# COURSE UNIT TITLE

: METHODS OF APPLIED MATHEMATICS

#### Description of Individual Course Units

Course Unit Code Course Unit Title Type Of Course D U L ECTS
MAT 5151 METHODS OF APPLIED MATHEMATICS ELECTIVE 3 0 0 9

#### Offered By

Graduate School of Natural and Applied Sciences

#### Level of Course Unit

Second Cycle Programmes (Master's Degree)

#### Course Coordinator

PROFESSOR VALERIY YAKHNO

#### Offered to

COMPUTER ENGINEERING
Computer Engineering Non-Thesis
COASTAL ENGINEERING
Nanoscience and Nanoengineering
Nanoscience and Nanoengineering
PHYSICAL OCEANOGRAPHY
GEOGRAPHICAL INFORMATION SYSTEMS - NON THESIS (EVENING PROGRAM)
NATURAL BUILDING STONES AND GEMSTONES
ENVIRONMENTAL EARTH SCIENCES
GEOGRAPHICAL INFORMATION SYSTEMS
Environmental Engineering
NAVAL ARCHITECTURE
ENVIRONMENTAL EARTH SCIENCES-NON THESIS
Geothermal Energy
M.Sc. Metallurgical and Material Engineering
Computer Engineering Non-Thesis
Mathematics
EARTHQUAKE MANAGEMENT - NON THESIS
Mathematics
Geophysical Engineering
ENVIRONMENTAL ENGINEERING
Geographical Information Systems (Non-Thesis)
COASTAL ENGINEERING
NAVAL ARCHITECTURE
EARTHQUAKE MANAGEMENT
M.Sc. Geothermal Energy (Non-Thesis-Evening)
Metallurgical and Material Engineering
GEOGRAPHIC INFORMATION SYSTEMS
Computer Engineering
GEOPHYSICAL ENGINEERING

#### Course Objective

The course brings an overview of the modern mathematical models of real processes and phenomena. Main problems are stated in terms of these models. General methods of solving problems are given. Modern study of wave phenomena in acoustic, elastic and electromagnetic media and materials is discussed.

#### Learning Outcomes of the Course Unit

 1 Ability to understand the modern mathematical models of real processes and phenomena 2 Ability to express the fundamental concepts of mathematical modeling by partial differential equations. 3 Ability to analyze mathematical models for the acoustic, electromagnetic, elastic waves in different media and materials. 4 Ability to correctly state the problems for the wave, heat-diffusion, Laplace, Poisson, Helmholtz, acoustic, and Maxwell's equations. 5 Ability to apply different methods for solving problems for the partial differential equations. 6 Ability to simulate the acoustic, elastic and electromagnetic waves in different materials.

Face -to- Face

None

None

#### Course Contents

 Week Subject Description 1 Introduction: Aspects of Mathematics: Theory, Methods and Applications. Real processes and phenomena: sound, diffusion, heating, electromagnetism (electricity), seismic waves (earthquake). Fields in terms of scalar and vector functions of one and several variables. 2 Vector analysis technique. Differential operators: grad, div, curl. Properties of these differential operators. Examples 3 Mathematical models of sound propagation, heating and diffusion processes. Acoustic system in homogeneous and inhomogeneous media. Reduction of the acoustic system to an acoustic equation. Diffusion equation. Heat equation. Initial value and initial boundary value problems. 4 Mathematical models of electrodynamics. The time-dependent Maxwells equations in homogeneous and inhomogeneous isotropic media. Maxwells equations in anisotropic dielectric and crystals. The reduction of Maxwells equation in homogeneous media to telegraph equations. 5 Mathematical models of elastodynamics. Dynamical system of anisotropic elasticity in homogeneous and inhomogeneous materials and media. Lame equation system of isotropic elasticity. Scalar and vector potentials. Reduction of Lames system to wave equations for P-wave and S-wave. 6 Method of characteristics. Method of characteristics for a scalar partial differential equation. Method of characteristics for solving initial value problem for acoustic system. 7 Midterm 8 Method of characteristics for solving initial value problem of Maxwells equations. Reduction Maxwells equation with initial data to symmetric hyperbolic system. Initial value problem solving by method of characteristics. 9 Ordinary differential equations of the first and second order with constant coefficients. Solving homogeneous first and second order ordinary differential equations. Variation of parameter method for solving inhomogeneous first and second order ordinary differential equations. Examples. 10 Eigenvalue-Eigenfunction problems for second order ordinary differential equations. Eigenvalues and corresponding to them eigenfunctions of boundary value problems with different boundary conditions. 11 Fourier series expansion method for solving initial boundary value problems. Solving the initial boundary value problems for the inhomogeneous wave equation. Solving the initial boundary value problems for the inhomogeneous heat-diffusion equation. 12 Greens functions metods. The Dirac delta function and its properties. Greens function of the initial value problem for the ordinary differential equation. 13 Construction of Greens function of initial value problem for the heat-diffusion equations. Fourier transform and its properties. Explicit formula of Greens functions of initial value problem for heat-diffusion equation. 14 Construction of Greens function of initial value problem for wave equations. Explicit formula for Greens functions of initial value problem for wave equation.

#### Recomended or Required Reading

1.Alan Jeffrey, Advanced Engineering Mathematics, Harcourt Academic Press, Massachusetts, 2002
2. Erwin Kreyszing, Advanced Engineering Mathematics, Wiley and Sons, 2006
3. Tayler A.B, Mathematical Models in Applied Mechanics, Oxford Applied Mathematics an Computing , Science Series, Clarendon Press, Oxford, 1986
4.Tikhonov, A.N., Samarskii, A.A., Equations of Mathematical Physics, Pergamon Press, 1996.

Lectures
Homeworks
Examinations
Assignments

#### Assessment Methods

 SORTING NUMBER SHORT CODE LONG CODE FORMULA 1 ASG ASSIGNMENT 2 MTE MIDTERM EXAM 3 FIN FINAL EXAM 4 FCG FINAL COURSE GRADE ASG * 0.30 + MTE * 0.30 + FIN * 0.40 5 RST RESIT 6 FCGR FINAL COURSE GRADE (RESIT) ASG * 0.30 + MTE * 0.30 + RST * 0.40

None

Examinations

English

To be announced.

#### Contact Details for the Lecturer(s)

valery.yakhno@deu.edu.tr

#### Office Hours

Will be determined in the beginning of the term

None

#### Workload Calculation

 Activities Number Time (hours) Total Work Load (hours) Lectures 13 3 39 Preparation for midterm exam 1 15 15 Preparation for final exam 1 25 25 Preparing assignments 10 5 50 Preparations before/after weekly lectures 13 7 91 Final 1 3 3 Midterm 1 3 3 TOTAL WORKLOAD (hours) 226

#### Contribution of Learning Outcomes to Programme Outcomes

PO/LOPO.1PO.2PO.3PO.4PO.5PO.6PO.7PO.8PO.9PO.10PO.11PO.12
LO.1
LO.2
LO.3432
LO.433
LO.52
LO.64