Numerical methods
in differential equations

Course for master students, Faculty of Electronics
Lectures are held on Fridays, 09:15 - 11:00
Accompanying labs are held also on Fridays.

1. Organization of the course/Syllabus

Attendance at lectures and classes is obligatory (one or two absences without excusing are allowed). Grading policy: your final grade will be formed on the basis of the sum of all points you will get for your tasks on labs according to the rule given in the following table, where the number M is equal to the amount of all possible points you will can get

ndst	dst	dst+	db	db+	bdb
[0 - 50% M)	[50% - 60% M)	[60% - 70% M)	[70% - 80% M)	[80% - 90% M)	[90% - 100% M]

For those of you who will want to improve your final grade there will be organized a test (in writing) of knowledge on theoretical aspects of the numerical methods in differential equations. It will take place on

(to be given)

Topics

1. Numerical solving the initial value problems. 
       Literature: [1], [2]
2. The consistency, stability and convergence of the approximate method.
3. Numerical solving the boundary value problems for ordinary differential equations.
4. Numerical schemes for the first order partial differential equations.
5. The finite-difference schemes for parabolic problems.
6. The stability and convergence properties of the finite-difference schemes for parabolic problems. 
7. The finite difference schemes for the wave equation.
8. The finite difference schemes for the Poisson equation with the Dirichlet type boundary condition.
9. The finite element method for the elliptic problems.

Exercises

I. The problems for self-studying
1. Instability in a computational calculations Example01.m

II. Problem sets for computer exercises on labs

1. Problem set 1  
2. Solution to Problem 1/List 1 in Matlab and Python codes problem01.m,  problem01.py 
3. Problem set 2,   
4. Problem set 3,   
Laboratory on 26.04.2024: Exercise,   

Lectures

1. lect 1, an approximation of the initial value problems
2. lect 2, the error of the explicit Euler method
3. lect 3, a slope field, stiff differential equations, the boundary value problems for second order differentaial equations: the bisection (shooting) method.
3. lect 4, periodic solutions, the finite difference method for the approximate solution to the second order differential problems..
3. lect 5, The Ritz-Galerkin method for boundary value problems
4. lect 6, The hyperbolic (wave) equations
5. lect 7, The parabolic type equations

Literature


            
[1] Z. Fortuna, B. Macukow, J. Wąsowicz, Metody Numeryczne, WNT Warszawa 2003
[2] J. Stoer, R. Bulirsch, Einfuehrung in die Numerische Mathematik I, II, Springer-Verlag 1978
[3] D. Kincaid, W. Cheney, Analiza numeryczna, WNT Warszawa 2005
[4] J. C. Butcher, Numerical Methods for Ordinary Differential Equations, John Wiley & Sons 2003
[5] A. Quarteroni, R. Sacco, F. Saleri, Numerical Mathematics, Springer Berlin Heidelberg 2007
[6] K. W. Morton, D. F. Mayers, Numerical Solution of Partial Differential Equations. An Introduction, Cambridge University Press 2005
[7] Richard L. Burden, J. Douglas Faires, Numerical Analysis
[8 ] R. M. Mattheij, S. W. Rienstra, J.H.M. ten Thije Boonkkamp, Partial Differential Equations. Modeling, Analysis, Computation