1 | 5 | 2/2 C+Ex | — | |

NMMA406 | Partial Differential Equations 2 | 6 | — | 3/1 C+Ex | |

NSZZ023 | Diploma Thesis I | 6 | — | 0/4 C | |

NMMO402 | Thermodynamics and Mechanics of Non-Newtonian Fluids | 5 | — | 2/1 C+Ex | |

NMMO403 | Computer Solutions of Continuum Physics Problems | 5 | — | 2/2 C+Ex | |

NMMO404 | Themodynamics and Mechanics of Solids | 5 | — | 2/1 C+Ex | |

Optional and Elective Courses | 1 |

*2nd year*

Code | Subject | Credits | Winter | Summer | |

NSZZ024 | Diploma Thesis II | 9 | 0/6 C | — | |

NMNV412 | Analysis of matrix iterative methods — principles and interconnections | 6 | — | 4/0 Ex | |

NSZZ025 | Diploma Thesis III | 15 | — | 0/10 C | |

Optional and Elective Courses | 30 |