The unparalleled talent of the country started from Harbin Institute of Technology

Chapter 30, Numerical Methods for Computing Periodic Solutions

"It's okay, take your time to revise it. It's normal that you can't finish it in one or two days. But you have to remember the requirements I told you, don't make random changes, otherwise I will definitely ask you to revise it again."

The next morning, Wang Duoyu returned to the editor-in-chief's office and took the initiative to report his work to Mrs. Zhou. As a result, the old lady said this to him.

This is even better. He can work more comfortably because he gets paid one dollar a day. After a while, he can just hand over the ending to the old lady for review.

At noon, when Wang Duoyu went to have lunch, he happened to see Fu Jixiang, so he asked him if he had any extra food coupons. The other party actually had some extra coupons and kindly exchanged them with him.

"When you return to Northeast China, if you want to buy these cakes, you can find me if you are short of tickets. I will help you exchange them in advance."

"Okay, thank you Captain Fu, thank you so much!"

With these food coupons, Wang Duoyu could buy pastries from Guixiang Village Food Store. One reason was that he was also greedy, and the other was that he wanted to repay Aunt Qian for the bowl of rice she gave him the night before.

Time passed quickly and it was evening. Just like yesterday, Wang Duoyu strolled to Xidan and planned to eat the famous roast duck there tonight.

I heard that the best restaurant for roast duck is Bian Yi Fang, and some say it is Quanjude, but to Wang Duoyu, they are all the same, because the food in this era is made with real ingredients, and it is rare to find a restaurant that does not take care of it.

Anyway, I have plenty of time, so I can just eat somewhere in the Xidan area. After I submit my manuscript and have time, I will try Bianyifang or Quanjude.

“Wow, so satisfying!”

He ate the entire roast duck, peeled off all the skin, fried half of the remaining duck bones, and made the other half into soup. How could he not be satisfied with eating the roast duck three times like this?

At the age of 22, he had a really good appetite and a really big belly. In his previous life, when he was only 30 years old, he only ate half a bowl of rice every day and not much vegetables because he was afraid of being overweight, and he often forgot to eat dinner.

Even so, his BMI index is still a little bit high. The physical examination report says that he is obese. On the surface, he looks strong, not fat.

After eating and drinking his fill, Wang Duoyu strolled out, strolled around Xidan for a while, bought a few books, and then came to the door of Guixiangcun Food Store to queue up to buy pastries.

There are many people who like to eat pastries, and they have to queue up to buy them.

"Comrade, please help me get a box of salt and pepper triangles, two boxes of mung bean cakes, two boxes of Taishi cakes, and a box of almond cakes."

When Wang Duoyu came, as soon as he opened his mouth, the other customers and salespeople in front and behind him opened their mouths wide. Others bought pastries by the dollar, but he came directly to purchase the goods.

There are six boxes in total, which is quite a lot.

"Comrade, do you have so many tickets? Oh, let me count them. OK, let me confirm with you again that they are one box each of triangular pastries and almond pastries, and two boxes each of mung bean cakes and Taishi cakes, right?" After the salesperson took the money and tickets from Wang Duoyu, he confirmed them again. The other customers queuing behind him stretched their necks and looked at them eagerly. Some children aged seven or eight saw this scene, pouted their lips and expressed their concerns to their parents in a wronged manner: that uncle bought all the cakes!

You know what, it was already very late at this time and the grocery stores were almost closing. There weren't many pastries on the shelves, but the line was still very long.

Wang Duoyu did not hear the children's complaints and wailing. He carried six boxes of pastries back to No. 3 Dafang Hutong. After entering the yard, he went straight to Aunt Qian's house and gave two boxes of pastries to her.

Return a courtesy in return, I believe the other party should be able to understand what he means.

Aunt Qian didn't want the two boxes of pastries, but Wang Duoyu held up the other four boxes of pastries in his hand, saying that he bought too many and he couldn't finish them all by himself. It's summer now and the pastries will easily go bad if left for too long.

This reason is impeccable, and even if Aunt Qian is not happy, there is nothing she can do.

Seeing this, neighbors like Aunt Zhang and Aunt Lu couldn't help but laugh. "Aunt Qian has finally gotten humiliated, hasn't she?" They thought Writer Wang was young and easy to manipulate, but hey, he's actually quite skilled and slippery. It seems like there's no way they can get him to do anything about it.

After closing the door, Wang Duoyu glanced at the manuscript on the table. It was the same as it was yesterday. He had not touched it at all. There was no need to read Han Chongqian's manuscript. It was definitely a waste of time.

Using candles, kerosene lamps, and electric lights, Wang Duoyu began his work for the evening, continuing the math paper he had left unfinished last night:

Since there is inherent instability in the selection of initial values ​​when solving the initial value problem, it is better to use multiplicate shooting, that is, dividing the original area (0, 1) into many sub-intervals, and solving the initial value problem on these sub-intervals will have better results.

The most clumsy method is to directly use any numerical integration method for solving the initial value problem of ordinary differential equations. For example, the simplest one is the Euler broken line method. Starting from any initial value that is not an equilibrium point, calculate forward one time step after another. It would be best if the calculation results of each step can be dynamically displayed on the screen.

Once a periodic trajectory is discovered, the calculation results are output and the numerical values ​​carefully observed. When approximately equal values ​​appear repeatedly, the periodic solution and the periodic size can be determined. This approach is obviously somewhat blind, resulting in a waste of computational time and requiring some manual intervention. While a bit clumsy, it effectively produces the result, but it is certainly not one of the methods we will be studying.

The paper written by Wang Duoyu is about a systematic description of the numerical method for calculating periodic solutions of autonomous systems of ordinary differential equations, because periodic oscillations are often encountered in problems such as physics, chemistry, biology, electronics, and engineering technology, and this phenomenon is also common in nature.

In mathematics, finding periodic solutions to differential equations is an old and difficult problem. As early as 16, David Hilbert's th problem was about determining the existence of periodic solutions to differential equations.

However, there has been no progress so far. It is not because no mathematicians have tried to decipher it in the past half century, but because the difficulty is far beyond imagination.

Especially after the 1960s, with the popularization of computers and the development of computer technology, people's research ideas and methods have changed. In the study of practical problems, such as the Brussels oscillator in biochemistry, people are not interested in the theoretical proof of the existence of periodic solutions, but are more concerned about the position, shape and size of the periodic solutions.

Therefore, Wang Duoyu's research on computational methods is particularly important.

(End of this chapter)