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Consider this phrase:

"To solution, an equation is derived to describe the particle motion."

And also this phrase:

"To this problem, the new proposed method can reduce the weight of the payload significantly."

The questions are: do the phrases in bold texts are valid and common ?

I have tried to check in Google Scholar, but could not find the result of text with such phrases -- let alone starting in the beginning.

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No, they are not grammatically correct.

To solution doesn't make any sense, because after to, we would normally expect a verb, like "To solve this problem..." but solution is a noun. Another problem is that solution is a countable noun, so you have to use it with a determiner, like the solution, or a solution, or my solution, but not just solution by itself.

To this problem isn't much better. It looks like what you want to say is that an equation is derived or a solution is proposed regarding the solution or problem, but we don't say that by beginning "To..." unless a verb is used after it, like

To solve this problem, an equation is derived to describe the particle motion.

or

To help with this problem, the new proposed method can reduce the weight of the payload significantly.

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