Let f \in C_{o}^{\infty} (\mathbb{R}^n). Propose formulas of the form

Molecca89g

Molecca89g

Answered question

2022-04-27

Let fCo(Rn). Propose formulas of the form u(x)=  f^(δ)eixδp(δ)dδ
to solve:
i) j=1n4u(x)xj4+u(x)=f(x)
and

ii) k,l 4u(x)xk2xl2-2·k=1n2u(x)xk2+u(x)=f(x)

Answer & Explanation

Yaretzi Odom

Yaretzi Odom

Beginner2022-04-28Added 16 answers

Step 1
For a general linear differential operator L[u] with constant coefficients you know that to get a particular solution to
L[u]=ceiδx
you make a trial approach with u=Aeiδx
and you get L[eitsx]=p(δ)eiδx
So A=cp(δ). If the right side is a sum of exponential terms,
L[u]=k=1nckeiδkx
then the particular solution is composed of the particular solutions for each term,
up(x)=k=1nckp(δk)eiδkx.
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)^eiδx,dδ.

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