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Description of Individual Course Units| Course Unit Code | Course Unit Title | Type of Course Unit | Year of Study | Semester | Number of ECTS Credits | | Fİ116 | Numerical Methods in Physics | Elective | 1 | 2 | 6 |
| | Level of Course Unit | | Second Cycle | | Objectives of the Course | | To gain the ability to solve problems that cannot be solved analytically or that are difficult to solve in physics by using numerical analysis methods. | | Name of Lecturer(s) | | Prof. Dr. Mehmet ÇINAR | | Learning Outcomes | | 1 | Can make vector differential and integral calculations.
Recognizes curvilinear coordinates, writes differential vector operators for all coordinate systems. | | 2 | Defines complex numbers, knows polar, trigonometric exponential forms.
Performs operations with complex functions, defines the analytic function, understands the importance of analytic functions in physics. | | 3 | Turns a function into a complex series, obtains the residue theorem, uses it to calculate integrals. It applies to some special functions (Legendre, Laguerre, Hermite polynomials). | | 4 | Recognizes the general forms of differential equations frequently used in physics (recognizes Laplace Equation, Poisson Equation, Helmholtz Equation, time dependent wave equation, Klein-Gordon Equation, Schrödinger Wave Equation and d Alembertian operator). | | 5 | Solve first and second order differential equations using the Green's Function method.
Recognizes and solves differential equations using the Frobenius method. | | 6 | Recognizes and solves differential equations using the Frobenius method. | | 7 | Knows the properties of Bessel functions, Legendre functions, Hermite polynomials, Laguerre polynomials. | | 8 | Makes and applies integral transforms, Fourier transforms, Laplace transforms. |
| | Mode of Delivery | | Normal Education | | Prerequisites and co-requisities | | None | | Recommended Optional Programme Components | | - | | Course Contents | | Vector analysis. Curved coordinates. Complex variables and functions. Complex integrals, complex series, residue theorem. Differential equations: First order differential equations. Second order differential equations, series solutions: Frobenius method. Inhomogeneous differential equations: Greens function. Bessel functions, Legendre functions, Hermite polynomials, Laguerre polynomials. Integral transforms: Fourier transforms, Laplace transforms. | | Weekly Detailed Course Contents | |
| 1 | Vector Analysis
Definitions, Elementary Approach
Rotation of the Coordinate Axes
Scalar or Dot Product
Vector or Cross Product
Triple Scalar Product, Triple Vector Product
Gradient
Divergence,
Curl,
Vector Integration
Gauss’ Theorem
Stokes’ Theorem
Potential Theory
Gauss’ Law, Poisson’s Equation
Dirac Delta Function
Helmholtz’s Theorem | | | | 2 | Vector Analysis in Curved Coordinates and Tensors
Orthogonal Coordinates in R3
Differential Vector Operators
Special Coordinate Systems: Introduction
Circular Cylinder Coordinates
Spherical Polar Coordinates
Tensor Analysis
Contraction, Direct Product .
Quotient Rule
Pseudotensors, Dual Tensors
General Tensors | | | | 3 | Determinants and Matrices
Determinants
Matrices
Orthogonal Matrices
Hermitian Matrices, Unitary Matrices
Diagonalization of Matrices
Normal Matrices | | | | 4 | Group Theory
Introduction to Group Theory
Generators of Continuous Groups
Orbital Angular Momentum
Angular Momentum Coupling
Homogeneous Lorentz Group
Lorentz Covariance of Maxwell’s Equations
Discrete Groups
Differential Forms | | | | 5 | Sonsuz seriler
Temel kavramlar
Yakınsama Testleri
Alternatif Seriler
Seri Cebiri
Fonksiyonlar Serisi
Taylor'ın Genişlemesi
Güç serisi
eliptik integraller
Bernoulli Sayıları, Euler-Maclaurin Formülü
Asimptotik Seriler | | | | 6 | Functions of a Complex Variable
Complex Algebra
Cauchy–Riemann Conditions
Cauchy’s Integral Theorem
Cauchy’s Integral Formula
Laurent Expansion
Singularities
Mapping
Conformal Mapping | | | | 7 | Functions of a Complex Variable
Calculus of Residues
Dispersion Relations
Method of Steepest Descents | | | | 8 | Midterm Exam | | | | 9 | The Gamma Function
Definitions, Simple Properties
Digamma and Polygamma Functions
Stirling’s Series
The Beta Function
Incomplete Gamma Function | | | | 10 | Differential Equations
Partial Differential Equations
First-Order Differential Equations
Separation of Variables
Singular Points
Series Solutions—Frobenius’ Method
A Second Solution
Nonhomogeneous Equation—Green’s Function
Heat Flow, or Diffusion | | | | 11 | Bessel Functions | | | | 12 | Legendre Functions | | | | 13 | Special Functions
Hermite Functions
Laguerre Functions
Chebyshev Polynomials
Hypergeometric Functions
Confluent Hypergeometric Functions
Mathieu Functions | | | | 14 | Fourier Series | | | | 15 | Integral Transforms
Development of the Fourier Integral
Fourier Transforms—Inversion Theorem
Fourier Transform of Derivatives
Convolution Theorem
Momentum Representation
Transfer Functions
Laplace Transforms | | | | 16 | Final Exam | | |
| | Recommended or Required Reading | | 1-Introduction to Methods of Applied Mathematics orAdvanced Mathematical Methods for Scientists and Engineers Sean Mauch http://www.its.caltech.edu/˜sean January 24, 2004
2-Karaoğlu B., (2004), Sayısal Fizik, Seçkin yayınları.
3-Mathematical methods for physics and engineering, A comprehensive guide, Second edition, K. F. Riley, M. P. Hobson and S. J. Bence, Cambridge University Press, 2004.
4- Mathematical Methods For Physicists, Sixth Edition George Arfken, Hans J. Weber, Elsevier Academic Press, 2005. | | Planned Learning Activities and Teaching Methods | | | Assessment Methods and Criteria | |
| Midterm Examination | 1 | 100 | | SUM | 100 | |
| Final Examination | 1 | 100 | | SUM | 100 | | Term (or Year) Learning Activities | 40 | | End Of Term (or Year) Learning Activities | 60 | | SUM | 100 |
| | Language of Instruction | | Turkish | | Work Placement(s) | | None |
| | Workload Calculation | |
| Midterm Examination | 1 | 1 | 1 | | Final Examination | 1 | 2 | 2 | | Self Study | 13 | 4 | 52 | | Individual Study for Mid term Examination | 7 | 5 | 35 | | Individual Study for Final Examination | 14 | 5 | 70 | | Homework | 7 | 3 | 21 | |
| Contribution of Learning Outcomes to Programme Outcomes | | LO1 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | | LO2 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | | LO3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | | LO4 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | | LO5 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | | LO6 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | | LO7 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | | LO8 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 |
| | * Contribution Level : 1 Very low 2 Low 3 Medium 4 High 5 Very High |
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