714-0321/01 – Partial Differential Equations (PDR)

Gurantor departmentDepartment of Mathematics and Descriptive GeometryCredits4
Subject guarantordoc. RNDr. Jarmila Doležalová, CSc.Subject version guarantordoc. RNDr. Jarmila Doležalová, CSc.
Study levelundergraduate or graduateRequirementOptional
Year1Semestersummer
Study languageCzech
Year of introduction1995/1996Year of cancellation
Intended for the facultiesFSIntended for study typesFollow-up Master
Instruction secured by
LoginNameTuitorTeacher giving lectures
DOL30 doc. RNDr. Jarmila Doležalová, CSc.
Extent of instruction for forms of study
Form of studyWay of compl.Extent
Full-time Credit and Examination 2+1
Combined Credit and Examination 12+2

Subject aims expressed by acquired skills and competences

Mathematics is essential part of education on technical universities. It should be considered rather the method in the study of technical courses than a goal. Thus the goal of mathematics is train logical reasoning than mere list of mathematical notions, algorithms and methods. Students should learn how to analyze problems, distinguish between important and unimportant, suggest a method of solution, verify each step of a method, generalize achieved results, analyze correctness of achieved results with respect to given conditions, apply these methods while solving technical problems, understand that mathematical methods and theoretical advancements outreach the field mathematics.

Teaching methods

Lectures
Individual consultations
Tutorials
Project work

Summary

Studenti se seznámí s Fourierovými řadami a s klasickými metodami řešení parciálních diferenciálních rovnic. V první kapitole jsou probírány Fourierovy řady v rozsahu potřebném pro další výuku. Ve druhé kapitole se studenti velmi stručně seznámí s parciálními diferenciálními rovnicemi prvního řádu. Teorie parciálních diferenciálních rovnic druhého řádu je zaměřena na Fourierovu metodu separace proměnných a metodu charakteristik. Jádro předmětu je v řešení lineárních parciálních diferenciálních rovnic druhého řádu (Laplaceova rovnice, vlnová rovnice, rovnice vedení tepla).

Compulsory literature:

James, G.: Advanced Modern Engineering Mathematics. Addison-Wesley, 1993

Recommended literature:

James, G.: Advanced Modern Engineering Mathematics. Addison-Wesley, 1993

Way of continuous check of knowledge in the course of semester

zápočet Účast ve cvičení je povinná, 20 % neúčasti lze omluvit - hodnocení 5 bodů. Odevzdání programů zadaných vedoucím cvičení v předepsané úpravě: 1. Fourierovy řady - hodnocení 0-15 bodů 2. PDR - hodnocení 0-20 bodů Student, který získá zápočet, bude hodnocen 5 – 40 body. zkouška Kombinovanou zkoušku tvoří praktická část (60 minut, příklady) a teoretická část (20 minut, teoretické otázky). Praktická část je hodnocena 0 - 50 body, teoretická část 0 - 10 body. Aby student u zkoušky uspěl musí získat v praktické části nejméně 25 bodů. klasifikace získané body známka 86 - 100 výborně 66 - 85 velmi dobře 51 - 65 dobře 0 - 50 nevyhověl Otázky k teoretické části zkoušky: 1. Fourierovy řady v reálném oboru - ortogonální systém funkcí, Fourierovy koeficienty 2. Řada sinová a kosinová, konvergence Fourierových řad 3. Parciální diferenciální rovnice - definice, řešení obecné, počátečního a okrajového problému 4. Některé typy PDR I. řádu - rovnice obsahující derivace pouze podle 1 proměnné, 5. PDR lineární a kvazilineární 6. PDR II. řádu - rovnice obsahující derivace pouze podle 1 proměnné, 7. Rovnice, u nichž lze snížit řád 8. Fourierova metoda separace proměnných 9. Lineární PDR II. řádu - klasifikace, charakteristiky 10. Kanonické tvary lineárních PDR II. řádu 11. Laplaceova rovnice - princip maxima a minima, Dirichletova úloha, vlastní čísla a funkce 12. Vlnová rovnice - počáteční problém kmitů struny, fázový tvar řešení 13. Okrajový problém kmitů struny 14. Rovnice vedení tepla - princip maxima a minima, počáteční a okrajová úloha

E-learning

Další požadavky na studenta

Prerequisities

Subject codeAbbreviationTitleRequirement
714-0369 MIV Mathematics IV Compulsory

Co-requisities

Subject has no co-requisities.

Subject syllabus:

Syllabus of lecture 1. Fourier series: periodic functions, Fourier coefficients, functions of the period 2, function of the period T, even and odd functions 2. Even and odd functions, half - range cosine and sine series, convergence of Fourier series 3. Solution of 2nd order linear differential equations with constant coefficients by Fourier series 4. Partial differential equations: general discussion 5. Methods of solutions of partial differential equations of the first order 6. Methods of solutions of partial differential equations of the second order 7. Fourier´s method of separation 8. 2nd order partial linear differential equations 9. Canonical form of 2nd order partial linear differential equations 10. Laplace equation: separated solutions, boundary conditions 11. Solution of the one-dimensional wave equation: d’Alembert solution, method of the separation of variables, 12. Solution of a boundary problem 13. Solution of the heat-conduction: one dimensional heat conduction equation, separation method. 14. Reserve

Conditions for subject completion

Full-time form (validity from: 2015/2016 Summer semester)
Task nameType of taskMax. number of points
(act. for subtasks)
Min. number of points
Credit and Examination Credit and Examination 100 (100) 51
        Credit Credit 40  20
        Examination Examination 60  31
Mandatory attendence parzicipation:

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Occurrence in special blocks

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