MAT 3724 Applied Analysis I

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MAT 3724Applied Analysis I

• 2.1 Part IIICauchy Problem for the Heat Equation

http//myhome.spu.edu/lauw
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Not to Myself

• Do not post the PPT until they finish the thought
  experiment.

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Heat Equation
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Example 1
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Thought Experiment

• Assume there is no energy in the rod initially.

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RecallThought Experiment 2.1 I

• What would happen to w(x,t) as the time moves on?

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RecallThought Experiment 2.1 I
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Experiment Time
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Answers

• Show Maple animations

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(a) 1 unit of heat is injected at x0
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(b) 2 units of heat is injected at x0
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(c) 2 units of heat is injected at x3
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(d) 2 units of heat is injected at x0 and 2
units of heat is injected at x3
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(e) 1 unit of heat is injected at x0 and 2
units of heat is injected at x3
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Greens Function (HW 2.1 II)

• Heat Kernel, Impulse Response Function

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Greens Function (HW 2.1 II)

• Heat Kernel, Impulse Response Function
• It satisfies the heat equation and

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Greens Function (HW 2.1 II)

• Heat Kernel, Impulse Response Function
• It satisfies the heat equation and
• It is considered as the solution of the heat
  equation with unit initial heat impulse

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Thought Experiment

• Consider the initial condition as
  continuous distribution of heat source.
• At any point y, units of energy is injected
  into the bar.

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Thought Experiment

• At any point y, the temperature distribution is
  given by
• The solution of (2.1) is the total sum of all
  temp. dist. for all

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Thought Experiment

• The solution of (2.1) is