Advanced Engineering Mathematics

Week 1: First Order Ordinary Differential Equations; Separable ODEs, Exact ODE’s, Reduction to exact form, Determining integration factors for nonexactness, Initial value problems, Linear ODE’s, Bernoulli Equation, Homogeneous ODEs, Nonhomeogeneous ODEs, Population Dynamics, Verhulst Equation, Picard’s existence and uniqueness theorems, Lipschitz condition

Week 2: Second Order Linear ODE’s; Homogeneous linear ODEs, Nonhomogeneous ODEs with second order, Reduction of order and basis, Differential operators, Modeling Free Oscillations (Mass-Spring System- Overdamping, Critical Damping, Under Damping), Euler-Cauchy Equations, Existence and Uniqueness of Solutions, Wronskian, Method of Undetermined Coefficients, Modeling Forced Oscillations (Mass-Spring System- Damped forced oscillations, Undamped forced oscillations-resonance), Solution by Variation of Parameters, Higher Order Linear ODEs; Homogeneous linear ODEs, Distinct real roots, Simple complex roots, Multiple real roots, Multiple complex roots, Non-homogeneous linear ODEs, Method of undetermined coefficients, Method of variation of parameters

Week 3: Systems of ODEs, Eigenvalues and eigenvectors, Conversion of an nth order ODE to a system, Basic theory of systems of ODEs, Constant coefficient systems and phase plane method, Qualitative Methods, Trajectories, 5 Types of critical points of the system (improper node, proper node, saddle point, center, spiral point), degenerate node, Criteria for critical points, Stability (stable, unstable and attractive and stable critical points), Qualitative methods for nonlinear systems, Linearization of nonlinear systems, Free Undamped Pendulum and linearization, Linearization of the damped pendulum, Non-homogeneous linear ODEs, Method of undetermined coefficients, Method of variation of parameters

Week 4: Series Solutions of ODEs; Power Series Method, Convergence interval, Radius of convergence, Legendre’s Equations, Legendre Polynomials, Frobenius Method, Bessel’s Equations, Bessel Functions,

Week 5: Bessel functions of the second kind, Sturm-Liouville Problems, Orthogonal Functions, Orthogonal Eigenfunction Expansions.

Week 6-7: Laplace Transforms, inverse transform, s-shifting, transforms of derivatives and integrals, Unit step function, t-shifting, Short impulses, Dirac’s Delta function, Partial Fractions, Convolution, Integral Equations, Differentiation and integration of transforms, Systems of ODEs, Laplace transform: general formulas.

Week 8: Linear Algebra, linear independence, rank, vector space, second and third order determinants, inverse of a matrix, Gauss-Jordan elimination, vector spaces, inner product spaces, linear transformations

Week 9: Matrix eigenvalue problems, eigenvalues, eigenvector, skew-symmetric and orthogonal matrices, eigenbases, diagonalization, quadratic forms

Week 10: Vector differential calculus, gradient, divergence, curl of a vector field, dot product, cross product, vector fields, directional derivatives, curves, arc length

Week 11: Vector integral calculus, line integrals, path independence of line integrals, double integrals, Green’s theorem in the plane, Surface integrals, Triple integrals, Divergence theorem of Gauss, Stokes Theorem.

Week 12-13: Fourier Series, integrals and transforms.

Week 13-14: Partial Differential Equations, Wave Equation, D’Alembert’s Solution of the Wave Equation, Heat equation and solution by Fourier Series, Heat equation and solution by Fourier İntegrals and transforms, Rectangular membrane, Double Fourier Series, Laplacian in Polar Coordinates, Circular Membrane, Fourier-Bessel Series, Solution of PDEs by Laplace Transforms