# Applied Mathematics

Course ID
6ΕΠ03
Επίπεδο
Είδος
Optional (compulsory)
Εξάμηνο
6
Περίοδος
Spring Semeter
ECTS
5
Ώρες Θεωρίας
4
Ώρες Εργαστηρίου
-

#### Description

Introduction to Differential Equations: Seperable differential equations. First Order Linear Differential Equations. Bernoulli’s equation. Exact and non-exact equations. Integrating factor technique. Second Order Linear differential equations with constant coefficients Method of variation of parameters. Method of determined coefficients. Euler equations. Linear Systems of differential equations. Homogeneous / Nonhomogeneous linear systems. Euler’s method for systems. Applications of Differential Equations in simple models of biological systems. Improper integral. Transformation Laplace and Fourier. • Introduction to complex analysis: Complex numbers and complex functions. Analytic functions. Cauchy-Riemann equations. • Introduction to stochastic analysis: In this part of the course, students are introduced to multivariate data analysis and to the use of multivariate normal sampling theory. They learn how to treat multivariate data and how to estimate the mean vector, the variance-covariance matrix and the correlation matrix, as well as, how to create linear transformations of random variables and graphical representations of multidimensional data. Moreover, in this course students are taught to perform one- and two-sample tests in multivariate data sets, profile analysis, partial and multiple correlation and multivariate ANOVA, discriminant analysis, principal components, factor analysis, and cluster analysis. In this part of the course, students are introduced to the basics of stochastic processes. Specifically, after learning basic terms, students are introduced to simple Markov chains (discrete time); recurrence, transience, stationary distributions; Poisson processes (continuous time) and their simple simulations using examples from biology and genetics.

#### Course objectives

Primary goal of this course is to introduce students to mathematical modelling of the problems and familiarise them with data analysis methods, in order to develop appropriate methodologies to support Department’s theses.

The course aims:

• at students’ comprehension and being familiarised with the description process of a problem (from any research area) by a differential equation or a system of differential equations or even the collection, organization, study and synthesis analysis of biological data,
• at the acquisition of knowledge and skills both for solving differential equations and for developing multivariable analysis methods.

#### Textbooks/Bibliography

• ΔΙΑΦΟΡΙΚΕΣ ΕΞΙΣΩΣΕΙΣ ΚΑΙ ΕΦΑΡΜΟΓΕΣ ΑΥΤΩΝ, ΕΥΓΕΝΙΑ Ν. ΠΕΤΡΟΠΟΥΛΟΥ, “ΓΚΟΤΣΗΣ ΚΩΝ/ΝΟΣ & ΣΙΑ Ε.Ε.”, 1η/2017, ΠΑΤΡΑ, 68383986
• Αριθμητικές Μέθοδοι και Εφαρμογές για Μηχανικούς, Σαρρής Ι.- Καρακασίδης Θ., ΕΚΔΟΣΕΙΣ Α. ΤΖΙΟΛΑ & ΥΙΟΙ Α.Ε., 4η/2017, ΘΕΣ/ΝΙΚΗ, 68373915
• Αριθμητική Ανάλυση: Συνήθεις Διαφορικές Εξισώσεις, ΜΙΧΑΗΛ Ν. ΒΡΑΧΑΤΗΣ, ΕΚΔΟΣΕΙΣ ΚΛΕΙΔΑΡΙΘΜΟΣ ΕΠΕ, 1η/2012, ΑΘΗΝΑ, 12867996
• Συνήθεις διαφορικές εξισώσεις, Αλικάκος Νικόλαος, Καλογερόπουλος Γρηγόρης, ΣΥΓΧΡΟΝΗ ΕΚΔΟΤΙΚΗ ΕΠΕ, 1η/2003, ΑΘΗΝΑ, 6848
• Διαφορικές Εξισώσεις και Εφαρμογές, ΓΕΩΡΓΙΟΣ Β. ΒΟΥΓΙΑΤΖΗΣ, ΓΕΩΡΓΙΟΣ Δ. ΜΠΟΖΗΣ, ΔΗΜΗΤΡΙΟΣ Β. ΠΑΠΑΔΟΠΟΥΛΟΣ, “ΕΚΔΟΣΕΙΣ ΚΛΕΙΔΑΡΙΘΜΟΣ ΕΠΕ”, 1η/2012, ΑΘΗΝΑ , 22768244
• ΑΝΑΛΥΣΗ ΜΗΤΡΩΩΝ ΓΙΑ ΕΠΙΣΤΗΜΟΝΕΣ ΚΑΙ ΜΗΧΑΝΙΚΟΥΣ, ALAN J. LAUB, ΕΚΔΟΣΕΙΣ ΚΛΕΙΔΑΡΙΘΜΟΣ ΕΠΕ , 1η/2010, ΑΘΗΝΑ, 21489
• ΔΙΑΦΟΡΙΚΕΣ ΕΞΙΣΩΣΕΙΣ, ΒΑΣΙΛΕΙΟΣ ΡΟΘΟΣ, ΧΡΥΣΟΒΑΛΑΝΤΗΣ ΣΦΥΡΑΚΗΣ, “Ελληνικά Ακαδημαϊκά Ηλεκτρονικά Συγγράμματα και Βοηθήματα – Αποθετήριο “”Κάλλιπος”””,  1η/2016, 320291
• ΜΑΘΗΜΑΤΙΚΗ ΜΟΝΤΕΛΟΠΟΙΗΣΗ, ΣΤΑΥΡΟΣ ΚΟΜΗΝΕΑΣ, “Ελληνικά Ακαδημαϊκά Ηλεκτρονικά Συγγράμματα και Βοηθήματα – Αποθετήριο “”Κάλλιπος”””, 1η/2016, 59303654
• ΜΑΘΗΜΑΤΙΚΗ ΑΝΑΛΥΣΗ, ΜΑΡΙΑ ΑΔΑΜ , “Ελληνικά Ακαδημαϊκά Ηλεκτρονικά Συγγράμματα και Βοηθήματα – Αποθετήριο “”Κάλλιπος”””, 1η/2016, 59303650

#### Assessment method

Optional semester project and mandatory written exams at the end of the semester.