Chapter 410

Chapter 413

any integral and derivative of real or complex order is usually called fractional derivative. Fractional calculus is widely used in viscoelastic mechanics, statistical and stochastic processes, dynamic system control and optical signal processing, and has rich theoretical connotation.

In this paper, we use the theory of continuous functions and Banach contractive mapping to study the existence of boundary value solutions for Nonlinear Differential Equations with fractional derivatives.

Do not need to do too deep understanding, Cheng Nuo only need to know a general, can calmly deal with any problem.

Cheng Nuo turned back page after page without haste. Although Cheng didn't deliberately speed up his speed, he seemed to have seen a ghost in the eyes of the three members of the same group.

He, is this really serious, not perfunctory us?

Thinking like this, the boy's eyes again fell on the Chali students, full of sorrow. It means, "this out of tune guy is your boy who invited him here?"

Once again, zari wanted to cry without tears.

As time goes by, more than ten minutes later, Cheng put the pile of A4 paper in his hand back on the desktop and said with a smile, "I just read your research content from the beginning to the end. If I guess well, you should be in trouble at the end of the differential equation boundary value analysis based on Banach compressed image?"

The three eyes fell on the students in Chali.

Why me again?!

"Don't look at me. I just told the God to ask him to help. I didn't say the specific problems we met. If you do not believe, ask the great God

Cheng Nuo took a few blank drafts from one side of the table and said, "zari 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 part, lacks a large part of the proof process. I think it should not be deliberately missed. "

The boy nodded, which was recognized by Cheng Nuo, "indeed, in this part, although we know what the desired result is, we have thought about the specific process for several days, but we have not come up with a result."

Just Cheng Nuo's performance has made the boys' impression of Cheng Nuo better.

This student, it seems, is not so common!

So he asked tentatively, "since you know the trouble we are in, is there any way to solve it?"

Cheng Nuo smile, erect a finger, swing, slowly spit out two words, "not difficult!"

Chali's face was happy.

The boy rocky took a puff from the corner of his mouth.

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

Rocky, the truth, is waiting for Cheng Nuo to speak.

"I think a large part of the reason why you have been writing on this issue for such a long time is due to the wrong method."

"In the wrong way?"

"Yes Cheng Nuo tapped the table top with his pen cap. "Let me ask you a question first. What is the nonlinear differential equation of fractional derivative?"

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

Cheng Nuo nodded with satisfaction, "that's right. But do you remember the fractional derivative and the conditions for its existence? "

Existence condition? Rocky was stunned.

Cheng Nuo explained, "if the Dirichlet boundary value is fixed, there will be such an existence condition for the differential equation of fractional derivative."

Cheng Nuo took up his 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 formulas written by Cheng Nuo and fell into deep thought.

Cheng Nuo did not give him time to think. He is not a teacher of several people, there is no need to follow their rhythm.

He went on to expound his point of view, "the way you try to prove the existence and uniqueness of the boundary value of the nonlinear differential equation with fractional derivative is directly through the derivation of the formula and using the theory of Banach contractive mapping to get the result."

"But according to the two existential conditions I wrote just now, this method is 100% wrong!" Cheng Nuo said in a determined tone.

"Well..." The boy couldn't help speaking.

Cheng Nuo's hands pressed down and said with a smile, "classmate, don't be so anxious. Steady gas field, stable gas field. I'll talk about the right way to prove it. "

Cheng Nuo wrote three key words on the sketch paper: Green function, Lipschitz compression condition and Banach space.

"My proof is very simple. In fact, as long as you understand my three key words, it is only a matter of time to understand them. However, in order to save time for both sides, I will deduce them directly." Cheng Nuo's tone is very insipid. After sorting out the thoughts in his mind, he is like giving a lecture and writing while talking."In the first step, the boundary value problem of differential equation with left and right fractional derivative is further studied by using disturbance method and Green function. The Dirichlet boundary value problem of homogeneous differential equation is given, then a U (x) =0, X ∈ (0, 1), y (0) =0=y (1)."

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

…………

“…… The boundary value problem can be obtained by the above theorem. It is known that operator T satisfies Lipschitz compression condition in continuous space c[o, 1]. Then according to Banach compressed image theory, operator T has a unique fixed point y ∈ c[o, 1], which is in conformity with... "

“…… The above definition and theorem can prove that there is a unique solution to the boundary value of the nonlinear differential equation of fractional derivative! "

As he said and wrote, cheno took nearly 20 minutes to prove that the boundary value only solved the problem to the four Chari people from the beginning to the end.

Except for the existence of the already immune system, all three were in the state of brain emergency.

That's what It's over?!

Think at first, they four blast liver blast seed study two days and two nights, also did not study out a so come.

But when we got to cheno, how could it have been 20 minutes?

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

Mickey looked at the inspector with a bitter face, staring at Cheng Nuo, who sat in his chair, and had five different tastes in his heart.

Look at it! I'm being beaten up!

He was in the unexpected, that legend of the century of the rare talent, but also really by him to meet!

He went to chanei and asked bitterly, "what is your friend's name, zari? Why have I never heard of our college and the number one? "

"It's normal that you haven't heard of it, because the gods are no longer interested in getting into the school," he shrugged. The recently fire of the cheno theorem, you know, is the God proposed! "

Hissing -!

Mickey was horrified!

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