1. clarity and novelty (in his view, a clear question can arouse the interest of peers, and its novel performance can stimulate the excitement of people's research. )
2. Methodology and Series (To study a problem, he always prepares a batch of tools to form a method, so as to extend the research problem to all corners and get a series of results. )
3. Ideality and essentiality (he appreciates the saying that "a good mathematical thought is better than ten methods", and thinks that a mathematical thought reveals the essence of the problem, just like a strategy in mathematical operation, guiding people to the other side of victory. )
Lu Jianke followed the above principles, sought and solved one problem after another, and made great achievements. Whenever he devotes himself to a problem, he always tries his best, perseveres and never gives up until he reaches his goal. Some questions are very important and difficult, and they have been puzzling for several years, but he is obsessed with thinking and finally has to break through. On the road to solving problems, he never sticks to the rules, always finds another way and gives full play to his great creative talents.
Lu Jianke's academic achievements and viewpoints can be roughly divided into four aspects:
(i) Boundary value problems of analytic functions;
(ii) Singular integral equation theory;
(3) Numerical theory of singular integral equation;
(4) Mathematical theory of plane elasticity.
More than 65,438+000 papers have been published in these works, among which (i) and (iii) the monograph "Boundary Value Problems of Analytic Functions" was published in 65,438+0,987, (iv) "Complex Variable Methods of Plane Elasticity" and "Periodic Problems of Plane Elasticity Theory" (co-authored with Cai Haitao). Two independent works have been translated into English by Singapore World Press, and the cooperative monograph has been translated into English by Gordan &; Gap Publishing House published an English translation.
The work in the four directions has formed an organic whole from theoretical research to practical application in this field, and it is hard to say which aspect is more important. In fact, his theory and application go hand in hand, and both are successful. Just give a few examples to illustrate its academic practice. 1962, Lu Jianke published a paper entitled "Compound Boundary Value Problem", which was the first one in his own research on boundary value theory of analytic functions. This first work shows his extraordinary skills and far-reaching influence in boundary value research, and this article is praised as a masterpiece by successors. The research on boundary value problems of analytic functions originated from the work of riemann sum Hilbert, a mathematician in the 9th century. In the forties and fifties of the 20th century, the Georgian School of the Soviet Union carried out its work in full swing, which pushed the research in this field to a period of prosperity and maturity. Professor MycXeлишвили, an academician of the Soviet Academy of Sciences, is the founder of the Georgian school. He collected all the achievements up to that time, plus many of his own ideas, and wrote a masterpiece "Singular Integral Equation". This book is a classic of boundary value research, which has been published three times and won the national award of the whole Soviet Union. The master's descriptions of various classical boundary value problems are concise, and their solutions are even standardized, which makes it difficult to broaden the work in this field. However, he did not find the so-called compound boundary value problem today. It is at this point that Lu Jian can show his profound insight. He further questioned the classical boundary value problem, that is, whether it is possible to find such a partition holomorphic function on a multi-layer partition region, which satisfies Riemann condition on some boundaries and Hilbert condition on others. Road view can be called a compound boundary value problem. Today, the form of this problem has evolved in various ways.
This complex boundary value problem cannot be solved by classical methods. Lu Jianke skillfully put forward a transformation, which transformed the compound boundary value problem into a classical problem by eliminating some conditions. So later colleagues called it "exclusion method". Since then, this method has been widely spread, and people have successfully applied it to various composite boundary value problems, so that the word "elimination method" has become a self-evident term in today's academic conference on boundary value problems.
Because of its obvious importance, in 1964, the original Journal of University Natural Science (Mathematics, Mechanics, Astronomy Edition) reprinted Lu Jianke's work. Then, China Science was translated into English and reprinted in 1965.
Four years later, another interesting thing happened. H.C. Rogorena (рогоина), a scholar in the former Soviet Union, had no idea that China scholars were pioneers and published similar research, but they demanded more and got less results. Compared with the two, the problems that the road view can solve are more universal and profound.
Lu Jianke's works
Direct solution of singular integral
By 1965, Lu Jianke has written 10 papers, and his work has covered all directions. At that time, it was in the prime of life and was in an excellent period of scientific research. He began to think about an extremely difficult but capable subject, which is called the direct solution of singular integral equation today. By 1960s, the theory of singular integral equation had been quite rich and complete, but in general, it was very difficult to really solve a singular integral equation. This is undoubtedly a soft spot for a highly applied subject. It can be predicted that if we take a step back and strengthen the input conditions, it will be possible to solve the singular integral equation.
Just as he put his ideas into research, in the summer of 1966, the "Cultural Revolution" began, and the Chinese nation was ruthlessly plunged into disaster. At that time, Lu Jianke had won considerable fame because of his outstanding achievements. A hat of reactionary bourgeois academic authority was naturally not spared, and legitimate academic research was terminated. He entered the "learning class" with such "charges". Children go to the countryside to jump the queue and settle down, and a complete family of four will be punished. At that time, let alone research, even basic personal freedom became a problem. The study of direct solutions of singular integral equations has been put on hold for 10 years. However, almost at the same time that Lu Jianke planned this topic, it may be earlier. An American scholar, A.S. Peters, and another scholar, K.M. Case, also noticed the same problem. They first opened the prelude to this research. Later, another former Soviet scholar, C. CaMKo, joined the work. Most of these works were published from the mid-1960s to the early 1970s, when China was in the midst of the "Cultural Revolution", and domestic scholars simply did not and could not get any information in this regard.
1975, Lu Jianke was deeply touched by the work of Petes and others. He carefully studied the work of these scholars, and found that although there were some shortcomings, they did open the clue of direct solution, but their work was too principled and lacked effective ways to realize the plan. In the final analysis, except for simple cases, they did not find the closed form of the solution (especially the solvable conditions). Therefore, the real "direct" solution must also tear down a "wall". It is for this reason that people give up the idea of a particularly effective solution. After the work of these people, the research in this field was silent for several years.
1975, Lu Jian could restart the study of direct solution of singular integral equation. He soon found that the essence of the problem was that singular integral equation must be a simple functional equation, and the crux here focused on how to get rid of the integral sign, which undoubtedly required a powerful tool to calculate singular integral. Soon, this tool came into being in Lu Jianke's work, which is the generalized residue theorem. The classical residue theorem tells us that to calculate the contour integral of analytic function, we only need to calculate its residue, but the singularity cannot fall on the contour. But the singular integral just has a singularity on the contour. At this time, Lu Jian can introduce the tension of the point, that is, the ratio of the internal angle between the point and the contour to the fillet, which vividly depicts the degree to which the boundary point faces the internal domain; Therefore, the residual amount at this point is calculated by the usual method and then multiplied by the tension. After this treatment, the residue theorem is generalized, and even the case of higher-order singular integrals has similar results.
By applying the extended residue theorem, a large class of singular integral equations with certain analytic coefficients and kernel density can be successfully transformed into simple functional equations, and then the equations and solvable conditions can be transformed into linear equations, and their solutions can be obtained directly. This is the principle and way that road view can point out for the direct solution of singular integral equations.
Lu Jianke's academic activities were imprisoned for 10 years, but when he went into battle in disguise, he showed great research vitality, which really blew a new wind for the scientific research atmosphere of Wuhan University at that time. Everyone admires his profound research skills, in fact, he also benefited from some mental thinking during the Cultural Revolution. He once revealed that in those years of imprisonment, he often thought about some problems in order to pass the boring time. This shows an upright intellectual's persistent pursuit of his career.
After Lu Jianke's work, the research on direct solutions of singular integral equations became active again, and various works followed, including convolution, displacement, and various periodic kernels. Of course, these tasks have their own skills and achievements. However, there is no doubt that the principles and methods followed are deeply branded with profound thoughts. After 1976, it was a sunny day for scientists in China. Lu Jianke's creation also reached a climax. He devoted considerable energy to the problem of periodicity, mainly the problem of double periodicity and double quasi-periodicity, and the problem of single periodicity was solved as early as the 1960 s. Lu Jianke's research on the cycle problem regularly follows three stages:
(1) Study on various boundary value problems in different periods;
(2) Study on singular integral equations with different periods;
(3) Research on various problems of periodic elasticity.
The motivation of these studies is practical problems. To solve various periodic plane elastic problems, it is necessary to establish a set of corresponding boundary value theory and singular integral equation theory. Lu Jianke is particularly good at choosing topics according to actual needs, so this kind of research has become one of his centers.
Periodic research is a hot topic, and many scholars in the world are engaged in this work. As far as time is concerned, Lu Jian may be in the stage of connecting the past with the future, but he is a pioneer who has made serious research on this issue and achieved systematic results. He mended many loopholes in previous studies, broadened the research field, added many new definitions, methods and achievements, and persistently cast others' and his own thoughts into a unified and complete theory.
Lu Jianke's research on the one-week problem was basically completed in the early 1960s, and the work of foreign scholars was earlier than him. However, from the application point of view, the results of general research and assembly to specific problems are not satisfactory. Lu Jianke continued this work. He started from the single-period Riemann boundary value problem and went step by step to various applications of single-period plane elasticity. 1963, he wrote a paper "Periodic Riemann Boundary Value Problem and Its Application in Elasticity", which was published in the Journal of Mathematics, and the full text was 46 pages long, which was really rare in this journal.
In the early 1960s, although some views on the dual-cycle problem had been formed, it was not until after the "Cultural Revolution" that Lu Jianke began to study this problem in depth. In 1950s, the Soviet scholar лиибрикова) studied the double cycle problem. Lu Jian can read this scholar's early works. He found that it may be because of too much imitation of the single loop situation that scholars neglected when choosing kernel functions to construct canonical functions. It is considered that since there is no single-pole elliptic function, we should either give up the periodicity requirement or keep double periodicity and allow the other pole when choosing the kernel function. He chose the latter, because it can not only correct her work, but also be easily extended to the situation of open arc segment that has not been studied by predecessors, and can also be used for reference in the subsequent study of bi-quasi-periodic problems, killing two birds with one stone. Lu Jian's attitude towards problems is often the same. He carefully examined every proof and every choice until he decided to choose a method that "this principle can be popularized and is most useful for further research".
After establishing boundary value theory and singular integral equation theory, Lu Jian can use these theories to solve many plane elastic problems. Of course, there is a very difficult but essential link that needs to be refined into a mathematical model, which is still full of mathematical methods and skills.
From 1980 to 198 1, he came to the United States as a visiting scholar and continued his research work at the University of Texas. He is willing to talk and discuss with colleagues in this circle to further enrich his achievements in elasticity theory and fracture mechanics. During 1 year, he published four papers in American magazines.
Mechanical quadrature of singular integral
During 198 1 visit to the United States, Lu Jianke also turned to the mechanical quadrature of singular integrals. All this is surprising. In fact, in order to shape his research into a complete whole, he has been interested in the numerical theory of singular integral equations for more than one day. The numerical solution of singular integral equation is a bridge between boundary value theory and practical application. But for a long time, mathematicians have not made much achievements in this respect; It is not that this kind of research is trivial, on the contrary, there are many thorns on the way forward. In 1950s, Academician A.H. krylov of the Soviet Union (Kpылов) put forward a preface for Academician Muskhelishvili's masterpiece "Several Basic Problems of Mathematical Elasticity", hoping that the second edition of the book can give numerical solutions. In later versions, he regretted that he could not realize the hope of developing numerical solutions. Since then, Lu Jianke has sprouted the idea of studying numerical solutions of singular integral equations. He noted the work of Georgian schools in this field, but failed to collect appropriate information. Because many of the school's works are published in local magazines in Georgia, it is really difficult for our country to get such magazines.
Since the 1970s, great progress has been made in the research of numerical methods for singular integral equations in the west, while the research in this field in China is almost blank. 198 1' s trip to the United States brought Lu Jianke an opportunity to start innovative research. He eagerly read and collected information, preparing to return to China to guide the students' work. In fact, during his visit to the United States, he had already started his first research. He started from the first line problem, and first engaged in the mechanical quadrature of singular integrals. He believes that it is not important to establish all kinds of specific formulas one by one, and there should be a unified idea for the establishment of all kinds of quadrature formulas. Soon, he proposed to establish a connection between the numerical quadrature of singular integral and the classical numerical quadrature of ordinary integral. This idea is undoubtedly of great significance, because the classical quadrature theory founded by Gauss and Markov is quite complete and rich, and if it can be cited, it will naturally get twice the result with half the effort. Subsequently, he created the separation singularity method and successfully realized his idea. By separating the singularity, he transformed the quadrature of singular integral into the classical quadrature, and the remaining problems were some technical treatments, which he solved. After returning home, the first doctoral student he supervised continued this work. Carrying forward his thinking method, the doctor proposed and established many kinds of quadrature formulas of singular integrals under very general conditions, and assembled them on some common weight functions, forming a large number of specific and applicable formulas; Since then, along with these achievements, many new concepts and arguments have been put forward for the numerical solution of the whole singular integral equation, and good work has been done.