MATLAB ®

Tutorial I:

Under the terms of the GNU General Public License GPL
  the First Course in Differential Equations

MATLAB ®

Tutorial I:

Under the terms of the GNU General Public License GPL
  the First Course in Differential Equations, part 1.1: Miscellany

Email: Prof. Vladimir Dobrushkin. (Sunday, April 6, 2025 10:28:11 AM)

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    • Applied Mathematics - I
    • Applied Mathematics - II
    • Computing information for the first course APMA0330
    • Computing information for the second course APMA0340
    • Matlab tutorial page for the second course
    • MuPad page for the second course
    • MuPad page for the first course
  • Preface
  • Introduction +
    1. Installing Matlab
    2. Getting started
    3. Publishing files
    4. Case sensitivity
    5. Equal signs
    6. Complex numbers
    7. Arrays and vectors
    8. Calculations
    9. Logical operators
    10. Functions
    11. Matlab Commands
    12. Chebfun
    13. Solving equations
    14. Derivative
    15. Existence
    16. Uniqueness
    17. Picard iterations
    18. Adomian iterations
  • Part I: Plotting +
    1. Plotting functions
    2. Plotting solutions to ODEs
    3. Discontinuous functions
    4. Direction fields
    5. Implicit plot
    6. Parametric plots
    7. Labeling figures
    8. Figures with arrows
    9. Electric circuits
    10. Plotting with filling
    11. Polar plot
    12. Some famous curves
    13. Cycloids
    14. Miscellany
  • Part II: First Order ODEs +
    1. Solving first order ODEs
    2. Singular solutions
    3. Plotting solutions to ODEs
    4. Direction fields
    5. Separable equations
    6. Autonomous equations
    7. Equations reducible to the separable equations
    8. Equations with linear fractions
    9. Exact equations
    10. Integrating factors
      1. Function of x
      2. Function of y
      3. Function of xy
      4. Function of x/y
      5. Function of x² + y²
      6. Common factors
      7. Homogeneous factors
      8. Special factors
      9. Approximation of integrating factors
    11. Linear equations
    12. RC circuits
    13. Bernoulli equations
    14. Riccati equations
    15. Qualitative analysis
      1. Bifurcations
      2. Landau theory
      3. Validity interval
    16. Orthogonal trajectories
    17. Population models
    18. Pursuit
    19. Applications
  • Part III: Numerical Methods +
    1. Numerical solution using ode23 and ode45
    2. Fixed point iteration
    3. Bracketing methods
    4. Secant methods
    5. Padé approximation
    6. Euler's methods
      1. Backward method
      2. Heun method
      3. Modified Euler method
    7. Runge--Kutta methods
      1. Runge--Kutta methods of order 2
      2. Runge--Kutta methods of order 3
      3. Runge--Kutta methods of order 4
    8. Polynomial approximations
    9. Error estimates
    10. Adomian Decomposition Method
    11. Modified Decomposition Method
    12. Finite Difference Schemes
    13. Variational iteration method
    14. Multistep methods
      1. Multistep methods of order 3
      2. Multistep methods of order 4
      3. Milne method
      4. Hamming method
    15. Applications
  • Part IV: Second and Higher Order ODEs +
    1. Differential equations of higher order
    2. Reduction higher order ODEs
    3. Linear operators
    4. Fundamental sets of solutions
    5. General solutions
    6. Complex roots
    7. Reduction of order
    8. Annihilator operators
    9. Method of undetermined coefficients
    10. Variation of parameters
    11. Operator methods
    12. Euler equations
      1. Factorization
      2. Inhomogeneous Euler equations
    • Part IV N: Nonlinear ODEs +
    1. Numerical solutions
    2. Boundedness of solutions
    3. Spring problems
      1. Free vibrations
      2. Damped vibrations
      3. Forced vibrations
      4. Resonance
      5. Nonlinear models
      6. Driven models
    4. Pendulum
      1. Simple pendulum
      2. Solution of pendulum equation
      3. Period of pendulum
      4. Real pendulum
      5. Driven pendulum
      6. Rocking pendilum
      7. Pumping swing
    5. Dyer model
    6. Electric circuits
    7. Adomian Decomposition Method
    8. Applications
  • Part V: Series and Recurrences +
    1. Recurrences
    2. Discrete logistic recurrence
    3. Generating functions
    4. Lagrange Inversion Theorem
    5. Review of power series
    6. Convergence acceleration
    7. Series solutions for the first order equations
    8. Examples for first order ODEs
    9. Series solutions for the second order equations
    10. Examples for second order ODEs
    11. Euler equations
    12. Regular singular points
      1. Distinct exponents
      2. Equal exponents
      3. Integer difference
      4. Complex exponents
    13. Polynomial solutions
    14. Bessel's equations
    15. Nonlinear ODEs
    16. Picard iterations for the second order ODEs
    17. ADM for first order ODEs
    18. ADM for second order ODEs
    19. Modified Decomposition Method
  • Part VI: Laplace Transformation +
    1. Definition of Laplace transform
    2. Heaviside and Dirac functions
    3. Laplace transform of discontinuous Functions
    4. Table of Laplace transforms
    5. Inverse Laplace transform
    6. Convolution integral
    7. Residue method
    8. Solving IVPs with Laplace transform
    9. Nonhomogeneous ODEs
    10. ODEs with discontinuous input
    11. Nonconstant coefficient IVP’s
    12. Bessel functions
    13. MLDM
    14. Elzaki transform
    15. Mechanical and Electrical applications
  • Part VII: Boundary Value Problems +
    1. Boundary Value Problems
    2. Green functions
    3. Picard iterations
    4. Finite difference schemes
    5. Shooting methods
    6. Tridiagonal linear systems
    7. Numerov method
    8. Adomian Decomposition
    9. Variational iteration method
    10. Blasius layer
    11. Falkner--Skan layer
    12. Heat transfer
    13. Singular BVPs
  • Glossary 
    • xxxxxxxxxx
     
    1
    x=4; y=16;
    2
    z = x + j*y
    Messages

    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, applied mathematics, computer science, biology, chemistry, and so much more. It is quite amazing at handling matrices, but has lots of competition with other programs such as Mathematica and Maple. Here is a code snippet plotting two lines (y vs. x and z vs. x) on the same graph. Click to view the code! 
    figure(1)
plot(x, y, 'Color', [1 0 0]) %blue line
hold on
plot(x, z, 'Color', [0 1 0]) %green line
    Two n-by-n matrices A and B are called similar if there exists an invertible n-by-n matrix S such that