sinovels Chapter 5 Freshman? You'd be too young to be a graduate student!
Chapter 5 Freshman? You'd be too young to be a graduate student!
"Begin the proof."
Lu Feng memorized the contents of the blueprint, closed the system panel, took out a draft paper from his bag, and began to work on it.
The mathematical foundation in the first part is fairly solid.
He had already seen the complete form of the Riemann-Liouville fractional derivative definition on the drawing. The core idea was to generalize the factorial of integer derivatives to the Gamma function, and then use an integral operator to define the differential operation of arbitrary order.
He can keep up with this step, after all, he just went through the points section systematically in the afternoon, and his foundation is still fresh.
The pen tip flew across the draft paper, filling the first page.
The pace slows down when we get to the Caputo derivative.
The difference between Caputo's definition and RL lies in the reversed order of differentiation and integration, which leads to completely different handling of initial conditions.
The derivation given on the drawing skipped several steps, and the intermediate transformation processes were omitted, which he had to complete himself.
Lu Feng stared at the crucial step of deriving the Laplace transform from the Caputo derivative, his pen hovering above the paper, hesitant to put it down.
This step requires the recursive relationship of the Gamma function and the integral representation of the Beta function, two things that were not covered in his advanced mathematics textbook that afternoon.
"We have no choice but to tough it out."
After pushing half a page and going around in circles, I finally proved the relationship Γ(α+1)=αΓ(α) myself.
The second draft paper was filled with writing.
When we entered the second part, physical modeling, the difficulty level suddenly increased.
The drawings replaced the classic spring-damping element with fractional differential equations, constructing a generalized Scott-Blair model.
The core idea of this model is that the mechanical response of real materials is neither purely elastic (stress is proportional to strain) nor purely viscous (stress is proportional to strain rate), but rather a kind of "fractional order" behavior that lies between the two.
Lu Feng can understand this concept.
In my past life, I dealt with too many problems related to rubber seals and polymer composites in factories. The stress relaxation curves of these materials could never be properly fitted using classical models; they were always off-target.
At the time, he only knew that "the classic model was not enough", but he didn't know how to fix it.
The drawings now tell him to use fractional derivatives instead of integer derivatives, and that a single parameter α can continuously describe the transition behavior between elasticity and viscosity.
"It's truly ingenious, but the derivation process is also incredibly difficult."
When it came to the core derivation in the third part, Lu Feng was completely stuck.
The drawing uses the Mittag-Leffler function as the fundamental solution to the fractional differential equation, a function he had never seen before.
Its definition is an infinite series, which in form resembles a generalization of the exponential function, but its convergence properties are completely different.
The conclusion is given directly on the drawing, skipping all the intermediate convergence proofs and Laplace inverse transformation processes.
Lu Feng tried to figure it out himself, writing down more than half a page of ideas on a draft paper, but found that he couldn't get it right no matter what he did.
Term-by-term integration of a series requires the condition of uniform convergence, but the convergence of the Mittag-Leffler function is not obvious for certain parameter values.
He changed his approach and tried to deduce the solution from the perspective of Laplace transform. He first assumed the form of the solution and then substituted it into the original equation to verify it.
I wrote another half page, and got stuck at a point where I needed to use the residue theorem.
He hasn't learned about complex functions yet.
The third and fourth sheets of draft paper were filled one after another, spreading out across the table.
Some places are marked with circles and labeled "to be verified," while others are marked with an X to indicate that the road is blocked.
Lu Feng put down his pen and rubbed his temples.
"An hour has passed, and we've only progressed less than a third of the core derivation."
The remaining part involves mathematical tools beyond his current knowledge base, and forcing it will only lead him further astray.
But he was not discouraged.
Rome wasn't built in a day.
There's no rush for the rest. He can come back to it after he's filled in the gaps in his knowledge of complex functions and mathematical analysis; that will be much more efficient.
The sky outside the window had already darkened, and the fluorescent lights in the library had been turned on at some point.
Lu Feng glanced at his phone; it was 19:47 PM.
"It's time to go have dinner."
He folded the draft paper and tucked it into his textbook.
Then his gaze fell on the advanced mathematics books on the table.
He had only finished the first volume of the Tongji University edition of "Advanced Mathematics" (both volumes) and the problem set that afternoon; he hadn't even touched the second volume's topics, such as multivariable calculus, line and surface integrals, and infinite series.
Moreover, he only looked at the questions in the first volume and didn't have time to write them down. Those questions were quite well-designed and much more difficult than the examples in the main text.
I took it back to my dorm and continued watching.
Lu Feng stacked the three books together and carried them to the management desk at the door.
The same girl with glasses who was sitting behind the management desk that afternoon was entering some data into the computer screen.
Hearing footsteps, she looked up.
"Borrowing books?"
"Yes, these three books." Lu Feng placed the books on the table.
"Student ID or student number".
"12031047".
The girl typed a few times on the keyboard, paused, and asked, "Starting with 12? A freshman?"
"Um."
Her gaze shifted from the screen to the stack of books Lu Feng was holding: "Advanced Mathematics" (Volumes 1 & 2) plus a problem set, the three books together being at least five or six centimeters thick.
In late September, a freshman in college had only just reached the second chapter of his advanced mathematics course, limits, when this person borrowed the entire set of books.
Another case of short-lived enthusiasm, something all freshmen can't avoid. They're full of fighting spirit at the start of the semester, but three months later they're just lying in their dorm beds.
When it came time to return the books, they were either lost or returned exactly as they were, never even opened.
The girl glanced at him and entered the borrowing information into the system.
"The maximum borrowing period is thirty days. Do not write or mark on the book, and do not fold pages. If there is any damage, you will need to pay the original price."
"no problem."
After signing the document, Lu Feng stuffed the book into his bag and turned to leave the library.
He didn't notice that a piece of draft paper had slipped out of his textbook and was lying quietly on the table in the window seat where he had just been sitting.
.........
Ye Guodong is on duty tonight.
The faculty duty system in the Department of Mathematics at Hong Kong City University is a long-standing tradition. It rotates weekly, and the faculty on duty stays in the office in the evening to answer students' questions and grade assignments.
But Ye Guodong doesn't like staying in the office; he prefers to bring his assignments to the library to be corrected.
The second-floor science and engineering area is quiet and less crowded at night, and the lighting is softer than the office fluorescent lights.
Ye Guodong arrived at around 7:30.
He chose a seat by the window and sat down.
He caught a glimpse of a piece of paper on the table next to him out of the corner of his eye.
A sheet of draft paper, A4 size, was covered with dense writing.
It must have been left by a student who sat in this seat before.
Ye Guodong didn't take it seriously at first; it was too common for students to leave draft paper in the library.
But his gaze inadvertently swept over a line of words on the paper, and he suddenly stopped.
Γ(α+1)=αΓ(α), followed by a complete derivation of integral by parts.
This wasn't copied directly from the textbook.
The textbook presents this recurrence relation as a single line of conclusion, with at most a sentence stating "easily obtained by integration by parts".
However, the derivation process was written on a full half-page on this draft paper, with each transformation clearly marked, and even the processing of the upper and lower limits of integration after the transformation was not omitted.
Ye Guodong picked up the draft paper and looked at it carefully, his brows furrowing unconsciously.
It's not because it's poorly written, quite the opposite.
That's so well written.
This content goes far beyond the scope of undergraduate advanced mathematics and belongs to the field of mathematical analysis or even functional analysis.
Moreover, I tried several different derivation paths while writing this draft.
This is a trace of a person's independent thinking.
Moreover, the direction of thinking is very accurate. Even those paths that are not feasible have a reasonable starting point and are not just blind attempts.
He taught in the Department of Mathematics at the University of Hong Kong for more than a decade, mentoring countless students.
Two or three out of ten students can derive the recursive relationship of the Gamma function on their own during their junior year.
Perhaps only one in ten people can understand the concept of fractional derivatives.
However, it was able to connect fractional derivatives with constitutive equations and also attempted to independently derive the Laplace transform properties of the Mittag-Leffler function.
Of all the students he taught, not a single one had any.
Who owns this draft paper?
Ye Guodong stood up, picked up the draft paper, and walked towards the management desk at the door.
The girl with glasses was still there, organizing the day's borrowing records.
"Excuse me, do you remember the seat in the second-to-last row by the window this afternoon?"
The girl thought for a moment: "I sat in that spot all day today, and I even borrowed three advanced mathematics books this afternoon."
Which department?
"Student ID starts with 12, freshman, Mechanical Engineering Class 1." The girl pulled up the borrowing record and glanced at it.
Ye Guodong was stunned.
A freshman?
You said this was done by a graduate student and I thought it was too small, but you're telling me it was done by a freshman?
Ye Guodong also took a deep breath, as if he had made up his mind.