Let f∈Co∞(Rn). Propose formulas of the form u(x)=displaysty≤∫{{f^(δ)eixδp(δ)dδto solve:i) ∑j=1n∂4u(x)∂xj4+u(x)=f(x)andii) ∑k,l{∂4u(x)∂xk2∂xl2−2⋅displaysty≤∑k=1n∂2u(x)∂xk2+u(x)=f(x)

Question

Nyla Trujillo · Accepted Answer

Step 1For a general linear differential operator L[u] with constant coefficients you know that to get a particular solution toL[u]=ceiδ⋅xyou make a trial approach with u=Aeiδ⋅xand you get L[ei∂ts⋅x]=p(δ)eiδ⋅xSo A=cp(δ). If the right side is a sum of exponential terms,L[u]=∑k=1nckeiδk⋅xthen the particular solution is composed of the particular solutions for each term,up(x)=∑k=1nckp(δk)eiδk⋅x.Now go to the limit in some fashion, for instance via step functions, to apply an analogous argument when the right side is a function with a Fourier transform,L[u](x)=f(x)=∫Df(δ)^eiδ⋅x,dδ.

Let \(\displaystyle{f}\in{{C}_{{{o}}}^{{\infty}}}{\left({\mathbb{{{R}}}}^{{n}}\right)}\). Propose formulas of the form

Answered question

Answer & Explanation

New Questions in Calculus 2