Omnipotent Data

Chapter 413

Arbitrary real-order or complex-order integrals and derivatives are usually called fractional calculus, and fractional calculus has applications in viscoelastic mechanics, statistics and stochastic processes, dynamical system control, and optical signal processing. Theoretical connotation.

Chali's research group is using continuous functions and Banah's compressive image theory to study the problem of boundary value existence of nonlinear differential equations with fractional derivatives.

There was no need for a deep understanding. Cheng Nuo only needed to know a general idea, and he could calmly deal with any problem.

Cheng Nuo flipped through the pages without haste. Although Cheng Nuo didn't deliberately speed up, in the eyes of the three fellow members of the research team in Chali, it was as if he had seen a ghost.

He, is he really looking at it seriously, instead of trying to fool us?

Thinking of this, the boy's eyes fell on Chali again, full of sorrow. Meaning, Is this incomprehensible guy invited by your kid?

Charlie wanted to cry again but had no tears.

Time passed by minute by minute, and ten minutes later, Cheng Nuo put the stack of A4 papers back on the table and said with a smile, I just read the content of your research from beginning to end, if I guess right If so, you should have run into trouble with the boundary value analysis of differential equations based on the Banah compressed image at the end, right?

The three of them all set their gazes on Chali.

Why me again? !

Chali rolled his eyes and said speechlessly, Don't look at me, I just told the Great God to ask him for a favor, and didn't mention the specific problems we encountered. If you don't believe me, you can ask the Great God?

Cheng Nuo took a few blank draft papers from one side of the table, and said, It's true that Chali didn't mention the specific content to me. But it's not hard to guess, your research report, at the final margin Analyzing that part, a large part of the proof process is missing, and I don’t think it was intentionally omitted.”

The boy nodded, approving Cheng Nuo's words, Indeed, in this part, although we know what the desired result is, we have been thinking about the specific process for several days, but we haven't come up with any results. .”

Cheng Nuo's performance just now has changed the boy's impression of Cheng Nuo.

This junior doesn't seem so ordinary!

So he tentatively asked,

Now that you know the trouble we're in, is there a way to solve it?

Cheng Nuo smiled, raised a finger and waved it, and slowly said two words, It's not difficult!

Chali's face brightened.

The corner of the boy Rocky's mouth twitched.

Why do I have a feeling of watching the live broadcast of the king?

Rocky, who had learned the truth, waited quietly for Cheng Nuo to speak.

I think that a large part of the reason why you have been inkling on this issue for so long is that you have used the wrong method.

Using the wrong method?

Yes! Cheng Nuo tapped the pen cap on the table lightly, Let me ask you a question first, what is a nonlinear differential equation with fractional derivatives?

The boy subconsciously replied, The nonlinear differential equation of the fractional derivative can be summarized by two formulas: f-(z)+(D+Dt)(z)-f(x, (z)), z∈( 0, 1), and y(0)=0=y(1).”

Cheng Nuo nodded in satisfaction, That's right. But do you still remember this fractional derivative and its existence condition?

existential conditions? Rocky was taken aback.

Cheng Nuo explained, When the Dirihlet boundary value is constant, there will be such an existence condition for the differential equation of the fractional derivative.

Cheng Nuo picked up the pen and wrote on the paper, (D0+y)(x)=(D1-y)(x), (D1-y)(x)=(D-y)(x).

The boy looked at the formula written by Cheng Nuo and fell into deep thought.

But Cheng Nuo didn't give him time to think. He is not the teacher of these people, and there is no need to follow their rhythm.

He went on to elaborate on his point of view, The way you try to prove that there is a unique solution to the boundary value of the nonlinear differential equation of the fractional derivative is to derive the result directly through the derivation of the formula and use Banah's compressive image theory.

But considering the two existence conditions I just wrote, this method is 100% wrong! Cheng Nuo said in a firm tone.

Then... the boy couldn't help but speak.

Cheng Nuo lowered his hands and said with a smile, Student, don't be in such a hurry, calm your aura, calm your aura. I'll explain the correct method of proof.

Cheng Nuo first wrote down three key words on the draft paper: Green function, Lipshitz compression condition, and Banah space.

My proof method is very simple. In fact, as long as you understand my three key words, it's only a matter of time before you understand. But in order to save time for both parties, I'll just deduce it directly. Cheng Nuo's tone was very flat, and he reasoned. The thoughts in my mind are like lecturing, writing while speaking.

The first step is to use the perturbation method combined with the Green function to further study the boundary value problem of differential equations with left and right fractional derivatives, and to give the boundary value problem of homogeneous differential equation Dirihlet, then - u(x)=0, x∈( 0, 1), y(0)=0=y(1).

Assuming that the function f(x, u) is continuous on [0, 1[×(+∞, -∞)-(-oo, +oo), then the homogeneous boundary value problem can be described as -u''( x)=f(x, u(x)), x∈(0, 1), u(0)=0=u(1). Where u(x) represents the solution of the boundary value problem.

…………

...According to the above theorem, it can be obtained that the boundary value problem has a unique solution on the continuous function space C[O, 1]. It can be known from the known conditions that on the continuous space C[O, 1], the operator T satisfies the Lipshitz compression Conditions, and according to Banah’s compressed image theory, operator T has a unique fixed point Y∈[o, 1] in space, which conforms to…”

...Through the above definitions and theorems, it can be proved that there is a unique solution to the boundary value of nonlinear differential equations with fractional derivatives!

While talking and writing, Cheng Nuo spent nearly 20 minutes deriving the problem of proving the unique solution of the marginal value to the four in Chali from beginning to end.

Except for Chali, who had already developed immunity, the other three were all in a state of brain failure.

Is this... the end? !

At the beginning, the four of them studied hard for two days and two nights, but they didn't figure out why.

But when it came to Cheng Nuo, why did it take twenty minutes?

Is this the difference between genius and mediocrity? But it's too realistic, right?

Mickey looked at Zhali with a bitter expression, staring at Cheng Nuo who was sitting in the chair with a calm expression, feeling mixed emotions in his heart.

Missed it! I was slapped in the face!

He didn't expect that the rare genius in the legend would actually be met by him!

He walked up to Chali and asked bitterly, Charli, what's the name of your friend? Why have I never heard of such a person in our college?

Charlie shrugged, It's normal that you haven't heard of it, because people like the Great God are no longer interested in making waves in the school. You know the Cheng Nuo Theorem that has become popular recently, it was proposed by the Great God!

hiss!

Mickey was terrified!