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Contents
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| Week 1: |
Introduction to Mathematical Models in Various Engineering Problems and Equation Formulation.
Examples on the Formulations of Linear and Nonlinear Equation Systems: Electronic Circuits, Load Bearing Mechanical Systems, Biomolecular Systems. |
| Week 2: |
Automation of Equation Formulation and Matlab Examples with Various Systems.
Review of Linear Algebra (Eigenvalues and Eigenvectors, Matrix Vector Products, Gaussian Elimination, Computational Complexity, Multivariate Linear Constant-Coefficient ODE Solutions, Companion Form). |
| Week 3: |
Numerical Solution of Linear Equation Systems (1).
Gaussian Elimination, Partial and Full Pivoting, LU Factors.
Sparse Matrices and Gaussian Elimination on Sparse Matrices, LU Factors. |
| Week 4: |
Numerical Solution of Linear Equation Systems (2).
Matlab Format for Storing Sparse Matrices, Related Operations.
Sparse Factors of Dense Matrices, Related Applications.
Iterative Methods for Solving Linear Equation Systems (Introduction to the GMRES Method).
Applications on Example Engineering Systems.
Information on Related Software Packages. |
| Week 5: |
Introduction to the Numerical Solution of Nonlinear Equation Systems and Related Mathematical Models.
Introduction to the Richardson and Newton Iteration Schemes.
Basic Information on Convergence Characteristics. |
| Week 6: |
Numerical Solution of Nonlinear Equation Systems - Newton's Scheme.
Derivations of Convergence Characteristics, Related Conditions.
Scalar and Multivariate Versions. Jacobian Definition. Comments and Derivations on Those Applications with Sparse Factors for Dense Jacobian Matrices. |
| Week 7: |
Numerical Solution of Nonlinear Equation Systems - Variations of Newton's Scheme.
Chord and Shamanskii Methods, Damped Newton. |
| Week 8: |
Numerical Solution of Nonlinear Equation Systems - Advanced Topics and Applications (Continuation Schemes).
Information on Related Software Packages.
Applications on Example Engineering Systems. |
| Week 9: |
Introduction to the Numerical Solutions of ODE (Ordinary Differential Equations) and One-Step Methods.
Forward / Backward Euler, Trapezoidal Methods. Convergence and Stability Derivations on Test Problems with Linear Systems. |
| Week 10: |
Numerical Solutions of ODE - Multistep Methods, Convergence, Stability.
Adams–Bashforth, Adams–Moulton, Backward Differentiation Formula (BDF) Solver Families.
Coefficient Computation Methods in These Solver Families, Information on Adaptive Time Step and Order Selection.
Convergence and Stability Issues in Solver Families. |
| Week 11: |
Numerical Solutions of ODE - Computations related to Stability.
Nonlinear Equations Systems Computations in Numerical Solutions of ODE.
Information on Related Software Packages.
Applications on Example Engineering Systems. |
| Week 12: |
Numerical Solutions of ODE - Runge-Kutta Schemes. |
| Week 13: |
Orthogonal Polynomials and Gaussian Quadrature Methods for the Numerical Solution of Integrals (1).
Lagrange Polynomials and Interpolation. Newton-Cotes Quadrature Formulas.
Polynomial Degrees Yielding Exact Theoretical Values in the Numerical Solution of Integrals. |
| Week 14: |
Orthogonal Polynomials and Gaussian Quadrature Methods for the Numerical Solution of Integrals (2).
Gaussian Quadrature. Weight Functions. Gaussian Quadrature Nodes.
Gauss–Hermite, Gauss-Legendre, Gauss–Laguerre, Gauss-Jacobi Quadrature Formulas and the Related Orthogonal Polynomials. Stochastic Integral Computations. |
| Week 15*: |
Preparation and Help for the students - Final Exam, Lecture Topics, Homeworks and Projects. |
| Week 16*: |
Final Exam. |
| Textbooks and materials: |
Ders notları ve slaytlardan yararlanılacaktır. Önerilen kaynaklara atıf yapılacak ve ilgili kitap bölümleri işaretlenecektir. |
| Recommended readings: |
Computer-Aided Analysis of Electronic Circuits, L. Chua and P. Lin.
Gilbert Strang. Linear Algebra and Its Applications, 4th Edition.
Lloyd N. Trefethen and David Bau III. Numerical Linear Algebra.
Iterative Solution of Nonlinear Equations in Several Variables, Ortega & Rheinboldt.
Solving Ordinary Differential Equations, Hairer, Norsett & Wanner.
Erwin Kreyszig. Advanced Engineering Mathematics.
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* Between 15th and 16th weeks is there a free week for students to prepare for final exam.
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