Youri Shestopalov, Jan Forsberg, 2000-08-18
Karlstads Universitet rum 3A 313, tfn 7001856
Kursplanering delkurs 3
5p.
Ma-Na 4-9, MAC 402 läsperiod 4 vt 2000.
Läsperiod 000821--000926 (veckorna 34-39)
Tentamen
Mån 000911, kl. 08:15-11.00
Omtentamen
den / 2000 kl
Kursen går på engelska. 27 lektioner.
Kursens mål
- tillägna sig kunskaper om metoder för att formulera och
bearbeta matematiska modeller samt att skriftligt och
muntligt presentera resultaten och motivera de
matematiska metodernas korrekthet.
Kurslitteratur
E. Pärt-Enander et. al, The MATLAB Handbook, New York:
Addison--Wesley, 1996. ISBN 0-201-877570 (MH)
M. Heath, Scientific Computing. An Introductory Survey. New
York: McGraw-Hill, 1997. ISBN 0-07-115336-5 (SC)
Recommended: E. Kreyszig, Advance Engineering Mathematics. 8th Edition. ISBN 0-471-33328-X
(Chapters 17, 18, 19) (AEM).
Kursinnehål
- MATLAB studium
Numeriska metoder:
- metoder för ekvationslösning
- numeriska metoder för integration
- numeriska metoder för ordinära differentialekvationer
- numeriska metoder för partiella differentialekvationer
- numeriska lösning för 1D och 2D värmeledningsekvationer
Inlämningsuppgifter (Normalt i 3-4 personers grupper )
- MATLAB laborationer, bevisuppgifter, modelleringsuppgiften.
Undervisning och examination
Undervisning består av lektioner, övninar, seminarier och
grupparbeten med handledning.
Obligatoriska moment: två laborationer med redovisningar.
Olika
examinationsformer
förekommer
såsom
seminarier,
redovisningar av arbetsuppgifter samt skriftliga och muntliga
prov.
Kursschema och rekommenderade uppgifter
34 Man 000821 08:15-11.00
Lecture 1: Introduction to MATLAB I.
Examples. Getting started. Matrix notations. Variables in
MATLAB. Arithmetic expressions and mathematical functions.
(Sec. 2 HM, pp. 18—28).
34 Tis 000822 12:15-15.00
Lecture 2: Introduction to MATLAB II.
Mathematical functions. Output format. Saving and loading.
Files. M-files. (Sec. 2 HM, pp. 28—45. Sec. 12.3 HM, pp. 206—
213).
34 Ons 000823 13:15-16.00
Lecture 3: Introduction to MATLAB III.
Matrix operations. Relational and logical operators and
functions. Creating new matrices. Some string commands(Sec. 3,
4, HM, pp. 46—84).
34 Tor 000824 08:15-11.00
Lecture 4: Programming in MATLAB.
Conditional statements. Loops. Functions.(Sec.
193—220).
12,
HM,
pp.
34 Fre 000825 08:15-11.00
Lecture 5: Numerical Methods in General. Solution of Nonlinear
Equations.
Floating-point form of numbers. Round-off rules. Errors of
numerical results. Loss of significant digits (AEM8: Sec.
17.1, pp. 831—836, Ex. 1.). Solution of equations by
iterations: fixed-point iteration. Newton’s method. Secant
method (AEM8: Sec. 17.2, pp. 838—846, Ex. 1—8).
Problems: AEM8: Problem set 17.1 (p. 836): 1, 5; Problem set
17.2 (p. 847): 3—5; 11, 15.
35
Man 000828 10:15-13.00 Lecture 6: Numerical Methods in
Linear Algebra.
Elements of linear algebra in MATLAB. (Sec. 7.1, HM, pp. 103—
220). Gauss elimination. LU decomposition.
35 Tis 000829 12:15-15.00
Lecture 7: Numerical Methods in Linear Algebra: Solution by
Iteration.
Gauss—Seidel iteration method. Convergence. Jacobi iteration
(AEM8: Sec. 18.3, pp. 900—904, Ex. 1, 2.). Ill-conditioning.
Matrix norms. Condition number of a matrix (AEM8: Sec. 18.4,
pp. 906—911, Ex. 1—6.).
Problems: AEM8: Problem set 18.3 (p. 905): 1, 3; 9. Problem
set 18.4 (pp. 912—913): 7, 15.
35 Ons 000830 13:15-16.00
Lecture 8: Numerical Integration.
Numerical integration: rectangular rule and trapezoidal rule.
Error bounds and estimate for the trapezoidal rule. Simpson’s
rule of integration. Integration in MATLAB .(Sec. 11.1, HM,
pp. 180—183).
Problems: AEM8: Problem set 17.5 (p. 880): 1—3; 7; 25.
35 Tor 000831 08:15-11.00
Lecture 9: Numerical Methods for First-Order Differential
Equations.
General: initial value problems for ODE. Euler method.
Accuracy and stability. Taylor series methods (SC: Sec. 9, pp.
269—279, 284—285). Improved Euler method (AEM8: Sec. 19.3, pp.
942—951, Ex. 1—4.). Runge—Kutta methods (SC: Sec. 9, pp. 285—
292; AEM8: Sec. 19.1, pp. 942—951, Ex. 1—4.). MATLAB functions
(Sec. 11.2, HM, pp. 184—192).
35 Fre 000901 08:15-11.00
Lecture 10: Modeling by the Runge—Kutta method. Formulation of
the Modeling Task 1.
Formulation of the modeling Task 1 for the Bungy—jump problem.
MATLAB functions and programming (Sec. 11.2, HM, pp. 184—192).
Problems: AEM8: Problem set 19.1 (p. 951): 1; 6; 17.
36 Mån 000904 13:15-16.00
Lecture 11: Multistep Methods and Methods for Systems of
Higher-Order Differential Equations.
Multistep methods (SC: Sec. 9, pp. 287—292; AEM8: Sec. 19.2,
pp. 952—955, Ex. 1.). Euler and Runge—Kutta methods (AEM8:
Sec. 19.3, pp. 942—951, Ex. 1—4.).
36 Tis 000905 12:15-15.00
Lecture 12: Numerical Solution to Boundary Value Problems for
ODE. The formulation. The shooting and finite-difference
methods. A MATLAB-based algorithm for the numerical solution
of a boundary value problem (SC: Sec. 10, pp. 301—311; HM:
Sec. 11, pp. 190—192).
36 Ons 000906 10:15-13.00
Lecture 13: Solving ODEs with MATLAB.
MATLAB software. Examples of solution. Modeling for different
examples(HM: Sec. 11, pp. 184—192).
36 Tor 000907 10:15-13.00
Lecture 14: Separation of Variables in PDE.
Formulation of the problem. Separation of variables and
obtaining the solution for one-dimensional heat equation and
two-dimensional wave equation (AEM8: Sec. 11.5, pp. 600—605,
Ex. 1—4).
Problems: AEM8: Problem set 11.5 (pp. 608): 3, 5.
36 Fre 000908 08:15-11.00
Lecture 15: Numerical Solution of PDE.
Finite difference and finite element methods for the Laplace
equation. Some methods for solving sparse systems (optional).
Report on Task 1 of modeling (SC: Sec. 11, pp. 315—333).
37 Mån 000911 13:15-16.00
Lecture 16: Report on Task 1 for the Bungy—jump Problem.
Intermediate Examination (Solution of Nonlinear Equations; Numerical
Integration; Numerical Methods for Linear Algebra and ODE).
37 Tis 000912 10:15-13.00
Lecture 17:
MATLAB for
Numerical Solution of PDE I.
Formulation of the Modeling Task 2.
MATLAB PDE Toolbox software. Introduction: demonstrations. PDE
algorithms and mesh generation. Exercises (based on the PDE
Toolbox Help). Formulation of the modeling Task 2: 2D timeindependent heat equation in a rectangular domain.
37 Ons 000913 13:15-16.00
Lecture 18: MATLAB for Numerical Solution of PDE. II.
PDE Toolbox interface algorithms and utilities. Exercises
using PDE Toolbox (based on the PDE Toolbox Help).
37 Tor 000914 08:15-11.00
Lecture 19: Graphics in MATLAB.
Two-dimensional
graphics.
Genereating
girds
and
drawing
contour plots. Three-dimensional graphics. Working with figure
files
(HM: Sec. 14, pp. 221—270).
37 Fre 000915 08:15-11.00
Lecture 20:
Numerical Solution to One-Dimensional Heat
Equation.
Finite difference and finite element methods for the numerical
solution to the one-dimensional heat equation. Grids. Errors.
(SC: Sec. 11, pp. 320—322).
38 Mån 000918 13:15-16.00
Lecture 21: Numerical Modeling for One-Dimensional Heat
Equation.
Practical application of finite difference and finite element
methods for the numerical solution to the one-dimensional heat
equation with particular initial data using MATLAB PDE Toolbox
(based on the PDE Toolbox Help).
38 Tis 000919 12:15-15.00
Lecture
22:
Problem
Settings
Equation.
for
Two-Dimensional
Heat
Formulation of the problem. Analysis of solution and reducing
to a standard form. Separation of variables and obtaining the solution in the
form of Fourier series (optional). (SC: Sec. 11, Ch. 11.3, pp. 325—
327). (AEM8: Sec. 19.4, pp. 962—975).
38 Ons 000920 08:15-11.00
Lecture 23:
Numerical Solution to Two-Dimensional Heat
Equation. II.
Formulation of the difference problem. Generating the grid:
rectangular boundary and curved boundary. (AEM8: Sec. 19.4,
pp. 962—975). Application of MATLAB and the MATLAB PDE
Toolbox.
38 Tor 000921 08:15-11.00
Lecture 24:
Numerical Solution to Two-Dimensional Heat
Equation. III. Formulation and solution for different boundary
value conditions. Application of MATLAB and the MATLAB PDE
Toolbox.
38 Fre 000922 08:15-11.00
Lecture 25: Numerical Modeling for Two-Dimensional Heat
Equation. IV.
Task 2 revisited: practical application of finite difference
methods for the numerical solution to the 2D heat equation
with particular initial data using MATLAB PDE Toolbox (based
on the PDE Toolbox Help).
39 Mån 000925 13:15-16.00
Lecture 26: Numerical Modeling for Task
Heat Equation. Numerical solution to the
different sets of the initial data using
according to particular tasks (based on the
2: Two-Dimensional
heat equation with
MATLAB PDE Toolbox
PDE Toolbox Help).
39 Tis 000926 12:15-15.00
Lecture 27: Final Lecture. Control of the Test amd Modeling
Results. Report on Task 2 for the Two-Dimensional Heat
Equation.