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Fractional Differential Equations in the Model of Vocational Education and Teaching Practice Environment

Publicado en línea: 15 Jul 2022
Volumen & Edición: AHEAD OF PRINT
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Recibido: 16 Feb 2022
Aceptado: 13 Apr 2022
Detalles de la revista
License
Formato
Revista
eISSN
2444-8656
Primera edición
01 Jan 2016
Calendario de la edición
2 veces al año
Idiomas
Inglés
Introduction

Most college students feel great pressure on studying mathematics. Failure to understand leads to low enthusiasm for learning. Students even changed their majors, were tired of studying and dropped out. This is because the methods and ways of thinking required for studying mathematics at the university level are completely different from those in middle schools [1]. The calculus at the university level has a large span, and its difficulty is qualitatively overstepped.

In the 1960s, American scholar Wilder first viewed that “mathematics is a kind of culture,” and then a wave of discussion and research on mathematics culture was launched at home and abroad. In 1980, Professor Morris Klein of the Courant Institute of Mathematics at New York University wrote in the preface of the book “Mathematics in Western Culture”: “Mathematics has always been a major cultural force in Western civilization. Almost everyone knows that mathematics has extremely important practical value in engineering design [2]. The most important thing is that as a precious and unparalleled human achievement, mathematics is pleasing to the eye and provides aesthetic value. At least it can be compared with any other cultural category. Comparable”. In 2005, the famous mathematician Academician Li Daqian pointed out at the “University Mathematics” course report forum: “Mathematics is an advanced culture and an important foundation of human civilization. Its generation and development play an important role in the process of human civilization. It plays a pivotal role.” Subsequently, many scholars put forward the meaning of mathematics culture. For example, Mr. Huang Qin’an's understanding of mathematics culture is as follows: “Mathematics culture is a dynamic system with specific functions that take mathematics science as the core and the relevant cultural fields radiated by mathematical thoughts, spirit, methods, content, etc., as organic components.” Teacher Gu Pei, a famous scholar of Nankai University, gave a more detailed explanation of mathematics culture: mathematics culture is the thought, spirit, method, and viewpoint of index science and its formation and development. In addition to the above connotations, mathematical culture also includes mathematicians, the history of mathematics, the beauty of mathematics, mathematics education, the human element in the development of mathematics, the relationship between mathematics and society, the relationship between mathematics and various cultures, and so on.

Although the description and understanding of mathematics culture are not the same, it has become a consensus reached by colleagues in mathematics circles to regard mathematics as a culture.

Through the scholars mentioned above and experts' expounding on mathematics culture, we can deeply realize that mathematics culture can be divided into broad and narrow senses. In a broad sense, mathematical culture refers to the relationship between mathematicians, educators, and all the humanities in the development of mathematics and the relationship between mathematics and various cultures [3]. Mathematics culture is the thought, spirit, and method contained in index science in a narrow sense. Compared with mathematics, the content of mathematics culture is richer, and the connotation is more profound. It is a high-level summary and embodiment of mathematical knowledge and ideas, mathematical ability, and quality.

The importance of penetrating mathematics culture in mathematics teaching
Can inspire students to be enterprising and patriotic

In the middle school stage, students' main goal is the college entrance examination because of the “baton” of the college entrance examination. Students learn whatever they take in the college entrance examination, and learning is often passive [4]. In addition, the different ways of thinking between middle school mathematics and university mathematics have caused students to have a negative attitude towards learning mathematics. In teaching “Mathematical Analysis,” teachers appropriately introduce some famous mathematician stories into the classroom. Introduce how they faced and solved some difficult problems. This can enhance students' determination and confidence in overcoming difficulties and stimulate their enthusiasm for learning. For example, the famous mathematician Zhang Yitang successfully proved the weak version of the twin prime conjecture for the first time in 2013. In the section on the limit of the sequence of numbers, we will mention the sentence in “Zhuangzi-Tianxia Pian,” “A hammer of a foot, half of the day will be inexhaustible.” This is the embryonic form of the concept of limit, which shows the wisdom of ancient Chinese thinkers. It is further associated with Mr. Chen Xingshen, a famous modern and contemporary mathematician. In his later years, he devoted himself to promoting the development of Chinese mathematics and founded the Chern Institute of Mathematics at his alma mater Nankai University [5]. At the same time, he contributed to the convening of the International Congress of Mathematicians held every four years in Beijing, China, in 2002 (the first time the department was held in a developing country). Mr. Chen has trained many famous mathematicians, including Liao Shantao, Wu Wenjun, Qiu Chengtong, Zheng Shaoyuan, and Li Weiguang. Qiu Chengtong was the first Chinese to win the Fields Medal of the International Mathematical Union and the second Chinese to win the Wolf Prize after Chern. Through these real examples, students' patriotism can be well stimulated, and national self-confidence can be enhanced.

Can improve teachers' mathematics cultural literacy

“Teachers need a bucket of water to give students a bowl of water.” This is a traditional saying in the education circle. With the progress of society and the development of science and technology, higher education is also developing rapidly. Just staying in a bowl of water and a bucket of water can no longer meet the needs of the development of society and higher education. Now we have reached a situation where “the teacher wants to have a bucket of water, and he should also point his students to a river; the teacher is a river, and he wants to guide the students to the sea.” Teachers must pursue and improve their professional development. The teacher's “water” should be a tireless stream that can be constantly replenished with spring water [6]. If you want to integrate the elements of mathematics culture into the teaching activities, you must select the teaching content in advance. Teachers need to understand which mathematical knowledge contains rich mathematical culture, which mathematical knowledge contains rich mathematical values, and which has the function of cultural transmission. Through this kind of thinking and learning, the teacher's cultural literacy will improve.

Can cultivate students' exploration and innovation ability

The ability to explore and innovate is the soul of national progress and the core of economic competition. The competition in today's society is not so much a competition of talents but a competition of human creativity. In today's rapid development of science and technology, innovation awareness and innovation capabilities have increasingly become the most important determinants of a country's international competitiveness and international status. As the main group for the inheritance and development of human history and culture, contemporary college students must cultivate their spirit of exploration and innovation [7]. For example, when calculus is mentioned in the “Mathematical Analysis” teaching process, fractional calculus is appropriately introduced to guide students to look up information. Only in this way can we further analyze their similarities and differences to get some useful conclusions. For example, when talking about the Gamma function, students will be guided to explore some of its basic properties and related equations and take further proof. If you can successfully prove some of this will enhance students' confidence in mathematics learning. This will gradually cultivate students' exploration and innovation abilities.

Ways and strategies to infiltrate mathematics culture in teaching
Introduce relevant knowledge of the history of mathematics in the classroom

There are many mathematical terms in the “Mathematical Analysis” course, and most of them have related historical backgrounds. For example, the term calculus can be introduced into its development process. In the 3rd century BC, the ancient Greek mathematician and mechanic Archimedes studied and solved the arcuate area under the parabola, the sphere and the spherical crown, the area under the spiral, and the volume obtained from the rotating hyperbola [8]. The idea of points. In the 7th century BC, the ancient Greek scientist and philosopher Thales also contained the idea of calculus in his research on the area, volume, and length of the sphere. During the Three Kingdoms period, Liu Huizenga also produced the budding thoughts of integrals in his ideas about the circle technique and the problem of finding the volume. In the 17th century, many famous mathematicians, astronomers, and physicists such as Fermat, Descartes, Robus, and De Sage in France, Barrow and Wallis in Britain, Kepler in Germany, and Karnataka in Italy Valery also produced the idea of calculus to solve practical problems. This contributed to the creation of calculus. In the second half of the 17th century, the national mathematician and physicist Newton and the German mathematician Leibnizying completed the initial creation of calculus independently based on the work of their predecessors. Their greatest achievement is closely linking two seemingly unrelated tangents (the central problem of differential calculus) and quadrature (the central problem of integral calculus).

Historically, the issue of the attribution and priority of the results of calculus has caused a long-term controversy in the mathematics circle. The importance of this debate is not the question of who wins and who loses but divides mathematicians into two factions. One group is a British mathematician who defends Newton. The other faction is the support of Leibniz by continental European mathematicians, especially the Bernoulli brothers. The two factions are opposed or even hostile to each other [9]. Later verification showed that although most of Newton's work was done before Leibniz, Leibniz was an independent inventor of the main idea of calculus. Until the beginning of the 19th century, scientists from the French Academy of Sciences, led by Cauchy, conducted serious research on the theory of calculus and established the limit theory. Later, the German mathematician Weierstrass further rigorously made the limit theory a firm basis for calculus. This enables the further development and growth of calculus. The introduction of this knowledge of the history of mathematics improves students' interest in learning and greatly benefits the improvement of students' humanistic quality.

Use mathematical methods to solve practical problems

Introduce some practical problems in the classroom and try to solve them by mathematical methods. This can not only enhance students' hands-on ability but also stimulate their interest in mathematics. For example, there is such a practical problem:

Example 1: An uncrewed aerial vehicle flies horizontally at an altitude of 3km at an altitude of 200km/h to the sky above the Nanguo Temple scenic spot in Tianshui for landscape photography. The altitude in the scenic area is between 1.2 km and 1.7 km. Try to find the angular velocity of the camera's rotation when the drone reaches the sky above the scenic spot. The problem can be easily solved after learning the function's derivative and its related rate of change [10]. The detailed process is given below: Assume that the horizontal distance between the drone and the scenic area is the angle between x(km), OA and OB as θ, as shown in Figure 1.

Figure 1

Schematic diagram of Example 1

But y=dxdt=200km/h,θ=arctan3x, \matrix{ {y = {{dx} \over {dt}} = - 200km/h,} \hfill \cr {\theta = \arctan {3 \over x},} \hfill \cr }

And the angular velocity is ω=dθdt=dθdx×dxdt=11+(2x)2(3x2)dxdt=3x2+9dxdt \omega = {{d\theta } \over {dt}} = {{d\theta } \over {dx}} \times {{dx} \over {dt}} = {1 \over {1 + {{\left( {{2 \over x}} \right)}^2}}}\left( { - {3 \over {{x^2}}}} \right){{dx} \over {dt}} = - {3 \over {{x^2} + 9}}{{dx} \over {dt}}

Therefore, when the drone flies over the scenic area, the angular velocity is ω|x=0=39(200)=2003(rad/h)=(2003×180π)/3600=103π(rad/s) \omega \left| {_{x = 0} = - {3 \over 9}\left( { - 200} \right) = {{200} \over 3}\left( {rad/h} \right) = \left( {{{200} \over 3} \times {{180} \over \pi }} \right)/3600 = {{10} \over {3\pi }}\left( {rad/s} \right)} \right.

Incorporate philosophical views “special and general” throughout the classroom

Special and general in philosophy are relative concepts and categories. They have a dialectical and unified relationship. From the perspective of materialism, the general and special of things exist objectively. They are all inherent to things and phenomena in the real world. Everything is a unity of commonality and individuality. In the teaching process of “Mathematical Analysis,” teachers should fully excavate the mathematics content of the philosophical viewpoints “special and general” contained in the textbook. If it can be used throughout the classroom, it will enable students to deepen their understanding of mathematical concepts and raise their understanding to a higher level [11]. This not only broadens the scope of knowledge but also cultivates good thinking habits. The following two examples illustrate how the mathematics material of the philosophical viewpoint of “special and general” is embodied in the classroom.

Example 2: For example, when we talk about the Gamma function, we get the following recurrence properties: Γ(s+1)=sΓ(s)=s(s1)(sn)Γ(sn) \Gamma \left( {s + 1} \right) = s\Gamma \left( s \right) = s\left( {s - 1} \right) \cdots \left( {s - n} \right)\Gamma \left( {s - n} \right)

If s takes a positive integer n + 1, the above formula becomes Γ(n+1)=n(n1)21Γ(1)=n!0+exdx=n! \Gamma \left( {n + 1} \right) = n\left( {n - 1} \right) \cdots 2\,1\;\Gamma \left( 1 \right) = n!\int_0^{ + \infty } {{e^{ - x}}dx = n!}

We will find that the Gamma function is an extension of factorial.

Example 3: For example, when talking about the derivative of a function, the first-order formula holds: f(x)=limx0f(x)f(xh)h f^\prime \left( x \right) = \mathop {\lim }\limits_{x \to 0} {{f\left( x \right) - f\left( {x - h} \right)} \over h}

Second-order formula: f(x)=limh0f(x)f(xh)h=limh0f(x)2f(xh)+f(x2h)h2 f^{''}\left( x \right) = \mathop {\lim }\limits_{h \to 0} {{f^\prime \left( x \right) - f^\prime \left( {x - h} \right)} \over h} = \mathop {\lim }\limits_{h \to 0} {{f\left( x \right) - 2f\left( {x - h} \right) + f\left( {x - 2h} \right)} \over {{h^2}}}

Combined with mathematical induction, the high-order derivative formula of function f(x) is obtained: f(n)(x)=limh0nj=0n(1)jn(n1)(nj+1)j!f(xjh) {f^{\left( n \right)}}\left( x \right) = {{\mathop {\lim }\limits_{h \to 0}} ^{ - n}}\sum\limits_{j = 0}^n {{{\left( { - 1} \right)}^j}} {{n\left( {n - 1} \right) \cdots \left( {n - j + 1} \right)} \over {j!}}f\left( {x - jh} \right)

If the natural number n is extended to the general positive real number α, the following left Grunwall–Liouvill fractional derivative is obtained: GLDa,xaf(x)=limh0aj=0[xah](1)ja(a1)(aj+1)j!f(xjh) _{GL}D_{a,x}^a\,f\left( x \right) = {{\mathop {\lim }\limits_{h \to 0}} ^{ - a}}\sum\limits_{j = 0}^{\left[ {{{x - a} \over h}} \right]} {{{\left( { - 1} \right)}^j}} {{a\left( {a - 1} \right) \cdots \left( {a - j + 1} \right)} \over {j!}}f\left( {x - jh} \right)

If the forward difference is used, the right Grunwall-Liouville fractional derivative definition can be similarly obtained. This shows that the fractional derivative is the generalization and extension of the general integer derivative. Furthermore, I can tell the students that fractional derivatives can provide a more profound and comprehensive explanation of abnormal phenomena such as historical memory and global spatial correlation involved in complex environments [12]. The fractional differential equation developed from it has become one of the important tools to describe various complex mechanical and physical behaviors. It has attracted more and more scholars' attention. If the above examples are used as the starting point, students can be exposed to contemporary mathematics's cutting-edge theories and hot topics. This broadens the students' knowledge and improves their ability to learn mathematics.

Cleverly associate certain mathematical concepts with ancient poetry

Literature focuses on thinking in images, while mathematics focuses more on logical thinking. The former is a representative of sensibility, while the latter is a representative of reason. The relationship between literature and mathematics is like a Tai Chi Bagua diagram, and a balance needs to be reached between sensibility and rationality [13]. Therefore, in the teaching process of “Mathematical Analysis,” if the elements of literature can be integrated into the classroom, it will deepen the memory and understanding of mathematical concepts and help shape a good personality and conduct. Here are two examples to illustrate how to combine ancient poems with some mathematical concepts skillfully.

Example 4: Number sequence and formula S=n=1+n=+ S = \sum\limits_{n = 1}^{ + \infty } {n = + \infty } , we can contact a digital poem appearing in “Prime Minister Liu Luoguo,” which is the famous Qing dynasty painter, painter, and writer Zheng Banqiao's famous essay “Yong Xue”:

One-piece, two pieces, three or four pieces, five, six, seven, eight, ninety pieces, thousands of pieces, countless pieces, flying into the plum blossom you will never see.

Here, n is used to represent the number of snowflakes, and its sum is used to indicate the process of snowflakes flying into the plum blossom. The final result reflects the final ablation of the snowflakes.

Example 5: When talking about integer-order derivatives, guide students to look up relevant information to summarize some common definitions of fractional-order derivatives [14]. For example, select the left and right Riemann-Liouville fractional derivatives as follows: RLD,xaf(x)=1Γ(na)dndxnxf(t)(xt)an+1dt, RLD_{ - \infty ,x}^a\,f\left( x \right) = {1 \over {\Gamma \left( {n - a} \right)}}{{{d^n}} \over {d{x^n}}}\int_{ - \infty }^x {{{f\left( t \right)} \over {{{\left( {x - t} \right)}^{a - n + 1}}}}dt,} RLDx,+af(x)=(1)nΓ(na)dndxnx+f(t)(tx)an+1dt RLD_{x,+ \infty }^a\,f\left( x \right) = {{{{\left( { - 1} \right)}^n}} \over {\Gamma \left( {n - a} \right)}}{{{d^n}} \over {d{x^n}}}\int_x^{ + \infty } {{{f\left( t \right)} \over {{{\left( {t - x} \right)}^{a - n + 1}}}}dt}

And this can also be compared with the famous work “Dengyouzhou Taige” by Chen Ziang, a writer and poet in the early Tang Dynasty, a representative of the reform of poetry and prose in the early Tang Dynasty: “You will not see the ancients before, and you will not see the people in the future [15]. The leisure of the world, alone, but tears down.” related. If x represents the current time node, the integration interval (−∞, x] of the left Riemann-Liouville derivative can correspond to the sentence “No one saw the ancients before,” and the integration interval (x, + ∞] of the right iemann-liouville derivative can correspond to the sentence “No one is seen later.”

Conclusion

“Mathematical Analysis” is the basic core course of mathematics and applied mathematics. Mathematical concepts are abstract and highly skilled, which makes it difficult for students to adapt during the learning process.

Figure 1

Schematic diagram of Example 1
Schematic diagram of Example 1

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