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Simplify a solution with the Chain Rule

I am trying to find a better solution for a problem that has popped up in my research work. I am familiar with the Chain rule already, but have ran into a problem that I am unclear how to resolve.
I am tasked with finding the solution to the following problem:
$$\frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}=0.01\frac{\partial u}{\partial x}$$
with the initial and boundary conditions
$$\begin{cases}
u(x,t=0)=\sin(x) \\
\frac{\partial u}{\partial x}(0,t)=0
\end{cases}
$$
$(1)$
$$\frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}=\frac{\partial^2u}{\partial x^2}$$
$(2)$
$$\frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}=0.01\frac{\partial u}{\partial x}$$
$(3)$
I want to solve the problem using the chain rule, while trying to keep the solution functions as simple as possible. I need to find $u(x,t)$, where $u(x,t)$ is the solution to the first problem. Please note that I can solve for $u(x,t)$ only using the chain rule. I can’t use the solution from the first problem, since it is written in terms of $u$ and I have to solve the second problem given the solution.
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