Loredana CASO | PARTIAL DIFFERENTIAL EQUATIONS
Loredana CASO PARTIAL DIFFERENTIAL EQUATIONS
cod. 0522200007
PARTIAL DIFFERENTIAL EQUATIONS
| 0522200007 | |
| DEPARTMENT OF MATHEMATICS | |
| EQF7 | |
| MATHEMATICS | |
| 2024/2025 |
| YEAR OF DIDACTIC SYSTEM 2018 | |
| AUTUMN SEMESTER |
| SSD | CFU | HOURS | ACTIVITY | |
|---|---|---|---|---|
| MAT/05 | 6 | 48 | LESSONS |
| Exam | Date | Session | |
|---|---|---|---|
| EQUAZIONI DIFF. ALLE DERIVATE PARZIALI | 10/01/2025 - 09:00 | SESSIONE ORDINARIA | |
| EQUAZIONI DIFF. ALLE DERIVATE PARZIALI | 10/01/2025 - 09:00 | SESSIONE DI RECUPERO | |
| EQUAZIONI DIFF. ALLE DERIVATE PARZIALI | 31/01/2025 - 09:00 | SESSIONE ORDINARIA | |
| EQUAZIONI DIFF. ALLE DERIVATE PARZIALI | 31/01/2025 - 09:00 | SESSIONE DI RECUPERO | |
| EQUAZIONI DIFF. ALLE DERIVATE PARZIALI | 25/02/2025 - 09:00 | SESSIONE ORDINARIA | |
| EQUAZIONI DIFF. ALLE DERIVATE PARZIALI | 25/02/2025 - 09:00 | SESSIONE DI RECUPERO |
| Objectives | |
|---|---|
| THIS COURSE IS INTENDED TO PRESENT AN INTRODUCTION TO THE FUNDAMENTAL PARTIAL DIFFERENTIAL EQUATION (PDE). IN PARTICULAR, THE MOST IMPORTANT EQUATIONS AND METHODS FOR APPLICATIONS WILL BE ANALYZED, INCLUDING STATIONARY PHENOMENA, PROPAGATION PHENOMENA BY TRANSPORT, BY DIFFUSION AND WAVE MOTION. KNOWLEDGE AND UNDERSTANDING: THE COURSE IS DEDICATED TO THE DEVELOPMENT OF THE THEORY OF CHARACTERISTICS AND TO THE KNOWLEDGE OF THE MOST IMPORTANT INITIAL /BOUNDARY VALUE PROBLEMS. ON THESE ISSUES THE MAIN RESULTS OF THE CLASSICAL THEORY AS WELL AS SOME METHODS OF RESOLUTION ARE DISCUSSED. APPLYING KNOWLEDGE AND UNDERSTANDING: THE STUDENT MUST LEARN TO RECOGNIZE AND CLASSIFY PDE, AND USE THE KNOWLEDGE ACQUIRED TO SOLVE SOME INITIAL/BOUNDARY VALUE PROBLEMS. MAKING JUDGEMENTS: THE STUDENT MUST KNOW THE RELATIONSHIP BETWEEN PDE AND APPLICATION PROBLEMS. HE WILL ALSO HAVE TO APPLY THE ANALYTICAL TECHNIQUES ACQUIRED FOR THE MODELING AND RESOLUTION OF SOME NATURAL PHENOMENA. COMMUNICATION SKILLS: THE STUDENTS MUST BE ABLE TO CLEARLY EXPLAIN THE DEFINITIONS AND THE STATEMENTS OF THEOREMS, DISTINGUISHING WITH CERTAINTY HYPOTHESIS AND THESIS AND HE WILL ALSO HAVE TO BE ABLE TO CONSISTENTLY EXPLAIN THE LOGICAL STEPS OF THE DEMONSTRATIONS. LEARNING SKILLS: THE STUDENT WILL HAVE TO CRITICALLY USE THE TOOLS AND CONCEPTS STUDIED FOR THE INTERPRETATION AND RESOLUTION OF SOME PHENOMENA MODELED BY A PDE, CORRECTLY INTERPRETING THE RESULTS OBTAINED. |
| Prerequisites | |
|---|---|
| BASIC TOPICS OF THE THEORY OF ORDINARY DIFFERENTIAL EQUATIONS AND THE THEORY OF MEASURE AND INTEGRATION FOR FUNCTIONS OF SEVERAL VARIABLES. |
| Contents | |
|---|---|
| PARTIAL DIFFERENTIAL EQUATIONS AND THEIR CLASSIFICATION. SOME PDE ELEMENTARY SOLVABLE. (2 HOURS OF THEORETICAL LESSONS + 3 HOURS OF EXERCISES) INTRODUCTION TO THE METHOD OF CHARACTERISTICS FOR QUASILINEAR FIRST ORDER PDES. SOLUTION OF SOME PDES WITH THE CHARACTERISTICS METHOD. (4 HOURS OF THEORETICAL LESSONS + 3 HOURS OF EXERCISES) MATHEMATICAL MODELS AND MASS BALANCE LAW. SCALAR CONSERVATION LAW. LINEAR TRANSPORT EQUATION: DISTRIBUTED SOURCE; EXTINCTION AND LOCALIZED SOURCE. TRAFFIC DYNAMICS MODEL: THE GREEN LIGHT PROBLEM AND RAREFACTION WAVES, TRAFFIC JAM AHEAD AND SHOCK WAVES. (6 HOURS OF THEORETICAL LESSONS + 4 HOURS OF EXERCISES) SECOND ORDER DIFFERENTIAL EQUATIONS. LAPLACE'S EQUATION: HARMONIC FUNCTIONS AND PROPERTIES, FUNDAMENTAL SOLUTION, MAXIMUM PRINCIPLE, REGULARITY. (8 HOURS OF THEORETICAL LESSONS) POISSON'S EQUATION: NEWTONIAN POTENTIAL, GREEN FUNCTION AND REPRESENTATION FORMULA. (6 HOURS OF THEORETICAL LESSONS) HEAT EQUATION: FUNDAMENTAL SOLUTION, MAXIMUM PRINCIPLE, REGULARITY, DUHAMEL’S METHOD. (6 HOURS OF THEORETICAL LESSONS) WAVE EQUATION: SPHERICAL MEANS AND EULER – POISSON – DARBOUX EQUATION, SOLUTION OF THE CAUCHY PROBLEM IN ODD DIMENSIONS AND IN EVEN DIMENSIONS WITH THE DESCENT METHOD, DUHAMEL'S PRINCIPLE. (6 HOURS OF THEORETICAL LESSONS) |
| Teaching Methods | |
|---|---|
| HE COURSE INCLUDES 48 HOURS OF CLASSROOM TEACHING DIVIDED IN 38 HOURS OF THEORETICAL LESSONS AND 10 HOURS OF PRACTICAL LESSON. THERE IS NO OBLIGATION TO ATTEND IT EVEN IF ATTENDANCE IS STRONGLY RECOMMENDED FOR THE ACHIEVEMENT OF THE EDUCATIONAL OBJECTIVE. |
| Verification of learning | |
|---|---|
| THE VERIFICATION OF LEARNING WILL BE MADE WITH FINAL EXAMINATION AIMED AT ASSESSING THE COMPLETE KNOWLEDGE AND UNDERSTANDING ABILITY OF THE CONCEPTS PRESENTED IN LESSONS. IN PARTICULAR, THE EXAM CONSISTS OF A NON-SELECTIVE WRITTEN TEST IN WHICH THE STUDENT IS ASKED TO SOLVE SIMPLE CAUCHY PROBLEMS WITH THE CHARACTERISTICS METHOD AND OF AN ORAL TEST IN WHICH THE STUDENT WILL HAVE TO DEMONSTRATE THAT HE KNOWS THE DEFINITIONS, STATEMENTS OF THEOREMS AND PROOFS PRESENTED DURING THE COURSE, CONCERNING THE CONSERVATION LAWS, THE CHARACTERISTICS METHOD AND ITS APPLICATIONS, THE LAPLACE EQUATION, THE DIFFUSION EQUATION, THE WAVE EQUATION. |
| Texts | |
|---|---|
| 1. SANDRO SALSA, FEDERICO M.G. VEGNI, ANNA ZARETTI, PAOLO ZUNINO, A PRIMER ON PDES - MODELS, METHODS, SIMULATIONS SPRINGER VERLAG, 2013 2. LAWRENCE C. EVANS, PARTIAL DIFFERENTIAL EQUATIONS, AMERICAN MATHEMATICAL SOCIETY, 2002. 3. FRITZ JOHN, PARTIAL DIFFERENTIAL EQUATIONS, SPRINGER VERLAG, 1991. |
| More Information | |
|---|---|
| LORCASO@UNISA.IT |
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