Many ecosystems and physical systems, such as the climate, can display strongly nonlinear behaviour, where the properties of the system suddenly change with changes in environmental conditions. A well-known and urgent example of this is the discussion about ‘tipping’ of the climate, thought to set off a dramatic change in the World’s climate if global temperatures start to exceed some threshold. Mathematically, ‘tipping’ is an example of a so-called bifurcation of the underlying system of equations. We will focus on the bifurcations in systems of ordinary differential equations (ODEs) and partial differential equations (PDEs) representing real-world systems, which can often only be solved numerically. Hence, this course focusses on numerical bifurcation theory: the study of understanding parameter sensitivity and bifurcations in numerical solutions. The course covers mathematical theory and papers applying the theory to real-life applications including climate science and ecosystems. We build on theory of ODEs, numerical methods and mathematical modelling. Basic programming experience is required for the practical assignments. After following this course the student is able to

Describe various non-linear phenomena appearing in systems of ordinary and partial differential equations. These include tipping, appearance of periodic solutions, pattern formation, flickering and critical slowing down.

Apply numerical continuation, stability analysis and branch switching around bifurcations for common elementary bifurcations (transcritical, pitchfork, limit point, Hopf) in the context of a simple model with physical applications.

Explain practical problems with respect to computation time, computer memory and numerical accuracy in the above techniques and various solutions to these problems.

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Partial differential equations (PDEs) are the mathematical representation of many phenomena in various branches of applied physics. Therefore it is important to be able to solve this type of equations and interpret the solution in terms of the physics they describe. This course provides an introduction into several analytical solution and analysis techniques for PDEs. These analytical solution techniques are often especially suitable for analysing the general physical behaviour of solutions to PDEs and are therefore indispensable tools in addition to numerical techniques (taught in the course AP3001-FE). Hence, the way these analytical techniques may be used to gain insight into the physical interpretation of solutions to PDEs are an important part of this course. The solution and analysis techniques that are discussed include:

Separation of variables

Method of eigenfunction expansion

Homogenisation for nonhomogeneous PDEs

Fourier transform method

Laplace transform method

Method of characteristics

Green's functions

Our main examples of PDEs to apply these techniques are the linear wave and heat equations in 1, 2, and 3 dimensions. Additionally we will discuss linear and nonlinear transport equations.

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T Differential equations occur in all branches of science and engineering as mathematical models for reality. Answering questions that may pop up in any of these branches often comes down to analysing and solving such differential equations. The aim of this course is to introduce you to a variety of techniques to solve differential equations of several commons types. In addition, we will discuss techniques to extract information from differential equations without solving them.

For a given (system of) differential equations the student can tell whether it is linear / non-linear, homogeneous / non-homogeneous, separable, exact or partial.

The student can make statements about existence, uniqueness and analyticity of solutions for boundary value problems and initial value problems without explicitly solving.

the student can make statements about qualitative properties of solutions such as limiting behavior, behavior near equilibrium points and radius of convergence of power series, without explicitly solving.

The student can apply various methods for constructing solutions to (a system of) DEs: Separation, variation of constants, reduction of order, using potentials, Laplace transform, series solutions, Fourier series.

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This course is about ordinary differential equations. For first order equations we will treat linear, separable, and exact equations. We will prove existence and uniqueness of solutions, using Picard iterations. For second order equations we will treat linear equations, power series solutions, and the method of Frobenius. The Laplace transform will also be treated, as will systems of linear differential equations. Stability and phase plane analysis will play an important role. After finalizing the course, you should be able to use the theory of ordinary differential equations (ODEs) to study (systems of) differential equations. In particular you should be able to:

Solve linear (systems of) ODEs.

Solve separable ODEs and exact ODEs.

Determine the stability of solutions of ODEs.

Prove or disprove local existence and uniqueness of solutions.

Solve second order ODEs using power series and Frobenius theory.

Solve ODEs using Laplace transforms.

Determine features (like periodic behaviour, asymptotic behaviour, boundedness) of (solutions of) nonlinear (systems of) ODEs and sketch their phase planes.

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The course focusses on modelling a practical mathematical physics problem.

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See more on the TU Delft course guide