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MA304: Topics in Applied Mathematics

Unit 5: PDE Methods   When the variance of the Gaussian convolution filter is replaced by ct, where c is a fixed positive constant and t is used as the time parameter, then the convolution filtering of any input function f(x) describes the heat diffusion process with initial temperature given by f(x).  More precisely, if u(x, t) denotes the temperature at the position x and time t, then u(x,t), obtained by the Gaussian convolution of the initial temperature f(x), is the solution of the  heat diffusion PDE with initial condition u(x, 0) = f(x), where the constant c is the heat conductivity constant.  However, this elegant example has little practical value, because the spatial domain is the entire x-axis,but it serves the purpose as a convincing motivation for the study of linear PDE methods, to be studied in this unit.  To solve the same heat diffusion PDE as in this example, but with initial heat source given on a bounded interval and with insulation at the two end-points to avoid any heat loss, the method of “separation of variables” is introduced.  This method separates the PDE into two ordinary differential equations (ODEs) that can be easily solved by appealing to the eigenvalue problem, studied in Unit 1, for linear differential operators with eigenfunctions given by the cosine function in x and with frequency governed by the eigenvalues, which also dictate the rate of exponential decay in the time variable t.  Superposition of the product of these corresponding eigenfunctions with coefficients given by the Fourier coefficients of the Fourier series representation of the initial heat content, studied in Unit 3, solves this heat equation.  In this unit, you will study an extension of the method of separation of variables to the study of boundary value problems on a rectangular spatial domain as well as the solution of other popular linear PDEs.  The diffusion process can be applied to image noise reduction.

Unit5 Learning Outcomes
Upon successful completion of this unit, the student will be able to:
 
- Apply the method of separation of variables to separate a given linear PDE into a finite family of ODEs. - Solve the corresponding eigenvalue problems for the spatial ODEs. - Apply the Fourier series of the input function to formulate the superposition solution of the boundary value problem.

5.1 From Gaussian Convolution to Diffusion Process   - Reading: From Gaussian Convolution to Diffusion Process The Saylor Foundation does not yet have materials for this portion of the course. If you are interested in contributing your content to fill this gap or aware of a resource that could be used here, please submit it here.

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5.1.1 Gaussian as Solution for Delta Heat Source   5.1.2 Gaussian Convolution as Solution of Heat Equation for the Real-Line   5.1.3 Gaussian Convolution as Solution of Heat Equation on the Euclidean Space   5.2 The Method of Separation of Variables   - Reading: University of Minnesota: Peter Olver’s Introduction to Partial Differential Equations: “Chapter 4: Separation of Variables: Introduction and the Diffusion and Heat Equations” Link: University of Minnesota: Professor Peter Olver’s Introduction to Partial Differential Equations: “Chapter 4: Separation of Variables: Introduction and the Diffusion and Heat Equations” (PDF)
 
Instructions: Please click on the link above, and then select the link for “Chapter 4: Separation of Variables” to download the text.  Study Chapter 4 on pages 103–109 to learn about the method of separation of variables (for the special case of one spatial variable), particularly for solving the heat equation.
Studying this reading should take approximately 1 hour and 30 minutes to complete.
 
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  • Reading: Cambridge University Press: Professor Marcus Pivato’s Linear Partial Differential Equations and Fourier Theory Link: Cambridge University Press: Professor Marcus Pivato’s Linear Partial Differential Equations and Fourier Theory (PDF)

    Instructions: Pleaseclick on the link above to download the PDF of the text.  Study Part I (on some motivating examples) and Part II (on the more general theory), particularly to learn about the abstract theory as a companion to the study of the reading by Professor Olver.
    Studying this reading should take approximately 1 hour to complete.
     
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5.2.1 Separation of Time and Spatial Variables   - Reading: Separation of Time and Spatial Variables The Saylor Foundation does not yet have materials for this portion of the course. If you are interested in contributing your content to fill this gap or aware of a resource that could be used here, please submit it here.

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5.2.2 Superposition Solution   - Reading: Superposition Solution The Saylor Foundation does not yet have materials for this portion of the course. If you are interested in contributing your content to fill this gap or aware of a resource that could be used here, please submit it here.

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5.3 Fourier Series Solution   - Reading: University of Minnesota: Peter Olver’s Introduction to Partial Differential Equations: Chapter 4: Separation of Variables: Introduction and the Diffusion and Heat Equations” Link: University of Minnesota: Professor Peter Olver’s Introduction to Partial Differential Equations: “Chapter 4: Separation of Variables: Introduction and the Diffusion and Heat Equations” (PDF)
 
Instructions: Please click on the link in above, and then select the link to download “Chapter 4: Separation of Variables.”  Study the Fourier series solution on pages 109–140.
Studying this reading should take approximately 2 hours to complete.
 
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

  • Reading: Cambridge University Press: Professor Marcus Pivato’s Linear Partial Differential Equations and Fourier Theory Link: Cambridge University Press: Professor Marcus Pivato’s Linear Partial Differential Equations and Fourier Theory (PDF)

    Instructions: Please click on the link above to access the PDF.  Study Part III (on Fourier series solutions) and Part IV (on Boundary value solutions).
    Studying this reading should take approximately 3 hours to complete.

    Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

5.3.1 Fourier Series Representation for Spatial Solution   Note: This topic is covered by the lectures assigned below subunit 5.3

5.3.2 Extension to Higher Dimensional Spatial Domain   - Reading: Extension to Higher Dimensional Spatial Domain The Saylor Foundation does not yet have materials for this portion of the course. If you are interested in contributing your content to fill this gap or aware of a resource that could be used here, please submit it here.

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5.4 Boundary Value Problems   - Reading: Boundary Value Problems The Saylor Foundation does not yet have materials for this portion of the course. If you are interested in contributing your content to fill this gap or aware of a resource that could be used here, please submit it here.

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5.4.1 The Neumann Boundary Value Problem   5.4.2 Anisotropic Diffusion   5.4.3 Solution in Terms of Eigenvalue Problems   5.5 Application to Image De-noising   - Reading: Application to Image De-noising The Saylor Foundation does not yet have materials for this portion of the course. If you are interested in contributing your content to fill this gap or aware of a resource that could be used here, please submit it here.

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5.5.1 Diffusion as Quantizer for Image Compression   5.5.2 Diffusion for Noise Reduction   5.5.3 Enhanced JPEG Image Compression