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 Differential Equations Ordinary and Partial Differential Equations Math Forum

 October 9th, 2018, 07:04 AM #1 Senior Member   Joined: Jan 2015 From: usa Posts: 104 Thanks: 1 A PDE solution Let $f\in C^2(\mathbb{R}^n)$. We define $$\phi(x,r)=\frac{1}{n\alpha(n)}\int_{\partial B(0,1)}f(x+rz)dS(z)$$ where $\alpha(n)$ is the volume of $B(0,1)$. I calculated $$\partial_r\phi=\frac{r}{n\alpha(n)} \int_{\partial B(0,1)}\Delta_xf(x+rz)dS(z)$$ Please help me to show that $$\partial_{rr}\phi-\frac{n-1}{r}\partial_r\phi=\Delta_x\phi$$ Thanks. Last edited by skipjack; October 9th, 2018 at 11:52 AM. October 9th, 2018, 08:20 AM   #2
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 Originally Posted by mona123 Let $f\in C^2(\mathbb{R}^n)$. We define $$\phi(x,r)=\frac{1}{n\alpha(n)}\int_{\partial B(0,1)}f(x+rz)dS(z)$$ where $\alpha(n)$ is the volume of $B(0,1)$. I calculated $$\partial_r\phi=\frac{r}{n\alpha(n)} \int_{B(0,1)}\Delta_xf(x+rz)dS(z)$$ Please help me to show that $$\partial_{rr}\phi-\frac{n-1}{r}\partial_r\phi=\Delta_x\phi$$ Thanks.

Last edited by skipjack; October 9th, 2018 at 11:53 AM. October 9th, 2018, 09:18 AM #3 Senior Member   Joined: Sep 2016 From: USA Posts: 635 Thanks: 401 Math Focus: Dynamical systems, analytic function theory, numerics Ahh so we have graduated from doing your abstract algebra HW to doing your PDE HW. It's nice to be promoted. Last edited by skipjack; October 9th, 2018 at 11:53 AM. November 2nd, 2018, 09:11 AM   #4
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 Originally Posted by SDK Ahh so we have graduated from doing your abstract algebra HW to doing your PDE HW. It's nice to be promoted.
Of course people are going to ask HW questions. If you don't know the answer don't reply. Tags pde, solution Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post rain Real Analysis 1 July 17th, 2013 12:28 PM niraj Algebra 5 December 14th, 2012 07:27 AM davedave Calculus 1 January 31st, 2012 02:48 PM bentick Algebra 11 April 14th, 2010 11:15 PM TreeTruffle Algebra 2 March 27th, 2010 01:22 AM

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