Master study programme **Mathematical Modelling in Physics and Technology** is designed for students of mathematics. (The programme for students of physics is called **Matematical and Computational Modelling in Physics**, details concerning this study programme can be found here.)

## Contacts

If you have any **questions concerning the study of mathematical modelling**, please contact Vít Průša (prusv@karlin.mff.cuni.cz) or Josef Málek (malek@karlin.mff.cuni.cz).

## Admission

Concerning the general overview of the admission procedure, please see the webpages of the Faculty of Mathematics and Physics.

Note also, that if you have a sufficiently strong academic record, you can apply for the exemption from the entrance examination.

**If you are not yet familiar with advanced subjects such as functional analysis but you are still strongly motivated to study mathematical modelling, please contact us, and we will try to recommend you textbooks and other material for self-study.**

It is expected that you have sufficient knowledge of the following topics.

**Differential and integral calculus**(single variable and multivariable). Limits, derivatives, line, surface and volume integrals. Foundations of calculus of variations.**Foundations of measure theory, Lebesgue integral**. These topics are taught, for example, in the undergraduate course Measure nad Integration Theory I, see the course syllabus for what is in this case meant by “foundations”.**Foundations of linear algebra**. Vector spaces, matrices, determinants, Jordan canonical form, orthogonalization, eigenfunctions and eigenvalues, multilinear algebra, quadratic forms.**Foundations of numerical solution of systems of linear algebraic equations**. Schur theorem, QR decomposition, LU decomposition, singular value decomposition, least squares method, conjugate gradient method, GMRES, backward error, sensitivity and numerical stability. These topics are taught, for example, in the undergraduate course Analysis of Matrix Calculations I, see the course syllabus for what is in this case meant by “foundations”.**Foundations of complex analysis**. Cauchy theorem, residue theorem, conformal mapping, Laplace transform. These topics are taught, for example, in the undergraduate course Introduction to Complex Analysis, see the course syllabus for what is in this case meant by “foundations”.**Foundations of functional analysis and theory of metric spaces**. Banach and Hilbert spaces, operators and functionals, Hahn-Banach theorem, dual spaces, bounded operators, compact operators, theory of distributions. These topics are taught, for example, in the undergraduate course Introduction to Functional Analysis, see the course syllabus for what is in this case meant by “foundations”.**Foundation of theory of ordinary differential equations**. Existence of solution, maximal solution, systems of linear ordinary differential equations, elements of stability theory. These topics are taught, for example, in the undergraduate course Ordinary Differential Equations, see the course syllabus for what is in this case meant by “foundations”.**Foundations of the classical theory of partial differential equations**. Quasilinear equations of the first order, Laplace equation and heat equation – fundamental solutions and maximum principle, wave equation – fundamental solution, finite differences method. These topics are taught, for example, in the undergraduate course Introduction to Partial Differential Equations, see the course syllabus for what is in this case meant by “foundations”.**Foundations of classical mechanics**. Newton laws of motion, Lagrange equation, Hamilton equations, variational principles, rigid body dynamics. These topics are taught, for example, in the undergraduate course Theoretical Mechanics, see the course syllabus for what is in this case meant by “foundations”.

## Curriculum

The expected length of the study is **two years**. You are expected to choose the topic of your thesis during the first year of your study. There are three sets of courses: **compulsory**, **compulsory elective** and **optional**.

Concerning the **compulsory** courses you **must complete all of them**. The strange term **compulsory elective** means that you **must complete some of these courses**. (The term “some” means that you need certain number of credits from these subjects, see faculty webpages for details.) The choice is yours, but it should reflect the topic of your thesis. Talk to your thesis supervisor if you want to check which courses are suitable for you! The choice of the **optional** courses is completely up to you. Ask your thesis supervisor, classmates or the people responsible for the study programme, see contacts, for a hint.

### Compulsory courses

You **must complete all these courses**. The next to the last column indicates the recommended year of study. Besides the courses listed below you must also complete three “virtual” one semester courses called “Diploma Thesis Seminar”. You will get the credit for this course provided that your thesis supervisor confirms that you work on the thesis.

Course | Annotation | Year | Semester |
---|---|---|---|

Functional Analysis 1 | topological spaces, weak topology, vector integration, spectral theory | 1st | winter |

Continuum Mechanics | introduction to continuum mechanics, kinematics, balance equations, linearized elasticity, Navier–Stokes fluid | 1st | winter |

Finite Element Method 1 | foundations of the theory of finite element method, linear elliptic equations | 1st | winter |

Partial Differential Equations 1 | notion of the weak solution, Lebesgue spaces, Sobolev spaces, linear elliptic equations | 1st | winter |

Thermodynamics and Statistical Physics | basic concepts, methods and results of classical thermodynamics and statistical physics | 1st | winter |

Partial Differential Equations 2 | Bochner spaces, parabolic and linear hyperbolic equations of the second order | 1st | summer |

Computer Solutions of Problems in Continuum Mechanics | basic commercial/open source software for numerical computation (Matlab, Femlab, FEniCS) and its application to solution of partial differential equations, basic numerical libraries (Blas, Lapack, Petsc), finite element libraries (Feat, Featflow), libraries for parallel computation (MPI, OpenMP) | 1st | summer |

Thermodynamics and Mechanics of non-Newtonian Fluids | non-Newtonian phenomena, constitutive relations for non-Newtonian fluids, thermodynamics of non-Newtonian fluids | 1st | summer |

Thermodynamics and Mechanics of Solids | mechanics and thermodynamics of nonlinear solids, constitutive relations for non-linear solids | 1st | summer |

Algorithms for Matrix Iterative Methods | matrix iterative methods, Krylov subspace methods, implementation | 2nd | winter |

### Compulsory elective courses

The courses are grouped into three categories — physic, theory of partial differential equations and numerical mathematics. You **must complete some of these courses**. (The term “some” means that you need certain number of credits from these subjects, see faculty webpages for details.) The choice is yours, but it should reflect the topic of your thesis. Talk to your thesis supervisor if you want to check which courses are suitable for you!

Course | Annotation | Year | Semester |
---|---|---|---|

High-performance Computing for Computational Science | concepts and tools for high-performance computing | 2nd | winter |

Analysis of matrix iterative methods – principles and interconnections | matrix iterative methods, Krylov subspace methods | 2nd | summer |

Ordinary Differential Equations 2 | dynamical systems, Poincaré-Bendixson theory, Carathéodory theory, optimal control, Pontryagin maximum principle, bifurcations, stable, unstable and central manifolds | 1st | winter |

Classical Electrodynamics | Maxwell equations, steady and quasisteady approximations, electromagnetic waves | 1st | summer |

Classical Problems in Continuum Mechanics | thermal convection, boundary layer theory, stability of fluid flows | 1st | summer |

Modelling in Biomechanics | applications of continuum mechanics in the description of biological systems | 2nd | winter |

Mathematical Methods in Mechanics of non-Newtonian Fluids | mathematical tools and techniques for establishing the existence, uniqueness, regularity and large time behaviour of weak solutions to partial differential equations governing the motion of non-Newtonian fluids | 2nd | winter |

Mathematical Methods in Mechanics of Solids | mathematical methods for analysis of boundary and initial value problems arising in mechanics and thermomechanics of solids | 2nd | winter |

Nonlinear Differential Equations and Inequalities 1 | pseudomonotone and monotone operators, set-valued mappings and applications to nonlinear elliptic partial differential equations and inequalities | 2nd | winter |

Numerical Methods in Mechanics of Fluids 1 | Stokes problem, Oseen problem, steady and unsteady Navier-Stokes equations, solvability of the discrete systems, Babuška-Brezzi condition | 2nd | winter |

Numerical Software 1 | implementation of numerical methods | 2nd | winter |

Numerical optimization methods 1 | theoretical and practical questions regarding the numerical solution of non-linear equations and minimization of functionals | 2nd | winter |

Mixture Theory | mechanics and thermodynamics of mixtures | 2nd | winter |

Electromagneic Field and Special Theory of Relativity | electrostatics, magnetostatics, electromagnetism (Maxwell equations, Lorentz force, electromagnetic waves, electric circuits), Minkowski spacetime, Lorentz transformation, relativistic particle dynamics, relativistic formulation of the theory of electromagnetic field | 2nd | summer |

Mathematical Theory of Navier-Stokes Equations | mathematical tools and techniques for establishing the existence, uniqueness, regularity and large time behaviour of weak solutions to partial differential equations governing the motion of incompressible Navier-Stokes fluid | 2nd | summer |

Mathematical Theory of Compressible Fluids | mathematical tools and techniques for establishing the existence, uniqueness, regularity and large time behaviour of weak solutions to partial differential equations governing the motion of compressible fluids | 2nd | summer |

Nonlinear Differential Equations and Inequalities 2 | pseudomonotone and monotone operators, set-valued mappings and applications to nonlinear parabolic partial differential equations and inequalities | 2nd | summer |

Numerical Methods in Mechanics of Fluids 2 | Euler equations, Cauchy problem for Euler equations, weak solution, finite volume method, construction of numerical fluxes, Godunov method | 2nd | summer |

Numerical Software 2 | implementation of numerical methods | 2nd | summer |

Parallel Matrix Computations | dense and sparse matrices, data structures, parallelization of direct methods for sparse matrices, parallel preconditioned Krylov subspace methods, domain decomposition | 2nd | summer |

Partial Differential Equations 3 | linear and nonlinear evolution equations, semigroup theory, asymptotic behaviour of the solutions to differential equations, pptimal control of evolution equations | 2nd | winter |

GENERIC — Non-equilibrium Thermodynamics | thermodynamics in GENERIC (General Equation for Non-equilibrium Reversible-Irreversible Coupling) formalism | 2nd | winter |

Non-equilibrium Thermodynamics of Electrochemistry | equations describing electrochemical processes | 2nd | summer |

Simulation and Theory of Biological and Soft Matter Systems I – Biopolymers, Ions and Small Molecules | introduction to simulation techniques and theoretical concepts relevant for biological systems | 2nd | winter |

Simulation and Theory of Biological and Soft Matter Systems II – Interfaces, Self-assembly and Networks | theory of interfaces, membranes and self-assembly and some phenomenological models of biological systems | 2nd | summer |

### Optional courses

The choice of the optional courses is completely up to you, you can choose any course taught at Charles University. (We encourage you to attend some lecture beyond your own specialisation, you might want to try economics, for example Fundamental social issues from the point of view of economics, biology and so forth.) Ask your thesis supervisor, classmates or the people responsible for the study programme, see contacts, for hint. We think that you should take into consideration these courses.

Course | Annotation | Year | Semester |
---|---|---|---|

Qualitative Properties of Weak Solutions to Partial Differential Equations | classical results about regularity and qualitative properties of weak solutions to partial differential equations | 2nd | winter |

Regularity of Solutions of Navier-Stokes Equations | recent results in the theory of evolutionary Navier-Stokes equations, special attention to the regularity of the solution in three space dimensions | 2nd | winter |

Regularity of Weak Solutions to Partial Differential Equations | classical results concerning regularity of weak solutions to elliptic partial differential equations | 2nd | summer |

Seminar on Partial Differential Equations | discussion of recent results in the theory of partial differential equations | both | both |

Seminar on Continuum Mechanics | traditional seminar founded by professor Jindřich Nečas, discussion of recent results in mathematical theory of continuous media | both | both |

Saddle Point Problems and Their Solution In 2024 the lecture was reclassified as optional. Prior to 2024 it was classified as “compulsory elective”. |
saddle point problems, iterative methods, preconditioning, implementation, numerical stability | 2nd | winter |

## Thesis topics

See the Study Information System for a list of thesis topics. You can also directly contact members of the Division of Mathematical Modelling or scientists from the other institutions participating on the study programme, and ask them to propose a new thesis topic.

## State exam

After two years the study is completed by the state exam that has two parts — thesis defense and oral part.

### Thesis defense

The student presents, usually in twenty minutes, the results of his/her thesis. After that he/she must discuss the issues pointed out in the referee reports. (The referee reports are available in the Study Information System beforehand. It is absolutely crucial to be ready to discuss referee’s comments in a highly qualified manner.) At the end the student answers question by the committee members and the audience.

### Oral part

The oral part of state exam consists of answering several question concerning the topics specific for the given study programme. In principle the student will be answering questions regarding the topics studied in the compulsory courses listed in the curriculum above.

The oral part of the state exam proceeds as follows. The student answers, after a preparation, in total six questions regarding

- theory of partial differential equations (one question),
- functional analysis (one question),
- finite element method (one question),
- theory of solution of systems of algebraic equations (one question),
- continuum kinematics and dynamics (one question),
- theory of constitutive relations (one question).

Answering one question is expected to take ten minutes. The students is expected to demonstrate deeper insight into the discussed topics and the ability to see the subject matter in a broad context. (The student is not expected to give *extreme* details of each proof. He/she should be able to describe the basic ideas behind the proofs and explain why the definitions/notions/theorems are stated as they are.) Detailed list of the topics for the state exam is listed below. **You can also download the list of topics as a PDF file**.

#### Partial differential equations

The topics are covered in courses Partial differential equations 1 a Partial differential equations 2. It is expected that the student is also familiar with the classical theory of partial differential equations at the level of the course Introduction to partial differential equations. (See Admission.)

##### Sobolev spaces

Weak derivative, definition and basic properties of Sobolev spaces W^{k,p} — reflexivity, separability, density of smooth function, extension operator for W^{1,p} functions and domain with Lipschitz boundary. Theorems concerning the continuous and compact embedding of Sobolev spaces into Lebesgue and Hölder spaces. Definition of the trace operator for functions in Sobolev spaces, trace theorem, inverse trace theorem.

##### Weak solution of linear elliptic partial differential equations in bounded domain

Definition of the weak solution to linear elliptic partial differential equation with various boundary conditions. Existence of a solution via Riesz representation theorem (symmetric operator), via Lax–Milgram lemma and via Galerkin method. Compactness of the solution operator, eigenvalues and eigenvectors. Fredholm alternative and its applications. Maximum principle for the weak solution. W^{2,2} regularity via finite differences technique. Selfadjoint operator, equivalence with the minimization problem for a quadratic functional.

##### Weak solution of nonlinear elliptic partial differential equations in bounded domain

Fundamental of calculus of variations, fundamental theorem of calculus of variations, dual formulation, relation to convexity. Existence and uniqueness of the solution to nonlinear problems via fixed point theorems (nonlinear Lax–Milgram). Existence via Galerkin method and Minty trick — monotone operator and semilinear term.

##### Second order linear parabolic partial differential equations

Bochner spaces and their basic properties, Gelfand triple, Aubin–Lions lemma. Definition of the weak solution. Initial conditions. Existence of a solution via Galerkin method, uniqueness and regularity of the solution (spatial and temporal), smoothing property, maximum principle.

##### Second order linear hyperbolic partial differential equations

Definition of the weak solution. Initial conditions. Existence of a solution via Galerkin method, uniqueness, regularity (spatial and temporal), finite propagation speed.

#### Numerical mathematics

The topics are covered in courses Finite element method and Matrix iterative methods 1. It is expected that the student is also familiar with fundamentals of numerical mathematics at the level of the course Analysis of matrix calculations 1 and Fundamentals of numerical mathematics. (See Admission.)

##### Finite element method for solution of linear elliptic partial differential equations

Galerkin and Ritz methods for solution of abstract linear elliptic equations. Estimate on the discretization error, Céa lemma. Defintion of the abstract finite element, simple examples of finite elements of Lagrange and Hermite type. Approximation theory in Sobolev spaces, approximation properties of polynomial preserving operators. Application of these results to Lagrange and Hermite type finite elements. Rate of convergence of approximate solutions to linear elliptic partial differential equations. Estimate of the rate of convergence in L^{2} norm, Nitsche lemma.

Fundamentals of numerical integration in finite element method.

##### Solution of systems of algebraic equations and eigenvalues computation

Methods for solution of systems of linear algebraic equations and eigenvalues computation. Spectral decomposition of operators and matrices. Invariant subspaces and spectral information, normality. Comparison of direct and iterative methods for solution of systems of linear algebraic equations. Projection process. Description of the convergence of the iterative methods. Relation between iterative methods for linear equations and methods for eigenvalues computation. Comparison of methods for solution of linear and nonlinear systems of algebraic equations. Numerical stability and algebraic error.

#### Functional analysis

The topics are partially covered in course Functional analysis I. Theory of function spaces is also partially discussed in courses Partial differential equations 1 and Partial differential equations 2. It is expected that the student is also familiar with fundamental of functional analysis at the level of the course Inroduction to functional analysis. (See Admission.)

##### Hilbert and Banach spaces

Definition, norm, scalar product, examples. Linear functionals. Hahn-Banach theorem. Dual space, representations of some dual spaces (Hilbert spaces, Lebesgue spaces). Riesz representation theorem. Weak and weak-* topology. Banach-Alaoglu theorem. Weak compactness. Reflexivity.

##### Continuous linear operators

Definition, basic properties, norm, space of linear operators, adjoint operator. Spectrum and its basic properties, spectral radius. Compact operators, symmetric operator, selfadjoint operator, closure, closed operator, definition and properties of adjoint operator. Eigenvalues and eigenvectors of symmetric elliptic operators.

##### Fixed point theorems

Banach theorem, Brouwer theorem, Schauder theorem, Schaefer theorem.

##### Integral transformations and fundamentals of the theory of distributions

Definition of Fourier transform in L^{1} and its basic properties, Fourier inversion theorem, Fourier transform of convolution and derivative. Space of test functions, characterization of a distribution, order of a distribution, operations with distributions (derivative, multiplication), Schwarz space and tempered distributions, Fourier transformation for functions in Schwarz space and tempered distributions, its basic properties. Fourier transform in L^{2}.

#### Continuum mechanics

The topics are covered in courses Continuum mechanics, Thermodynamics and mechanics of nonnewtonian fluids and Thermodynamics and mechanics of solids.

##### Kinematics

Description of the motion of continuous media. Deformation of line, surface and volume elements, deformation, deformation gradient, polar decomposition of deformation gradient and its interpretation, right and left Cauchy–Green tensor, Green–Saint-Venant tensor. Rate of deformation of line, surface and volume elements. Velocity, velocity gradient, symmetric velocity gradient, material time derivative. Isochoric deformation. Streamlines and pathlines. Kinematic condition for material surface. Lagrange and Euler description. Compatibility conditions for linearized strain tensor. Isotropic tensor functions, representation theorem for isotropic tensor functions.

##### Dynamics

Balance equations (mass, momentum, angular momentum, total energy) in Euler and Lagrange description. Integral form for the balance equations, localization principle. Cauchy stress tensor, first Piola–Kirchhoff stress tensor, Piola transformation. Balance equations in non-inertial reference frame.

##### Simple constitutive relations

Compressible and incompressible Navier–Stokes–Fourier model (viscous heat conducting fluid), equation of state for ideal gas. Geometric linearization, linearized elasticity. Boundary conditions, displacement and traction boundary conditions.

##### Nonnewtonian fluids

Balance equations in the case of non-newtonian fluids, identification of entropy production. Clausius–Duhem inequality. Assumption on the maximization of the entropy production and its application in the design of mathematical models for fluids, concept of natural configuration. Overview of non-newtonian phenomena — shear dependent viscosity, normal stress differences, activation/deactivation criteria, stress relaxation, non-linear creep. Principle of frame indifference and its consequences, frame indifferent quantities in fluid mechanics, frame indifferent rates. Application of representation theorem for isotropic tensorial functions. Overview of standard models for non-newtonian fluids. Power-law fluids, fluids with pressure dependent viscosity, Bingham type fluids. Viscoelastic fluids and simplified spring-dashpot models. Korteweg fluids.

##### Solids

Principle of frame indifference and its consequences, frame indifferent quantities in solid mechanics. Elastic materials in finite elasticity theory, linearized elasticity. Incompressible materials in finite elasticity and in linearized elasticity. Elastic material as a material that does not produce entropy, relation between the stress tensor and free energy. Hyperelastic materials, examples of hyperelastic materials, behaviour with respect to the determinant of deformation gradient. Variational formulation of the static problem for deformation of hyperelastic solids. Viscoelastic solids — Kelvin–Voigt model — and simplified spring-dashpot models.