Chapter 413

Integral and derivative of any real or complex order is often referred to as fractional calculus, and fractional calculus has applications in viscoelastic mechanics, statistical and stochastic processes, dynamical system control, and optical signal processing.

Tsari and his group used continuous functions and Ba nac h compressed image theory to study the problem of the existence of solutions to the boundary values of nonlinear differential equations of fractional derivatives.

There is no need to do too in-depth understanding, Cheng Nuo only needs to know a rough idea, and he can calmly deal with any problem.

Cheng Nuo turned back page by page without hurrying, although Cheng Nuo did not deliberately speed up, but in the eyes of the three members of the same research group in Chali, it was like seeing a ghost.

He, is this really looking at it seriously, not perfunctory us?

Thinking like this, the boy's gaze fell on Tsari-chan again, full of grievances. Meaning, "This unobtrusive guy was invited by your kid?"

Chari wanted to cry again without tears.

Time passed minute by minute, and after more than ten minutes, Cheng Nuo put the stack of A4 paper in his hand back on the table, and said with a smile, "I just read the content of your research from beginning to end, if my guess is correct, you should be in trouble with the boundary value analysis of differential equations based on Banach compressed images at the end, right?"

The three of them all looked at Zari classmates.

Why me again?!

Chari rolled his eyes and said speechlessly, "Don't look at me, I just told the great god to ask him for a favor, and I didn't say anything about the specific problems we encountered." If you don't believe me, ask the great god?"

Cheng Nuo took a few blank pieces of scratch paper from the table on the side and said, "Chali really didn't mention the specific content to me. However, it is not difficult to guess that your research report, in the final boundary value analysis, is missing a large part of the proof process, and I don't think it was deliberately omitted. ”

The boy nodded, which was regarded as Cheng Nuo's words of approval, "Indeed, in this part, although we know what the desired result is, but the specific process, we have been thinking about it for several days, but we have not come up with a result." ”

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

This junior brother doesn't seem so ordinary!

So he asked tentatively, "Now that you know the trouble we have, is there a way to solve it?"

Cheng Nuo smiled, raised a finger and swung it, and slowly spit out two words, "It's not difficult!"

Tsari's face was overjoyed.

The corner of the boy's mouth twitched.

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

Rocky, who knew the truth, waited for Cheng Nuo to speak.

"I think a large part of the reason why you've been inking on this issue for so long is because you're using the wrong method. ”

"Wrong way?"

"That's right!" Cheng Nuo tapped the desktop lightly with the cap of his pen, "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 fractional derivatives can be summarized by two formulas: f a (z) + (D + Dt) (z) a f(x, (z)), z∈ (0, 1), and y(0) = 0 = y (1). ”

Cheng Nuo nodded with great satisfaction, "That's right." But do you remember that this derivative, and its existential conditions?"

Existential conditions?Lodge was stunned.

Cheng Nuo explains, "If the boundary value of Di ri chle t is constant, there will be such an existential condition for the differential equation of the order 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 line of formula written by Cheng Nuo and fell into deep thought.

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

He went on to elaborate on his point, "The way you are trying to prove that there is a unique solution to the boundary values of nonlinear differential equations of fractional derivatives is to derive the results directly from the formula, using the Ba nac h compressed image theory. ”

"But judging from the two existential conditions I wrote about just now, this method is 100% wrong!" Cheng Nuo said in a firm tone.

"That's ......," the boy couldn't help but speak.

Cheng Nuo pressed down with his hands and said with a smile, "Classmate, don't be in such a hurry, stabilize your aura, stabilize your aura." The correct way to prove it, I'll talk about it right away. ”

Cheng Nuo first wrote down three key words on scratch paper: Green function, Lipschitz compression condition, and Banac h-space.

"My proof method is very simple, in fact, as long as you understand my three key words, it is only a matter of time before you understand, but in order to save the time of both parties, I will directly deduce it. Cheng Nuo's tone was very flat, and after sorting out the ideas in his mind, it was like a lecture, writing while speaking.

"In the first step, the perturbation method combined with the Gr een function is used to further study the boundary value problem of the differential equation with the left and right fractional derivatives, and the homogeneous differential equation Di r i chl e t is given the boundary value problem, then a u ( x) = 0,x∈(0,1), y(0)=0=y(1). ”

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

............

“...... From the known conditions, it can be seen that on the continuous space C[O, 1], the operator T satisfies the Li ps chi t z compression condition, and according to the Ba nac h compression image theory, the operator T has a unique fixed point Y ∈c[o, 1] in space, which conforms to the ......"

“...... Through the above definitions and theorems, it can be proved that there is a unique solution to the boundary value of the nonlinear differential equation of the fractional derivative!"

While talking and writing, Cheng Nuo spent nearly twenty minutes to deduce the problem of proving the unique solution of the boundary value to the four of Chali from beginning to end.

Except for Tsarri, an existence that has developed immunity, the other three are all in a state of brain failure.

That's ...... It's over?!

I think that at the beginning, the four of them studied for two days and two nights, but they didn't come up with a reason.

But when it comes to Cheng Nuo, why does it become a matter of twenty minutes?

Is this the gap between genius and mediocrity? But it's too special, isn't it?

Mickey looked at Chali with a bitter face, staring at Cheng Nuo who was sitting on the chair with a calm expression, and his heart was mixed.

I've been slapped in the face!

He didn't expect that the legendary genius that was rare in a hundred years was really met by him!

He walked up to Chali and asked bitterly, "Chari, what's the name of this friend of yours? Why have I never heard of this person in our academy?"

Charri shrugged, "It's normal that you haven't heard of it, because the great god has no interest in making a fuss in the school. I know the Cheng Nuo theorem that has become popular recently, it was proposed by the great god!"

Hiss——!

Mickey is terrified!