
About Course
Transform Techniques.
Course Intro: The aim of the course is to teach various transform techniques that are essential for a student of physical sciences and engineering. They include Fourier series, Fourier transforms, Laplace transform,s and Z-transforms.
Course outline
1-Introduction to Fourier series
2-Finding Fourier series of a periodic function
3-Fourier transforms over the real line
4-Fourier transform and its properties
5-Fourier transform and its applications
6-Preliminaries on complex variable techniques
7-Introduction to Laplace transform
8-Laplace transform and its properties
9-Laplace transform and its properties
10-Laplace transform and its applications to ODE and PDEs
11-Z-transforms
12-Z-transforms and its properties and applications
Course Content
Transform Techniques for Engineers
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Use of Delta function in the Fourier series convergence
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Fourier series – Examples
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Complex Fourier series
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Conditions for the Convergence of Fourier Series
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Conditions for the Convergence of Fourier Series(continued)
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Introduction to Fourier series
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More Examples on Fourier Series of a Periodic Signal
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Gibb’s Phenomenon in the Computation of Fourier Series
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Properties of Fourier Transform of a Periodic Signal
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Properties of Fourier transform (continued)
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Parseval’s Identity and Recap of Fourier series
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Fourier integral theorem-an informal proof
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Definition of Fourier transforms
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Fourier transform of a Heavyside function
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Use of Fourier transforms to evaluate some integrals
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Evaluation of an integral- Recall of complex function theory
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Properties of Fourier transforms of non-periodic signals
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More properties of Fourier transforms
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Fourier integral theorem – proof
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Application of Fourier transform to ODE’s
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Application of Fourier transforms to differential and integral equations
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Evaluation of integrals by Fourier transforms
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D’Alembert’s solution by Fourier transform
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Solution of Heat equation by Fourier transform
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Solution of Heat and Laplace equations by Fourier transform
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Introduction to Laplace transform
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Laplace transform of elementary functions
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Properties of Laplace transforms
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Properties of Laplace transforms (continued)
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Methods of finding inverse Laplace transform
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Heavyside expansion theorem
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Review of complex function theory
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Inverse Laplace transform by contour integration
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Application of Laplace transforms-ODEs’
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Solutions of initial or boundary value problems for ODEs’
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Solving first order PDE’s by Laplace transform
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Solution of wave equation by Laplace transform
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Solving hyperbolic equations by Laplace transform
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Solving heat equation by Laplace transform
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Initial boundary value problems for heat equations
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Solution of integral equations by Laplace transform
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Evaluation of integrals by Laplace transform
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Introduction to Z-transforms
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Properties of Z-transforms
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conclusions
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Evaluation of infinite sums by Z-transformson
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Solution of difference equations by Z-transforms
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Inverse Z-transforms
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