===== Partial Differential Equations with Applications in Physics and Industry =====

During the course we will learn about many applications of PDEs and methods of solving them. 

==== Grades ====
During the semester you will collect points from various activities. 

=== Tutorial (30 points )===

  * **Solving problems (10 points)**. During each tutorial class you will present solutions of problems from the published sets (see below). Probably not every problem from each set will be presented due to time limitations (but we shall see). 

If it will physically be impossible to score 10 points from the problem solving part due to a large number of volunteers (approximately an uniform distribution of points), then we will rescale the total number of points so that everyone working systematically will be awarded. 

/*
Your points collected during the tutorial $x$ will be rescaled according to the formula: 

$\displaystyle{\min\left(15, \lceil 1.5 x\rceil \right)}$,

where $\lceil \cdot \rceil$ is the ceiling function. 

*/

  * **Test (20 points)**. There will be **one** test during the semester approximately in the middle.

=== Project (15 points) === 

This will concern application and implementation of a chosen PDE problem. The details will be given in March. Projects are **necessary** to pass the course. 

{{ ::projects.pdf |Project rules}}

Choosing a topic: **2 April**\\
Project deadline: **9 June**\\
Project upload: [[https://eportal.pwr.edu.pl/|ePortal]]

/*
<box 90% round green|Projects>

{{ ::projects.pdf |Project instructions}} In particular, we use **Jupyter** and choose a topic and a two-person group until **1 April**. The final deadline is **18 June**. 

Please upload your projects to the [[https://eportal.pwr.edu.pl|ePortal]] under a suitable folder in the PDE course (only **one** person from the team does that). In case of many files please compress it into a ZIP format. 

Note that we **we do not have** a defense or presentation of your projects. 

</box>

*/

**{{ :lecturenum.pdf |Notes concerning numerical methods.}}**

=== Exam (25 points) ===
There will be two possible examinations. **Everyone** has to take the primary exam. The secondary is only for those who will fail to score the minimal passing number of points throughout the semester. The successful secondary exam grants the minimal grade, that is $3.0$. 
  - **Primary**. 27.06 from 09:00 until 11:00 in 1.27@C13.
  - **Secondary**. 04.07 from 09:00 until 11:00 in 1.27@C13.

Details will be given in the future.

=== Marks ===
In total you can obtain $S \leq 70 = 30 + 15 + 25$ points. The final mark will be derived from the table below. I reserve the right to change these rules but only for your convenience.

In order to pass it is obligatory to **return the project**.

^   $S$  ^   Mark   ^
|   $30\leq S < 39$ |  $3.0$   |
|   $39\leq S < 45$   |   $3.5$   |
|   $45\leq S < 52$   |   $4.0$   |
|   $52\leq S < 60$   |   $4.5$   |
|   $60\leq S \leq 70$   |   $5.0$   |  

==== Syllabus ====
Here you can find the contents of the course that has been covered so far. 

/*

  - **03.03**: Introduction to PDEs. Conservation laws.
  - **10.03**: First order equations: further examples, method of characteristics.
  - **17.03**: First order equations: method of characteristics cont'd.
  - **24.03**: First order equations: shock waves, Rankine-Hugoniot's condition, voids, fans, and rarefaction waves.
  - **31.03**: Fourier's method.
  - **14.04**: Heat equation: derivation, formulation, and solution of the initial-value problem.
  - **28.04**: Heat equation: derivation, formulation, and solution of the initial-value problem.
  - **12.05**: Heat equation: nonhomogeneous source and boundary condition, Green's function, uniqueness.
  - **19.05**: Heat equation: problems on the real line.
  - **26.05**: Heat equation: further examples. Laplace's and Poisson's equations: examples.
  - **02.06**: Laplace's and Poisson's equations: Green's functions and fundamental solution.
  - **16.06**: Wave equation.
  - **23.06**: Wave equation.
  - **27.06**: TBA

*/
/*


  - **24.03**: 
  - **31.03**: 
  - **07.03**: Heat equation: derivation, formulation, and solution of the initial-value problem.
  - **13.04**: Heat equation: nonhomogeneous source and boundary condition, Green's function, uniqueness.
  - **21.04**: Heat equation: problems on the real line.
  - **28.04**: Heat equation: half-line and further examples.
  - **05.05**: Laplace's and Poisson's equations: introductory examples.
  - **12.05**: Laplace's and Poisson's equations: problems on a circle, mean-value property, and Bessel functions.
  - **19.05**: Laplace's and Poisson's equations: Green's functions and fundamental solution.
  - **26.05**: Wave equation: derivation.
  - **02.06**: Wave equation: examples of d'Alembert's solution.
  - **09.06**: Wave equation: plane and spherical waves. 

*/

/* 
  - **22.04**: Heat equation: 
  - **30.04**: Heat equation: uniqueness and problems on the real line.
  - **06.05**: Heat equation: half-line and further examples.
  - **13.05**: Laplace's and Poisson's equations: derivations and problems on the circle. //Optional (but highly advisable): read about other problems with separated variables //.
  - **20.05**: Laplace's and Poisson's equations: Green's functions and fundamental solution.
  - **27.05**: Laplace's and Poisson's equations: Method of reflections for Green's functions. Examples of wave equation.
  - **02.06**: Wave equation: separation of variables and d'Alembert's solution.
  - **10.06**: Wave equation: examples of d'Alembert's solution and plane waves.
  - **17.06**: Wave equation: spherical waves and solution of an initial value problem in $\mathbb{R}^3$. 

*/
==== Lecture Notes ====
{{ ::pdelecture.pdf |Lecture Notes}} contain more material that we will be able to cover. I will indicate what is obligatory, however, going through all the book is certainly beneficial.  

==== Problem sets ====
There are many more problems on the list than you will be able to solve during your classes. However, it is **strongly recommended** to practice as much exercises as possible on your own. Moreover, during the tutorial you will be required to solve also the modelling or more difficult problems (and not only the standard calculation practice).  

{{ :problem_set_1_-_first_order_equations_and_shock_waves.pdf |Problem Set 1 - First order equations and shock waves}}
{{ ::problem_set_2_-_fourier_s_method.pdf |Problem Set 2 - Fourier's method}}
{{ ::problem_set_3_-_heat_equation.pdf |Problem Set 3 - Heat equation}}
{{ :problem_set_4_-_laplace_and_poisson_equations.pdf |Problem Set 4 - Laplace's and Poisson's equation}}