How do I get an estimate for this nonlocal ODE?Consider the following nonlocal ODE on&nbsp;[1,∞):r2f&nbsp;″(r)+2rf′(r)−l(l+1)f(r)=−(f′(1)+f(1))r2f(1)=αlimr→∞f(r)=0where l is a positive integer and&nbsp;α&nbsp;is a real number.Define the following norm&nbsp;‖·‖‖f‖2:=∫1∞r2f'(r)2dr+l(l+1)∫1∞f(r)2drI want to prove the estimate:‖f‖≤Cl(l+1)|α|for some constant C independent of&nbsp;α, l and f. But I am stuck.Here is what I tried. Multiply both sides by f and integrate by parts to get:‖f‖2=∫1∞r2f2+∫1∞l(l+1)f2&nbsp;=f′(1)f(1)+(f′(1)+f(1))∫1∞fr2&nbsp;&nbsp;≤f′(1)α+(f′(1)+α)3∫1∞f2&nbsp;&nbsp;≤f′(1)α+(f′(1)+α)3‖f‖where I used Cauchy-Schwartz in the before last line. I am not sure how to continue and how to get rid of the f'(1) term.Any help is appreciated.

Question

How do I get an estimate for this nonlocal ODE?Consider the following nonlocal ODE on&amp;nbsp;[1,∞):r2f&amp;nbsp;″(r)+2rf′(r)−l(l+1)f(r)=−(f′(1)+f(1))r2f(1)=αlimr→∞f(r)=0where l is a positive integer and&amp;nbsp;α&amp;nbsp;is a real number.Define the following norm&amp;nbsp;‖·‖‖f‖2:=∫1∞r2f&#039;(r)2dr+l(l+1)∫1∞f(r)2drI want to prove the estimate:‖f‖≤Cl(l+1)|α|for some constant C independent of&amp;nbsp;α, l and f. But I am stuck.Here is what I tried. Multiply both sides by f and integrate by parts to get:‖f‖2=∫1∞r2f2+∫1∞l(l+1)f2&amp;nbsp;=f′(1)f(1)+(f′(1)+f(1))∫1∞fr2&amp;nbsp;&amp;nbsp;≤f′(1)α+(f′(1)+α)3∫1∞f2&amp;nbsp;&amp;nbsp;≤f′(1)α+(f′(1)+α)3‖f‖where I used Cauchy-Schwartz in the before last line. I am not sure how to continue and how to get rid of the f&#039;(1) term.Any help is appreciated.

Jonathon Hanson · Accepted Answer

Step 1Here is how we deal with the f&#039;(1) term:                                        f            ′                    (          1          )                                    =          −                      ∫            1                          ∞                                            f            ″                    (          r          )          d          r                                                        =          −                      ∫            1                          ∞                                            [            −                                          f                (                1                )                +                                  f                  ′                                (                1                )                                            r                4                                      +                                          l                (                l                +                1                )                                            r                2                                      f            (            r            )            −                          1              r                                      f              ′                        (            r            )            ]                    d          r                    Step 2Since&amp;nbsp;∫1∞1r4dr&amp;lt;1, we can just solve for f&#039;(1) and use Cauchy Schwartz to estimate it by |f(1)| and ∥f∥. Then I can easily get the estimate I wanted.

How do I get an estimate for this

Answered question

Answer & Explanation

New Questions in Calculus 2