definition A functional differential equation is a differential equation in which the derivative $u'(t)$ of an unknown function u has a value at t that is related to u as a function of some other function at t. A general first-order functional differential equation is therefore given by $u'(t)=f(t,y(t),y(u(t)))$
I have not have the classes about functional. I think it is but I am not sure. So please give me a answer. Thank you.
Let $F(u)$ be a functional ,$f(u)=0$ is its Euler-Lagrange equation. We need to calculate the $u$ which minimize the $F$. So, as usually, we have the equation as follows: $$\frac{\partial u}{\partial t}=-f(u)$$ Through iteration, we get the final u (the original $u_{0}$ is arbitrary,ps1): $$\frac{u^{n+1}-u^{n}}{\triangle t}=-f(u)$$ $$\frac{u^{n+1}\left(x,y\right)-u^{n}\left(x,y\right)}{\triangle t}=-f\left(u\right)=\iint H(u(a,b))K(g\left(x,y\right)-g\left(a,b\right))dadb \qquad (1)$$ $$u,g:R^{2}\rightarrow R$$
$$\begin{eqnarray*} K:\, R & \rightarrow & R\ K(a) & = & \begin{cases} 1 & a=0\ 0 & otherwise \end{cases} \end{eqnarray*} \begin{eqnarray*} H:\, R & \rightarrow & R\ H(a) & = & \begin{cases} 1 & a\geq0\ 0 & otherwise \end{cases} \end{eqnarray*}
$$
In the right side of equation 1, there is no derivative. And in the left side of equation 2, is $\partial u/\partial t$ a derivative according to t? Is equation 1 a functional differential equation?
1.how did you arrive at that formula with the double integral? To a specific point (x,y), for example x=1,y=6, then the equation becomes:$$\frac{u^{n+1}\left(1,6\right)-u^{n}\left(1,6\right)}{\triangle t}=-f\left(u\right)=\iint H(u(a,b))K(g\left(1,6\right)-g\left(a,b\right))dadb \qquad (2)$$ In the right side of the equation(2), we get a real number,so we could update $u^{n+1}\left(1,6\right)$. To all points in our plane coordinate, we use the same way to update every point.
2. how do you actually want to use this expression?
I want to update values of u，I mean calculate the function u by iteration. $R^{2}\rightarrow R$
ps1:$u_{0}$ I initialize $u_{0}$ at random. In my experiments, it makes no difference to get the final results. I tested this, the final results were almost same. So I believe the difference between two results is very samll and can be neglect. As a result, I did not calculate the error. So in here, I can not say the results are exactly same. They are almost same to human's eyes.
waiting answer(@﹏@)~
