How to Numerically Solve a 1D Heat Equaiton ? ( MATLAB Script in Description)

***** Correction: At 1:33, In the green box the following text would be more appropriate, "The Divergence of Gradient or the Flow of Gradient of Temeperature" instead of "The spatial Gradient of Temperature".
This is a tutorial on how to solve a 1D heat equation using Finite Difference Approach for a case of Dirichlet Boundary conditions. 1D Heat equation is a Parabolic Partial Differential Equation. It is an initial value problem solved using FTCS scheme and by time marching technique. Explicit method is used.

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Physical undestanding of the Heat Equaiton:
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The time rate of change in temperature is proportional to the Laplacian of temperature.
The Laplacian of temperature is the Divergence of gradient, i.e, how the gradient flows.
The Gradient of temperature at a given point, is actually a vector which speaks on how the temeperature varies as we move out from that point in different directions.

Now if the Laplacian is positive at a point, that means the surrounding points are at higher temperature than that particular point. So the heat flows into the point. Hence, we will have the temperature rising with time, at that particular point.

On the other hand if the Laplacian is negative at a point, that means the surrounding points are at lower temperature than that particular point. So the heat flows out of the point. Hence, We will have the temperature dropping with time, at that particular point.

This video tries to explain the key concepts and math graphically.
A MATLAB script is shown, to implement the algorithm.
An animation of temperature evolution inside an frozen aluminum rod that is heated from ends.
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Link to Code:

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A detailed code that produces the images for animation is availabe at this link : https://drive.google.com/file/d/1mlqSrWhH5w0TKnFpwC9rYMnDhAVpLXyX/view?usp=sharing
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Видео How to Numerically Solve a 1D Heat Equaiton ? ( MATLAB Script in Description) канала praveenkch