Analytical solutions to the heat equation in 1D

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Analytical solutions to the heat equation in 1D

• A very brief introduction

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Dimensionless quenching problem
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0.5
0
0
1
0.5
X
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Propose a solutionCourse notes show the formal
procedure for obtaining this solution
This generic form satisfies the diffusion
equation. A, B, and lambda are arbitrary
constants (for now). Heat equation is linear so
true solution can be a sum of many!
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BC at x0
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BC at x1
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Initial condition
How to find the coefficients, A ?
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Orthogonality property of sin
We can check these relationships visually.
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Use orthogonality
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Putting it all together
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How many terms in the sum?
Time is fixed, animation is adding more terms to
the sum
X
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Orthogonal functions
Function space 1 is the known function
Normal vector space F is known vector
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Later time
X
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Substitute assumption into heat eqn.
What is the only possible solution?
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The constant gives 2 ordinary diff. equations
Note the negative sign and power of two are
because I know the answer!
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Solve time part
Note Negative sign keeps the solution at
infinite time bounded
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Solve x part
Note power of two removes carrying sqrt around!