Q6P Substitute (8.25) into (8.22) an... [FREE SOLUTION]

Chapter 13: Q6P (page 658)

Substitute (8.25) into (8.22) and use (8.23) and (8.24) to show that (8.25) is a solution of (8.22).

Short Answer

Expert verified

It has been proved that (8.25) from the book $u (r) = 鈭� G (r, r^{'}) f (r^{'}) d 蟿^{'}$ is a solution to (8.22): ${鈭�}^{2} u = f (r)$ .

Step by step solution

Given Information:

It has been given that for reference follow example 1.

Uses of the Green function:

The green function is a function that is used to solve a corresponding partial differential equation in three dimensions, namely Poisson鈥檚 equation.

$\begin{matrix} {鈭�}^{2} u = f (r) \\ = f (x, y, z) \end{matrix}$

Step 3:Take the equation and start solving:

To prove that (8.25) from the book $u (r) = 鈭� G (r, r^{'}) f (r^{'}) d 蟿^{'}$ is a solution to (8.22):

${鈭�}^{2} u = f (r)$

Use equation (8.23) to write the following equation.

${鈭�}^{2} G (r, r^{'}) = 未 (r 鈭� r^{'})$

And equation (8.24).

$鈭� f (r^{'}) 未 (r 鈭� r^{'}) d 蟿^{'} = f (r)$

Now prove the fact:

Change in (8.25) into the left-hand side of (8.22), and try to calculate the expression.

$\begin{matrix} {鈭�}^{2} u = {鈭�}^{2} 鈭� G (r, r^{'}) f (r^{'}) d 蟿^{'} \\ = 鈭� {鈭�}^{2} G (r, r^{'}) f (r^{'}) d 蟿^{'} \\ = 鈭� 未 (r 鈭� r^{'}) f (r^{'}) d 蟿^{'} \end{matrix}$

${鈭�}^{2} u = f (r)$

Hence it has been proved.

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91影视

Substitute (8.25) into (8.22) and use (8.23) and (8.24) to show that (8.25) is a solution of (8.22).

Short Answer

Step by step solution

Given Information:

Uses of the Green function:

Step 3:Take the equation and start solving:

Now prove the fact:

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