I'm having trouble understanding how I'm supposed to show the following for this question.
Let \(\displaystyle u(x)\) be an arbitrary \(\displaystyle C^{1}\) function defined for \(\displaystyle x\geq 0\) such that \(\displaystyle u(0)=0\). Consider the ordinary differential operator \(\displaystyle \frac{d}{dx}\) which assigns to each such function \(\displaystyle u\) the new continuous function \(\displaystyle {u}'(x)\). Show that the inverse operator, say B, assigns to each continuous function \(\displaystyle f(x)\), defined for \(\displaystyle x\geqslant 0\), the function
\(\displaystyle B[f](x)\equiv \int_{0}^{\infty}g(x,z)f(z)dz\), where \(\displaystyle g(x,z)=\left\{\begin{matrix}
1&0\leq z\leq x \\
0&z > x
\end{matrix}\right.\)
Consequently, the solution of the problem \(\displaystyle {u}'(x)=f(x) (x\geq 0)\) with boundary condition \(\displaystyle u(0)=0\), is given in terms of the integral operator B with Green's function g(x,z).
I don't really understand what any of this means. So, I don't think my "work" can be considered that.
My thought is that we have
\(\displaystyle {u}'(x)=f(x)\)
Which is the same thing as \(\displaystyle \frac{d}{dx}[u(x)] = f(x)\)
By using the inverse operator B on both sides, I get
\(\displaystyle B\frac{d}{dx}[u(x)] =Bf(x)\) Which gives
\(\displaystyle u(x) =Bf(x)=\int_{0}^{\infty}g(x,z)f(z)dz\) So,
\(\displaystyle u(x)=\int_{0}^{\infty}g(x,z)f(z)dz\).
But the way the question is posed, makes me think I'm supposed to somehow derive this integral. Am I making this overly complicated? Could someone explain how they got this integral?
Any help is appreciated. I'm just trying to understand the material.
Let \(\displaystyle u(x)\) be an arbitrary \(\displaystyle C^{1}\) function defined for \(\displaystyle x\geq 0\) such that \(\displaystyle u(0)=0\). Consider the ordinary differential operator \(\displaystyle \frac{d}{dx}\) which assigns to each such function \(\displaystyle u\) the new continuous function \(\displaystyle {u}'(x)\). Show that the inverse operator, say B, assigns to each continuous function \(\displaystyle f(x)\), defined for \(\displaystyle x\geqslant 0\), the function
\(\displaystyle B[f](x)\equiv \int_{0}^{\infty}g(x,z)f(z)dz\), where \(\displaystyle g(x,z)=\left\{\begin{matrix}
1&0\leq z\leq x \\
0&z > x
\end{matrix}\right.\)
Consequently, the solution of the problem \(\displaystyle {u}'(x)=f(x) (x\geq 0)\) with boundary condition \(\displaystyle u(0)=0\), is given in terms of the integral operator B with Green's function g(x,z).
I don't really understand what any of this means. So, I don't think my "work" can be considered that.
My thought is that we have
\(\displaystyle {u}'(x)=f(x)\)
Which is the same thing as \(\displaystyle \frac{d}{dx}[u(x)] = f(x)\)
By using the inverse operator B on both sides, I get
\(\displaystyle B\frac{d}{dx}[u(x)] =Bf(x)\) Which gives
\(\displaystyle u(x) =Bf(x)=\int_{0}^{\infty}g(x,z)f(z)dz\) So,
\(\displaystyle u(x)=\int_{0}^{\infty}g(x,z)f(z)dz\).
But the way the question is posed, makes me think I'm supposed to somehow derive this integral. Am I making this overly complicated? Could someone explain how they got this integral?
Any help is appreciated. I'm just trying to understand the material.