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

CodeSubjectCreditsWinterSummer
NMMA401Functional Analysis 1 84/2 C+Ex
NMMA405Partial Differential Equations 1 63/1 C+Ex
NMMA406Partial Differential Equations 2 63/1 C+Ex
NMMO401Continuum Mechanics 62/2 C+Ex
NMMO402Thermodynamics and Mechanics of Non-Newtonian Fluids 52/1 C+Ex
NMMO403Computer Solutions of Continuum Physics Problems 52/2 C+Ex
NMMO404Themodynamics and Mechanics of Solids 52/1 C+Ex
NMNV405Finite Element Method 1 52/2 C+Ex
NMNV411Algorithms for matrix iterative methods*52/2 C+Ex
NOFY036Thermodynamics and Statistical Physics 63/2 C+Ex
NSZZ023Diploma Thesis I 60/4 C
NSZZ024Diploma Thesis II 90/6 C
NSZZ025Diploma Thesis III 150/10 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.

CodeSubjectCreditsWinterSummer
NMMA407Ordinary Differential Equations 2 52/2 C+Ex
NMMA531Partial Differential Equations 3 42/0 Ex
NMMO432Classical Problems of Continuum Mechanics 42/1 C+Ex
NMMO463GENERIC — non-equilibrium thermodynamics 42/1 C+Ex
NMMO531Biothermodynamics*52/2 C+Ex
NMMO532Mathematical Theory of Navier-Stokes Equations 32/0 Ex
NMMO533Nonlinear Differential Equations and Inequalities 1 63/1 C+Ex
NMMO534Nonlinear Differential Equations and Inequalities 2 63/1 C+Ex
NMMO535Mathematical Methods in Mechanics of Solids 32/0 Ex
NMMO536Mathematical Methods in Mechanics of Compressible Fluids 32/0 Ex
NMMO537Saddle Point Problems and Their Solution 52/2 C+Ex
NMMO539Mathematical Methods in Mechanics of Non-Newtonian Fluids 32/0 Ex
NMMO541Theory of Mixtures 42/1 C+Ex
NMMO543Modelling in biomechanics*53/0 C+Ex
NMMO567Simulation and Theory of Biological and Soft Matter Systems I - Biopolymers, Ions and Small Molecules 32/0 Ex
NMMO568Simulation and Theory of Biological and Soft Matter Systems II — Interfaces, Self-Assembly and Networks 32/0 Ex
NMMO660Non-equilibrium thermodynamics of electrochemistry 42/1 C+Ex
NMNV403Numerical Software 1 52/2 C+Ex
NMNV404Numerical Software 2 52/2 C+Ex
NMNV412Analysis of matrix iterative methods — principles and interconnections 64/0 Ex
NMNV501Solution of Nonlinear Algebraic Equations*52/2 C+Ex
NMNV503Numerical Optimization Methods 1 63/1 C+Ex
NMNV532Parallel Matrix Computations 52/2 C+Ex
NMNV537Mathematical Methods in Fluid Mechanics 1 32/0 Ex
NMNV538Mathematical Methods in Fluid Mechanics 2 32/0 Ex
NMNV565High-Performance Computing for Computational Science 52/2 C+Ex
NOFY026Classical Electrodynamics 52/2 C+Ex
NTMF034Electromagnetic Field and Special Theory of Relativity 52/1 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

CodeSubjectCreditsWinterSummer
NMMA452Seminar on Partial Differential Equations 30/2 C0/2 C
NMMA461Regularity of Navier — Stokes Equations 30/2 C0/2 C
NMMA583Qualitative Properties of Weak Solutions to Partial Differential Equations 32/0 Ex
NMMA584Regularity of Weak Solutions to Partial Differential Equations 30/2 C
NMMO461Seminar in Continuum Mechanics 20/2 C0/2 C
NMMO463GENERIC — non-equilibrium thermodynamics 42/1 C+Ex
NMMO561Regularity of solutions of Navier-Stokes equations 32/0 Ex
NMMO564Selected Problems in Mathematical Modelling 30/2 C
NMMO660Non-equilibrium thermodynamics of electrochemistry 42/1 C+Ex
NMNV406Nonlinear differential equations 52/2 C+Ex
NMNV541Shape and Material Optimisation 1 32/0 Ex
NMNV542Shape and Material Optimisation 2 32/0 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

CodeSubjectCreditsWinterSummer
NMMA401Functional Analysis 1 84/2 C+Ex
NMMA405Partial Differential Equations 1 63/1 C+Ex
NMMO401Continuum Mechanics 62/2 C+Ex
NOFY036Thermodynamics and Statistical Physics 63/2 C+Ex
NMNV405Finite Element Method 1 52/2 C+Ex
NMMA406Partial Differential Equations 2 63/1 C+Ex
NSZZ023Diploma Thesis I 60/4 C
NMMO402Thermodynamics and Mechanics of Non-Newtonian Fluids 52/1 C+Ex
NMMO403Computer Solutions of Continuum Physics Problems 52/2 C+Ex
NMMO404Themodynamics and Mechanics of Solids 52/1 C+Ex
 Optional and Elective Courses 1  

2nd year

CodeSubjectCreditsWinterSummer
NSZZ024Diploma Thesis II 90/6 C
NMNV412Analysis of matrix iterative methods — principles and interconnections 64/0 Ex
NSZZ025Diploma Thesis III 150/10 C
 Optional and Elective Courses 30