Sage

Tutorial:

Under the terms of the GNU General Public License GPL

the Second Course in Differential Equations: Picard's iterations

Email: Prof. Vladimir Dobrushkin. (Friday, September 20, 2019 11:03:36 AM)

Applied Mathematics - I

Applied Mathematics - II

Computing information for the first course APMA0330

Computing information for the second course APMA0340

Sage tutorial for the first course APMA0330

Sage tutorial for the second course APMA0340

Linear Algebra with Sage

Preface

Introduction +

3D plotting
Existence and Uniqueness
Picard iterations
Adomian iterations
Euler--Lagrange equations
Lagrange multipliers
Hamilton principle
Converting to a system

Part I: Matrix Algebra +

How to define vectors
How to define matrices
Basic operations with matrices
Linear systems of equations
Determinants and Inverses
Special matrices
Eigenvalues and Eigenvectors
Diagonalization Procedure
Sylvester formula
The Resolvent method
Polynomial interpolation
Positive matrices
Roots
Miscellany

Part II: Linear Systems of ODEs +

Motivation
Variable coefficient systems of ODEs
Floquet theory
Constant coefficient systems of ODEs
Planar Phase Portrait
Euler systems of equations
Fundamental matrices
Reduction to a single equation
Method of undetermined Coefficients
Variation of parameters
Laplace transform
Stiff ODEs
Second order ODEs
Spring-mass systems
Electric circuits
Applications

Part III: Non-linear Systems of ODEs +

Planar autonomous systems
Numerical solutions
Stability
Linearization
Spring-mass systems ▼

Anharmonic motion
Forced anharmonic motion
Duffing equations
Forced Duffing equation
Driven hard spring
Driven soft spring
Coulomb damping
Quadratic damping

Conservative systems
Gradient systems
Competing species
Predator-Prey equations
Harvesting species
Lyapunov second method
HIV models
Periodic solutions
Asynchronous solutions
Limited Cycles
van der Pol equations
Neuroscience
Biochemistry
Miscellany

Part III C: Chaos +

Lorenz equations
Rössler attractor
Electric circuits
Chua circuits
Pendulum ▼

Phase Portrait
Moving pivot
Numerical Simulation
Spring Pendulum
Elastic Pendulum
Double Pendulum
Escapement
Inverted Pendulum
ADM approximation
Fourier series

Mechanical problems
Miscellany

Part IV: Numerical Methods +

Iterative Methods
Iterations for Nonlinear Systems
Numerical Solutions using ode45
System conversion
Power series method
Modified Decomposition Method
Euler's methods
Runge--Kutta methods
Finite Difference Methods
Adomian Decomposition Method
Variational iteration method
Finite Element Method
Second order ODEs
Applications

Part V: Fourier Series +

Sturm--Liouville problems
Fourier transform
Fourier series
Periodic Extension
Complex Fourier series
Even and odd functions
Examples
Modes of convergence
Gibbs phenomena
Convergence of Fourier series
Cesàro summation
Square wave functions
Orthogonal expansions
Chebyshev expansions
Legendre expansion
Hermite expansion
Laguerre expansion
Bessel expansion
Motivated Examples

Part VI: PDEs +

First order PDEs
Separation of variables
Green's functions
Blasius equation
Fluid problems
Applications
Miscellany

Part VI P: Parabolic Equations+

Heat conduction problems
Boundary Value Problems for heat equation
Other heat transfer problems
Fourier transform
Fokas method
Resolvent method
Fokker--Planck equation
Numerical solutions of heat equation
Black Scholes model
Monte Carlo for Parabolic

Part VI H: Hyperbolic equations +

Wave equations
IBVPs
2D wave equations
Forced wave equations
Transverse vibrations of beams
Numerical solutions of wave equation
Klein–Gordon equation
3D wave equations
Cagniard method

Part VI E: Elliptic equations +

Laplace equation
Dirichlet problem
Neumann problems for Laplace equation
Mixed problems for Laplace equation
Laplace equation in infinite domain
Laplace equation in infinite stripe
Laplace equation in infinite semi-stripe
Numerical solutions of Laplace equation
Laplace equation in polar coordinates
Laplace equation in a corner
Laplace equation in spherical coordinates
Poisson's equation
Helmholtz equation
Liouville's equation
Monte Carlo for Elliptic

Part VII: Special Functions +

Orthogonal polynomials
Gamma function
Bessel functions ▼

Generating functions
Modified Bessel functions
Hunkel functions
Kelvin functions
Recurrences
Orthogonality of Bessel functions
Airy functions
Applications

Chebyshev functions ▼

Generating functions
Orthogonality
Recurrences
Zeroes
Applications

Legendre functions ▼

Generating functions
Orthogonality
Recurrences
Zeroes
Applications

Lambert function
Mathieu function
Elliptic functions
Hypergeometric functions
Kummer's equation
Miscellany

Glossary

Preface

Example: **DESCRIPTION OF PROBLEM GOES HERE** This is a description for some MATLAB code. MATLAB is an extremely useful tool for many different areas in engineering

Glossary