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 $u (r) = ∭ G (r, r^{'}) f (r^{'}) d τ^{'}$ is a solution to (8.22): $\nabla^{2} u = f (r)$ .

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’s equation.

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

Step 3:Take the equation and start solving:

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

$\nabla^{2} u = f (r)$

Use equation (8.23) to write the following equation.

$\nabla^{2} G (r, r^{'}) = δ (r - r^{'})$

And equation (8.24).

$∭ f (r^{'}) δ (r - r^{'}) d τ^{'} = f (r)$

Now prove the fact:

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

$\begin{matrix} \nabla^{2} u = \nabla^{2} ∭ G (r, r^{'}) f (r^{'}) d τ^{'} \\ = ∭ \nabla^{2} G (r, r^{'}) f (r^{'}) d τ^{'} \\ = ∭ δ (r - r^{'}) f (r^{'}) d τ^{'} \end{matrix}$

$\nabla^{2} u = f (r)$

Hence it has been proved.

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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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