Solve PDE with direct integration. Hello friends, today it’s about how to solve PDE with direct integration.

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### Solve PDE with direct integration

#### Examples of how to solve PDE with direct integration

Note: None of these examples is mine. I have chosen these from some books. I have also given the due reference at the end of the post.

So here is the first example.

##### Example 1

According to stroud and Booth (2011), “Solve the following equation:

”

##### Solution

Now here the given partial differential equation is:

(1)

And the initial value is at Also the boundary value is at .

As I can see, here is a function of both and .

First of all, I’ll integrate equation (1) partially with respect to to get

Now here is an arbitrary function of .

(2)

As I already know that the boundary value of this PDE is at .

So I can substitute and at equation (2) to get

Thus I can rewrite equation (2) as

(3)

Next, I’ll integrate equation (3) partially with respect to to get

Now here is another arbitrary function of .

(4)

As I already know that the initial value of this PDE is at .

So I can substitute and at equation (4) to get

Thus I can rewrite equation (4) as

Hence I can conclude that this is the answer to this example.

Now comes my other example.

##### Example 2

According to stroud and Booth (2011), “Solve the following equation:

”

##### Solution

Now here the given partial differential equation is:

(5)

And the initial values are at and at .

As I can see, here is a function of both and .

So I’ll start with the integration of the differential equation.

###### Step 1

First of all, I’ll integrate equation (5) partially with respect to to get

(6)

Now here is an arbitrary function of .

As I already know that the boundary value of this PDE is at .

So I can substitute and at equation (6) to get

Hence the value of is

Thus I can rewrite equation (6) as

(7)

Then I’ll do the next integration.

###### Step 2

Next, I’ll integrate equation (7) partially with respect to to get

Now here is an arbitrary function of .

(8)

As I already know that the other initial value of this PDE is at .

So I can substitute and at equation (8) to get

Now I already know that

Thus I can say that

Thus I can rewrite equation (8) as

which means

Hence I can conclude that this is the answer to this example.

Dear friends, this is the end of my today’s post on how to solve PDE with direct integration. Thank you very much for reading this. Please let me know how you feel about it. Soon I will be back again with a new post. Till then, bye, bye!!

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