**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.

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: 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.

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

**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**
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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**
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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**
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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**
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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
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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**