Transform Methods and Numerical Analysis 6H3704
Required reading:
Part 1:
Textbooks for part 1.There are many books available for Laplace, Fourier and Z-transforms
transforms. This part is designed from many sources – so advice is to check handouts, which
will be supplied during the lectures.
The following book is recommended but not obligatory for part 2.
1. Differential Equations with Boundary-Value Problems, Denis G. Zill , Michael R
Cullen,THOMSON LEARNING, edition 6 ( or 5) ISBN: 0534418872
Par1 : Transform Methods
LAPLACE TRANSFORM
Recommended
Differential
exercises:
Equations with
Bound-VP, Denis
G. Zill , Michael R
Cullen, edition 6
1 Definition, Properties and Table of Laplace
Transform. Inverse Transform. Translation
Theorems and Derivatives of a Transform
7.1
7.2
7.3
2 Transform of Derivatives and Integrals.
System of Linear Differential Equations.
3 Applications: Linear Dynamical Systems.
Heaviside’s step function. Dirac Delta
Function. The Transfer Function (Impulse
Response). Step Response. Stable Systems.
Z- TRANSFORM
7.4
7.5
4 Definition, Properties and Table of the zTransform.
5 Difference (Recurrence Equations.
6 Discrete System. Unit Pulse and Response.
Stable Systems.
FOURIER SERIES
7 Periodic Functions. Orthogonallity. Even and
Odd Functions. Trigonometric Fourier series.
8 Cosine and Sine Series.
9 Amplitude-phase Form. Complex Form
10 Fourier Method for Ordinary and Partial
Differential Equations.
FOURIER TRANSFORMS
11 Definition, Properties and Table of Fourier
Transform.
12 Applications
7.6
7.1: 1, 9, 23, 27,
31, 33, 35
7.2: 1,3,5…17
7.3:
1,3,5…17,37,43
7.4: 1,3,9,19,27,
37,39
7.5: 1,3
7.6: 1, 313,15
Course material will
be given during the
lessons
Handouts
Diff.Eq. Zill: 11.1,
11.2
Alternative:
Calculus, 9.9:
Exercises 1..5
11.3
11.1: 1,3
11.2: 1,3,7,9,17
11.3: 1,3,5, 7,
11,13
Handouts
Handouts+
Diff.Eq. Zill: 12.5
12.6
Handouts
Handouts
12.5 Exercise 7
12.6 Example 2
Textbooks for part 2. The material in this part of the course is taken from many sources;
particular references will be given in the lectures.
The following books are recommended but not obligatory for part 2.
Part 2:
1. Calculus, A Complete Course; Adams, Robert A; edition 6 (or 5) ; Addison Wesley ; ISBN:
321270002:
2. Differential Equations with Boundary-Value Problems, Denis G. Zill , Michael R
Cullen,THOMSON LEARNING, edition 6 ( or 5) ISBN: 0534418872
Par2 : Numerical Analysis
1
Non-linear Equations.
Solving an Equation by Bisection.
Fixed-Point Iteration.
The secant method (Regula falsi).
The Newton-Raphson method.
Recommended
exercises:
Course material will Handouts
be handed out
CA;4.6 Examples
during the lessons 1, 2,3
+
CA;6.6 Exercises
Calculus, Adams:
1,3,5
4.6
2 Systems of Linear and Nonlinear Equations.
Calculus, Adams:
13.6
+ Handouts
Handouts
CA;13.6 Example
1
Exercises 1,3,5
3 Approximations. Taylor’s formula.
Calculus, Adams:
4.7, 4.8
+ Handouts
CA;4.7 Example 1
CA;4.8 Examples
1, 2, 5
Calculus, Adams:
6.6, 6.7, 6.8
CA;6.6 Examples
1,2,3
CA;6.6 Exercises
1,3
CA;6.7 Examples
1,2,3
6.7 Exercises 1,3
6.8 Example 4
CA;17.3Examples
1,2,3,4
CA;17.3
Exercises 1,3, 5
Polynomial Interpolation.
Lagrange’s Interpolations Formula
4 Cubic spline interpolation.
Numerical Differentiation
Numerical Integration.
The Trapezoidal and Midpoint Rules.
Simpson’s Rule. Approximate Integration
Using Taylor formula
5 Numerical Treatment of Initial Value Problems for
6
Calculus Adams:
17.3
Ordinary Differential Equations.
Euler’s Method. Euler’s Method. The Fourth-Order
Runge-Kutta Method.
Boundary Value Problems for Ordinary Differential Diff Eq, Zill :
9.5
Equations. Finite-Difference method
7 Boundary Value Problems for Partial Differential
Handouts
8 Periodic Functions. Orthogonallity. Even and
Odd Functions. Trigonometric Fourier series
Diff.Eq. Zill: 11.1,
11.2
Alternative:
Calculus, 9.9:
Exercises 1..5
Diff.Eq. Zill: 11.3
Diff Eq; 9.5,
Examples 1,2
Exercises 3,5,7
Equations. Finite-Difference method. FiniteVolume method.
9 Cosine and Sine Series.
10 Fourier Method for Ordinary and Partial
Differential Equations.
Handouts+
Diff.Eq. Zill: 12.5
12.6
11.1: 1,3
11.2: 1,3,7,9
11.3: 1,3,5, 7,
11,13
12.5 Exercise 7
12.6 Example 2