pjbooks Chapter 7, seeking guidance?
Chapter 7, seeking guidance?
Lu Feng looked at the message on the screen and couldn't help but shake his head.
It was just a piece of scratch paper, but this teacher Ye managed to find the library management desk from the handwriting and derivation process, then find his student ID through the borrowing records, and finally get his contact information from the counselor.
Such keen insight and decisive action are truly rare qualities.
"Yes, I didn't notice when I left and left it behind."
After typing this line, Lu Feng leaned back in his chair.
The derivations written on the draft paper were, in his view, merely basic calculations for the warm-up phase.
What surprised him was Mr. Ye's keen sense of smell.
The other party replied quickly.
"I have that draft paper with me now. Don't worry, I don't mean anything by it. I just have something I'd like to ask you."
consult?
Lu Feng stared at those two words for two seconds.
An associate professor in the mathematics department used the word "ask for advice" when speaking to a freshman.
"You're too kind, Teacher Ye. Please go ahead."
The intervals between the words on the other end of the line have become longer; they're probably organizing their thoughts.
Almost a minute later, a message popped up.
"When you were deriving the Laplace transform property from the Caputo derivative in your draft, you made a big leap in one step. You first made a variable substitution on the integration interval, and then directly reversed the order of operations for differentiation and integration."
"I've reviewed this step several times, and the result is correct, but I don't understand why you chose to make the replacement in that position. The conventional approach is to integrate by parts first and then change the order, but you did it the other way around."
Could you explain your thought process at the time?
Lu Feng enlarged the chat window and reread Ye Guodong's question.
He was referring to the lower middle section of the draft paper; that step was indeed skipped because there was absolutely no need to write anything afterward.
In vibration analysis, there is a method called time-frequency transformation. The core operation is to reparameterize the variables before integral transformation. The purpose is to separate the physical quantities that were originally coupled together, so that the subsequent transformation is cleaner.
This technique is common in signal processing, but it is not the mainstream approach in pure mathematical derivation.
When he wrote the draft, he went entirely by intuition and didn't consider whether it was mathematically standardized.
For someone who works in pure mathematics, this step can indeed feel abrupt.
Lu Feng thought for a moment and then started typing on the keyboard.
"Professor Ye, the reason I changed the order in that step is this: if we follow the conventional path and perform integration by parts first, it will produce a boundary term, which will not converge when the fractional order α takes certain values."
"But if we first perform variable substitution, normalize the integration interval to [0, 1], and then change the order, the boundary terms will become the standard form of the beta function, and convergence is naturally guaranteed."
After sending it, he waited for about ten seconds and then added another message.
"To put it simply, it avoids the problem of boundary term divergence. Substituting variables first and then reordering them is simpler than reordering them first and then dealing with the boundary terms."
There was silence on the other end for almost two minutes.
Lu Feng assumed Ye Guodong was processing the information.
Zhang Wei lost another game in the dormitory, and his wails echoed throughout the room.
The message popped up.
"Wait a minute, let me find a pen and paper to write this down."
Another half minute passed.
"Okay, you can continue."
Lu Feng's lips twitched slightly. Was it really necessary to be so serious about something that wasn't just a draft?
An associate professor, around 9 p.m., was discussing mathematics with a freshman on QQ with paper and pen.
He reorganized his thoughts and sent them in three parts.
The original integration interval [0, t] is mapped to [0, 1] through τ = t·s, while the order of the differential operator is retained as a parameter.
The normalized integrand satisfies the conditions of Fubini's theorem on [0, 1], and no further proof is needed for the reordering.
We can directly obtain a product form of the gamma function and the power function, and then substitute it back using the definition of the Caputo derivative, thus obtaining the entire Laplace transform property in one step.
He used the most concise language possible in each paragraph, avoiding beating around the bush and piling up technical terms.
After sending it, Lu Feng read the three paragraphs again.
He only realized after writing it down that this approach was indeed unconventional.
The derivation in the standard textbook requires at least five or six steps, and convergence must be handled separately in between.
He solved this problem in three steps, the key being that one variable substitution.
The other side remained quiet for a long time.
Lu Feng had already switched back to the paper page to continue reading the literature when Ye Guodong's message popped up.
"I've checked it, and every step is correct."
"Lu, I have benefited greatly from exchanging ideas with you. I have been doing analysis for so many years, and I never thought of this way of handling variable substitution."
"Teacher Ye, you flatter me. I just happened to think of a clever way."
"It's not about taking shortcuts."
Ye Guodong replied quickly.
"Good mathematics never rejects shortcuts; the shortest proof is the best proof. Your line of thinking shows that you have a very solid geometric intuition for integral transformations, which cannot be developed by simply doing problems."
"Lu, it was a very pleasant conversation with you today. This is my phone number. If you have any questions about mathematics or life in the future, you can come to room 412 of the Mathematics Department office building anytime."
Ye Guodong sent a string of mobile phone numbers.
"Okay, thank you, Teacher Ye."
You're welcome, goodnight.
After closing the chat window, Lu Feng leaned back in his chair and let out a long sigh. He turned his head and glanced behind him. Zhang Wei was throwing a bad hand of cards onto his suitcase, accusing the classmate who dealt the cards of cheating.
I never imagined that a piece of scrap paper casually tossed in the library could attract such a great figure.
Lu Feng refocused his attention on the computer screen. He immediately opened the documents he had downloaded earlier and, referring to his previous line of thought, began to organize his thoughts again.
Numerical solution of fractional derivatives, and proof of convergence of the Mittag-Leffler function.
Until there was a "snap".
The dormitory lights went out instantly, plunging the entire room into darkness, with only a few laptop screens still glowing faintly.
"Oh no! Lights out!"
"Finally, I can go to sleep!"
In the darkness, several relieved cheers rang out.
Zhang Wei's voice was the loudest, filled with a relieved excitement: "Family! No classes tomorrow! I can finally sleep until noon!"
Lu Feng pulled himself out of the sea of documents and stretched his stiff neck.
He closed his notebook, and his voice rang out clearly in the darkness.
"Don't worry, I'll get up early tomorrow and make sure I drag you out of bed at 7:30 sharp."
The dormitory was silent for a second.
"No!" Zhang Wei's wail pierced the night sky. "Brother Lu! My dear brother! You can't be so cruel! I finally managed to arrange a meeting with the fairy sister in my dreams!"
Lu Feng picked up the washbasin on the table, stood up, and walked towards the door.
He left without turning his head.
"That's just a wet dream. Stop reading fantasy novels."