Numerical Simulation of two-

dimensional Heat Equation

using Finite difference Method

Prepared By:Besan Shqirat

Figure 1 :heat
distribution when
T=2000
1
Objective:

1 Find the solution of heat equation in

two dimension
2 Simulate the solution using python

2
Ch.1 Introduction:

In this work we solve the heat

equation and find the general solution
using finite difference method , and
after that we need to simulate the
solution
So what we mean in heat equation ?
And how we can solve it using the
finite difference method ? And how we
can do the simulation
3
 Ch.2 Literature
Review:
In this chapter we need to defined the
heat equation what is it ? And we need
to defined the finite difference method
to start solving the equation

4
The heat
equation
Heat equation(also Finite
known as the
Its difference
a class of
diffusion
method
numerical
equation) is a
techniques for
specific partial
solving
differential
differential
equation. This
equations by
describes how
approximating
the distribution of
derivatives with
heat (or
finite differences.
temperature)
evolves over time
in a given region

5
 Ch.3 Methodology:

In this chapter we need to start solving the heat

equation
…(1)
so let us review the heat equation :

To solve this equation in the finite difference method we

2
𝑢
want
2 to assume that the equation is a steady state
𝜕 𝑢
2
+ 2
=0 … ( 2 )
𝑥 𝜕 𝑦

6
Now we want to see the finite difference method . So
we can write the first derivative on this formula:
…(3)

Also its equal to :

…(4)

Figure 2:finite difference

Now for to simplify let us replace it for some indexing in method
forward finite difference the equation become :

…(5)
In central :
…(7)
And in backward become :
…(6)
Figure 3:finite difference
7
method
 So we write the first derivative, for the second one to
′ ′′ (3)
write it we
𝑢 ( 𝑥+ 𝑎 ) =𝑢 ( 𝑥 ) +
𝑢 need
( 𝑥 )
𝑎+
𝑢to ( 𝑥 )remember
2
𝑎+
𝑢 ( 𝑥 ) 3 the Taylor
𝑎 +… … ( 8 ) expansion:
1! 2! 8!

Let so the Taylor expansion become :

…(9)

And if so the Taylor expansion become :

…(10)

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 Now let us sum them and neglect the higher order
because they approaching
′′
to zero:
𝑢 (𝑥)
𝑢 ( 𝑥+ 𝛥 𝑥 ) +𝑢 ( 𝑥 − 𝛥 𝑥 )=2 𝑢 ( 𝑥 ) + 2 𝛥 𝑥 2 … ( 11 )
2!

Make some simple operation:

… (12)

So the second derivative is:

…(13)

So we can write it in this formula:

…(14)

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Similarly for the y axis :

…(15)

So we can write it in this formula :

…(16)

So let us recall the main eq. sub eq.16 and eq.14 in it :

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We now want to unify the denominators and sum them :

…(18)

… (19)

… (20)

… (21)

And this is the general numerical

solution for our square

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 Ch.4 Results
and T top changed
Discussion:
In the previous chapter we find the
general solution so to customize this
solution we need to assume the

T right =0

T left =0
boundary and initial condition T inside =0
In this picture that describe the
equation in 2d ,Our boundary and
initial condition are :
T everywhere zero, T in the left side
T bottom =0
of the square 0C , T in the right side
Figure 4 :square with its boundary
of the square 0C , T in the bottom condition
side of the square 0C
T in the top side of the square will 12
 Figure 5:heat Figure 6 :heat
distribution when distribution when
T=1000 T=1200

Figure 7 :heat 13
distribution when
Figure 8: animation for the
temperature changed 14
Ch.5 Conclusion :

In this work, we found the general solution to the heat equation in two dimensions using the
difference method. We derived the solution step by step in two dimensions and saw the
method for finding it. After that, to customize the solution, we imposed some initial and
boundary conditions. Then we created the simulation for different conditions, the first of
which is that the temperature be 1000, then 1200, and finally 2000, and see the clear
difference that occurs.

Finally, we created an animated simulation describing the temperature change after some time
had passed from the beginning of the flow

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 Applicatio
n:
Visualizing 3D Representation of
Temperature Temperature
Distributions Explore the heat
Leverage powerful
data visualization equation solution in
techniques to three dimensions,
create intuitive using a surface plot
and informative to depict the
heat maps that temperature values
reveal the as a continuous
temperature function of the spatial
patterns across coordinates, Figure 9:application on
providing a more heat distribution
the 2D domain
over time, comprehensive
enabling deeper understanding of the
insights into the heat transfer process. 16
Thanks

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