CC BY
12
Вестник РУДН. Серия: Информатизация образования
RUDN Journal of Informatization in Education
2017 Том 14 № 3 348-356
http://journals.rudn.ru/informatization-education
DOI 10.22363/2312-8631-2017-14-3-348-356 UDK 378+517.9
THE DIDACTIC ANALYSIS OF STUDIES ON THE INVERSE PROBLEMS FOR THE DIFFERENTIAL EQUATIONS
In article results of the didactic analysis of the organization and carrying out seminar classes in the inverse problems for the differential equations for students of higher educational institutions of the physical and mathematical directions of preparation are discussed. Such analysis includes a general characteristic of mathematical content of seminar occupations, the analysis of structure of seminar occupation, the analysis of realization of the developing and educational purposes, allocation of didactic units and informative means which have to be acquired by students when training each section of content of training in the inverse problems and other important psychology and pedagogical aspects. The attention to establishment of compliance to those of seminar occupations to lecture material and identification of functions in teaching and educational process which are carried out at the solution of the inverse problems, and also is paid to need to show various mathematical receptions and methods of their decision.
Such didactic analysis helps not only to reveal such inverse problems at which solution students can collectively join in creative process of search of their decision, but also effectively organize control of assimilation of knowledge and abilities of students on the inverse problems for the differential equations.
Key words: training in the inverse problems for the differential equations, the didactic analysis of studies, the student
Efficiency and effectiveness ofpedagogical activities in higher educational institutions in many respects depends on the formulated didactic purposes and tasks, the realized didactic principles of training, selection and formation of content of training, methods oftraining, forms of the organization of studies, on scheduled ways of their implementation (look, for example, [1; 19; 21; 22; 25]). Therefore, tasks of enhancement of characteristics of educational process, upgrade of training, education and development of the identity of students on the basis of the modern achievements of world science are set for the modern Russian higher school.
Content of training in the inverse problems for differential equations (IPDE) is guided more by students of the higher education institutions studying at the physical and mathematical directions of preparation and shall consider their professional directivity of preparation. Training in IPDE shan't degenerate in the simplified and concise presentation of the appropriate courses. When training in IPDE it is important to pay attention to detection of application-oriented aspects of such mathematical tasks and physical interpretation of the received decision; on formation of skills of students
V.S. Kornilov
Moscow city pedagogical university Sheremetyevskaya str., 29, Moscow, Russia, 127521
independently to formulate logical outputs of application-oriented and humanitarian character from the received decision of the inverse problem (look, for example, [2—18; 20; 23; 24]).
Planning, development of a technique of training and implementation of the process of training of students in IPDE has to be carried out by experts in this scientific area. For the correct statement of training in IPDE it is necessary to reach a certain level of mutual understanding when planning content of training of students in physical and mathematical disciplines, the considering content of training in IPDE both in department, and between departments of physical and mathematical profiles of the corresponding directions of training of students in higher education institution. Content of training in IPDE includes knowledge not only general education physical and mathematical disciplines, but also special sections, such as the theory of the generalized and analytical functions, the tensor analysis, methods of the solution of the nonlinear integrated equations and other special sections which not always enter the content of training in physical and mathematical disciplines.
Training in IPDE has to be carried out on older years when students have already mastered basic mathematical disciplines, such as algebra and geometry, the mathematical and functional analysis, the ordinary differential equations and the differential equations in private derivatives, the integrated equations, optimization methods, numerical methods and other subject matters. Situation when the volume of knowledge of IPDE , extent of possession and the nature of the skills acquired by students is defined by them together with leading experts in area of future specialization of students is reasonable. Time allowed for training in IPDE has to be fixed in the appropriate working programs in which besides the content of training the purposes, the principles, tasks and basic provisions of training in IPDE have to be stated.
One of the main forms of the organization of studies for IPDE is seminar occupation. Feature of seminar classes in IPDE consists in a possibility of the active involvement of each student in discussion of setting of the inverse problem, approaches and methods of its decision of the logical analysis of results of its decision in applied researches. On seminar occupations students, mastering the theory and methodology to IPDE, solve various mathematical problems. For example, construct an integrable equation to which satisfies the decision of the direct task; prove theorems of a correctness of the inverse problem; explain the idea of search of approximate solution to IPDE; write out the difference analog IPDE; construct a computing algorithm by means ofwhich approximate solution to IPDE can be found, carry out the analysis of its properties (convergence, stability and other properties), solve also other mathematical problems.
As the educational job, maybe, it is entrusted to students to explain the idea of the proof of a correctness or the conditional correctness to IPDE or, for example, according to the found solution to IPDE to formulate logical outputs of application-oriented or humanitarian character.
On seminar occupations by the teacher the results of independent and personal operation of students on assimilation of the theory and practice of the decision can be summed up of IPDE. It is necessary to mark that independent operation of students is the important and necessary educational activities promoting effective training in IPDE. The organization of independent operation of students it is necessary to fix and deepen
fundamental knowledge in the field of the theory and methodology to IPDE, to gain experience of an independent research of such tasks and a statement of logical outputs of application-oriented and humanitarian character.
Within independent operation, for example, it can be offered to students to execute mathematical calculations which don't require new knowledge of the inverse problems which was lowered on lecture occupation; to decide to IPDE, logically connected to the explained sections of the inverse problems; to read the specific scientific article across IPDE which on setting and a method of a research is close to considered earlier on lecture occupation, to understand a method of its decision, to analyze a physical sense of the most application-oriented task, to express the opinion on advantage and advantages of the approach to its decision explained in article; to perform term or final qualification paper on IPDE.
Hours and subjects for independent operation of students shall be stated in the working program for IPDE by the teacher. Their quantity shall be coordinated with the head of the department for the purpose of prevention of educational overloads of students. The diagram of its carrying out shall be developed and agreed by the teacher with students.
We will mark that the marked didactic analysis allows to reveal a role of seminar classes in IPDE in formation at students of skills of application of mathematical methods of the decision of application-oriented tasks; judgment by students of scientific and cognitive potential of the theory to IPDE; formation at students of analytical skills and abilities to formulate similar settings to IPDE and to generalize them; acquisition by students of skills it is reasoned and clearly to explain solutions to IPDE.
The similar didactic analysis includes a general characteristic of mathematical content ofseminar occupations (the analysis ofa subject ofseminar occupation, her communication with the stated material on the previous lecture occupation); judgment of mathematical content of seminar occupation (the analysis of the studied physical process or the phenomenon; analysis of statement to IPDE; the analysis of structure of auxiliary data from the physical and mathematical disciplines necessary for the decision for IPDE; the analysis of possible approaches to the decision to IPDE and the expected results of her decision; analysis of methods of the proof of the corresponding theorems of existence, uniqueness and conditional stability of the decision to IPDE, their constructibility; the analysis of novelty of the considered return task for students, etc.); the stages of the organization of the developing training of students determined by mathematical contents.
In the course of the didactic analysis it is necessary to pay attention to statement of the purposes of seminar occupation which are defined by the mathematical content of this occupation, a role and the place of seminar occupation in the section of content of training in IPDE.
Besides, it is necessary to carry out the analysis of structure of seminar occupation which includes:
— training of students for active, conscious assimilation of knowledge (updating of knowledge and her receptions; ways of creation of motivation of training in IPDE or problem situations; statement is more whole than seminar occupation);
— the analysis of the scientific character, completeness and sequence of a research considered to IPDE; methods of activization of activity of students at the decision to
IPDE, degree of their independence, methods of management of cognitive activity of students, implementation of feedback;
— the analysis of system of mathematical tasks from special sections of mathematical disciplines (the generalized functions, analytical functions, fundamental solutions of the differential equations, the nonlinear integrated equations, etc.) which decision promotes the successful decision to IPDE;
— the analysis of forms and control methods assimilation of knowledge in the field of the theory and methodology to IPDE students;
— the analysis of logic of seminar occupation, interrelation of his stages, logic in transition from one stage of occupation to another;
— the analysis of registration of results of solutions of mathematical tasks from special sections of mathematical disciplines or decision IPDE to a board;
— analysis of summing up seminar occupation;
— analysis distribution of time for various stages of carrying out seminar occupation.
It is also necessary to carry out the analysis of realization of the developing and
educational purposes which, in particular, assumes:
— development of all-educational abilities of students (ability to work with educational and scientific literature across IPDE);
— development of intellectual abilities of students (mathematical, logical, speech);
— formation of scientific outlook of students (ability and skills to reveal application to IPDE in applied researches);
— development ofindependence ofstudents in mastering ofthe theory and methodology to IPDE;
— accounting of specific features of students;
— development in students of motivation of the doctrine;
— development in students of logical, algorithmic and information thinking;
— detection of level of insistence of the teacher, his objectivity in an assessment of knowledge and abilities of students;
— formation of culture of communication of the teacher with students.
In the course of carrying out seminar classes in IPDE it is expedient to fix and analyze didactic units and informative means which have to be acquired by students when training each section of training in IPDE. Mathematical and physical concepts, definitions, axioms, lemmas, theorems of existence, uniqueness and stability, constructive algorithms of the decision to IPDE are among such didactic units. Their assimilation at the level of knowledge, understanding, application in standard situations is reached by means of the decision of protozoa to IPDE. Mastering students of such didactic units is the initial stage of mastering methods and receptions of the decision to IPDE according to the concrete section of content of training in IPDE.
It is necessary to observe coherence of subjects to IPDE of seminar occupations lecture not only on types and identity of the return tasks, but also to consider a professional orientation of training. It is expedient to reveal functions in teaching and educational process which are performed at the decision to IPDE among which motivational informative, developing, bringing up, operating, illustrative.
LITERATURE
[1] Бондаревская Е.В. Теория и практика личностного образования. Ростов н/Д: Изд-во Ростовского педагогического университета, 2000. 320 с.
[2] Ватульян А.О., Беляк О.А., Сухов Д.Ю., Явруян О.В. Обратные и некорректные задачи: учебное пособие. Ростов н/Д: Изд-во Южного федерального университета, 2011. 232 с.
[3] Денисов А.М. Введение в теорию обратных задач: учеб. пособие. М.: Изд-во МГУ им. М.В. Ломоносова, 1994. 207 с.
[4] Кабанихин С.И. Обратные и некорректные задачи: учеб. пособие. Новосибирск: Сибирское научное издательство, 2009. 458 с.
[5] Корнилов В.С. К вопросу о типовой программе по дисциплине «Обратные задачи для дифференциальных уравнений» // Вестник Московского городского педагогического университета. Серия «Информатика и информатизация образования». 2004. № 1 (2). С. 79—83.
[6] Корнилов В.С. О междисциплинарном характере исследований причинно-следственных обратных задач // Вестник Московского городского педагогического университета. Серия «Информатика и информатизация образования». 2004. № 1 (2). С. 80—83.
[7] Корнилов В.С. Основы методической системы обучения дисциплине «Обратные задачи для дифференциальных уравнений» // Вестник Самарского государственного экономического университета. Самара: СГЭУ, 2005. № 3 (18). С. 190—196.
[8] Корнилов В.С. Некоторые обратные задачи идентификации параметров математических моделей: учебное пособие. М.: МГПУ, 2005. 359 с.
[9] Корнилов В.С. История развития теории обратных задач для дифференциальных уравнений — составляющая гуманитарного потенциала обучения прикладной математике // Вестник Московского городского педагогического университета. Серия «Информатика и информатизация образования». 2009. № 1 (17). С. 108—113.
[10] Корнилов В.С. Психологические аспекты обучения студентов вузов фрактальным множествам // Вестник Российского университета дружбы народов. Серия «Информатизация образования». 2011. № 4. С. 79—82.
[11] Корнилов В.С. Лабораторные занятия как форма организации обучения студентов фрактальным множествам // Вестник Московского городского педагогического университета. Серия «Информатика и информатизация образования». 2012. № 1 (23). С. 60—63.
[12] Корнилов В.С. Обратные задачи в содержании обучения прикладной математике // Вестник Российского университета дружбы народов. Серия «Информатизация образования». 2014. № 2. С. 109—118.
[13] Корнилов В.С. Обучение студентов обратным задачам математической физики как фактор формирования фундаментальных знаний по интегральным уравнениям // Бюллетень лаборатории математического, естественнонаучного образования и информатизации. Рецензируемый сборник научных трудов. Самара: Самарский филиал МГПУ, 2015. Т. VI. С. 251—257.
[14] Корнилов В.С. Формирование фундаментальных знаний будущих учителей информатики и математики по функциональному анализу при обучении обратным задачам математической физики // Вестник Московского городского педагогического университета. Серия «Информатика и информатизация образования». 2015. № 3 (33). С. 72—82.
[15] Корнилов В.С. Реализация научно-образовательного потенциала обучения студентов вузов обратным задачам для дифференциальных уравнений // Казанский педагогический журнал. 2016. № 6. С. 55—59.
[16] Корнилов В.С. Базовые понятия информатики в содержании обучения обратным задачам для дифференциальных уравнений // Вестник Российского университета дружбы народов. Серия «Информатизация образования». 2016. № 1. С. 70—84.
[17] Корнилов В.С. Теория и методика обучения обратным задачам для дифференциальных уравнений: монография. М.: Изд-во «ОнтоПринт», 2017. 500 с.
[18] Корнилов В.С. Формирование фундаментальных знаний по математическому моделированию при обучении обратным задачам для дифференциальных уравнений // Вестник Московского городского педагогического университета. Серия «Информатика и информатизация образования». 2017. № 1 (39). С. 92—99.
[19] Краевский В.В., Хуторской А.В. Основы обучения. Дидактика и методика: учеб. пособие. М.: Академия, 2007. 352 с.
[20] Левченко И.В., КорниловВ.С., БеликовВ.В. Роль информатики в подготовке специалистов по прикладной математике // Вестник Московского городского педагогического университета. Серия «Информатика и информатизация образования». 2009. № 2 (18). С. 108— 112.
[21] Пидкасистый П.И. Педагогика. М.: Просвещение, 1996. 602 с.
[22] Подласый И.П. Педагогика: 100 вопросов — 100 ответов: учеб. пособие для вузов. М.: ВЛАДОС-пресс, 2004. 365 с
[23] Романов В.Г. Обратные задачи математической физики: монография. М.: Наука, 1984. 264 с.
[24] Самарский А.А., Вабишевич П.Н. Численные методы решения обратных задач математической физики: монография. М.: УРСС, 2004. 478 с.
[25] Хуторской А.В. Современная дидактика: учебное пособие. М.: Высшая школа, 2007. 639 с.
© Kornilov V.S., 2017
Article history:
Received: 19 April, 2017 Accepted: 22 May, 2017
For citation:
Kornilov V.S. (2017) The didactic analysis of studies on the inverse problems for the differential equations. RUDN Journal of Informatization Education, 14 (3), 348—356. DOI 10.22363/23128631-2017-14-3-348-356
Bio Note:
Kornilov Viktor Semenovich, doctor of pedagogical sciences, candidate of physical and mathematical sciences, full professor, deputy head of the department of informatization of education of the Moscow city pedagogical university. Contact information: e-mail: vs_kornilov@mail.ru
ДИДАКТИЧЕСКИЙ АНАЛИЗ УЧЕБНЫХ ЗАНЯТИЙ ПО ОБРАТНЫМ ЗАДАЧАМ ДЛЯ ДИФФЕРЕНЦИАЛЬНЫХ УРАВНЕНИЙ
В.С. Корнилов
Московский городской педагогический университет Шереметьевская ул., 29, Москва, Россия, 127521
В статье обсуждаются результаты дидактического анализа организации и проведения семинарских занятий по обратным задачам для дифференциальных уравнений для студентов высших учебных заведений физико-математических направлений подготовки. Такой анализ включает: общую характеристику математического содержания семинарских занятий; изучение структуры семинарского занятия; реализацию развивающих и воспитательных целей,
выделение дидактических единиц и познавательных средств, которые должны быть усвоены студентами при изучении каждого раздела содержания обучения обратным задачам и другие важные психолого-педагогические аспекты. Обращается внимание на установление соответствия тем семинарских занятий лекционному материалу и выявление функций в учебно-воспитательном процессе, которые выполняются при решении обратных задач, а также на необходимость демонстрировать различные математические приемы и методы их решения.
Такой дидактический анализ помогает не только выявить такие обратные задачи, при решении которых студенты могут коллективно включиться в творческий процесс поиска их решения, но и эффективно организовать контроль усвоения знаний и умений студентов по обратным задачам для дифференциальных уравнений.
Ключевые слова: обучение обратным задачам для дифференциальных уравнений, дидактический анализ учебных занятий, студент
REFERENCES
[ 1] Bondarevskaya E.V Teorija ipraktika lichnostnogo obrazovanija [Theory and practice of personal education]. Rostov-na-Donu, 2000. 320 p.
[2] Vatulyan A.O., Belyak O.A., Sukhov D.Yu., Yavruyan O.V Obratnye inekorrektnyezadachi [Inverse and incorrect tasks]: ucheb. posobie. Rostov-na-Donu: Izd-vo Juzhnogo federal'nogo universiteta, 2011. 232 p.
[3] Denisov A.M. Vvedenie v teoriju obratnyh zadach [Introduction to the theory of the inverse problems]: ucheb. posobie. M.: Izd-vo MGU im. M.V Lomonosova, 1994. 207 p.
[4] Kabanihin S.I. Obratnye i nekorrektnye zadachi [Inverse and incorrect tasks]: uchebnik dlja studentov vuzov. Novosibirsk: Sibirskoe nauchnoe izdatel'stvo, 2009. 458 p.
[5] Kornilov V.S. Kvoprosu o tipovojprogrammepo discipline «Obratnye zadachi dlja differencial'nyh uravnenij» [To a question of the standard program for discipline "The inverse problems for the differential equations"]. Vestnik Moskovskogo gorodskogo pedagogicheskogo universiteta. Serija «Informatika i informatizacija obrazovanija» [Bulletin of the Moscow city pedagogical university. "Informatics and Informatization of Education" series]. 2004. No. 1 (2). Pp. 79—83.
[6] Kornilov VS. O mezhdisciplinarnom haraktere issledovanij prichinno-sledstvennyh obratnyh zadach [About cross-disciplinary character of researches of cause and effect inverse problems]. Vestnik Moskovskogo gorodskogo pedagogicheskogo universiteta. Serija «Informatika i informatizacija obrazovanija» [Bulletin ofthe Moscow city pedagogical university. "Informatics and Informatization of Education" series]. 2004. No. 1 (2). Pp. 80—83.
[7] Kornilov V.S. Osnovy metodicheskoj sistemy obuchenija discipline «Obratnye zadachi dlja differencial'nyh uravnenij» [Bases of methodical system of training in discipline "The inverse problems for the differential equations"]. Vestnik Samarskogo gosudarstvennogo jekonomicheskogo universiteta [Bulletin of the Samara state economic university]. 2005. No. 3 (18). Pp. 190—196.
[8] Kornilov VS. Nekotorye obratnye zadachi identifikacii parametrov matematicheskih modelej [Some inverse problems of identification of parameters of mathematical models]: ucheb. posobie. M.: MGPU, 2005. 359 p.
[9] Kornilov VS. Istorija razvitija teorii obratnyh zadach dlja differencial'nyh uravnenij — sostavljajushhaja gumanitarnogo potenciala obuchenija prikladnoj matematike [History of development of the theory of the inverse problems for differential equations — a component of humanitarian potential of training in applied mathematics]. Vestnik Moskovskogo gorodskogo pedagogicheskogo universiteta. Serija «Informatika i informatizacija obrazovanija» [Bulletin of the Moscow city pedagogical university. "Informatics and Informatization of Education" series]. 2009. No. 1 (17). Pp. 108—113.
[10] Kornilov V.S. Psihologicheskie aspekty obuchenija studentov vuzovfraktal'nym mnozhestvam [Psychological aspects of training of students of higher education institutions in fractal sets]. Vestnik Rossijskogo universiteta druzhby narodov. Serija «Informatizacija obrazovanija» [Bulletin of the Russian university of friendship of the people. "Education Informatization" series]. 2011. No. 4. Pp. 79—82.
[11] Kornilov V.S. Laboratornye zanjatija kakforma organizacii obuchenija studentov fraktal'nym mnozhestvam [Laboratory researches as form of the organization of training of students in fractal sets]. Vestnik Moskovskogo gorodskogo pedagogicheskogo universiteta. Serija «Informatika i informatizacija obrazovanija» [Bulletin of the Moscow city pedagogical university. "Informatics and Informatization of Education" series]. 2012. No. 1 (23). Pp. 60—63.
[12] Kornilov V.S. Obratnye zadachi v soderzhanii obuchenijaprikladnoj matematike [The inverse problems in the content of training in applied mathematics]. Vestnik Rossijskogo universiteta druzhby narodov. Serija «Informatizacija obrazovanija» [Bulletin of Peoples' Friendship University of Russia. "Education Informatization" series]. 2014. No. 2. Pp. 109—118.
[13] Kornilov VS. Obuchenie studentov obratnym zadacham matematicheskoj fiz,iki kakfactor formirovanija fundamental'nyh znanij po integral'nym uravnenijam [Training of students in the inverse problems of mathematical physics as factor of formation of fundamental knowledge of the integrated equations]. Bjulleten' laboratorii matematicheskogo, estestvennonauchnogo obrazovanija i informatizacii. Recenziruemyj sbornik nauchnyh trudov [Bulletin of laboratory of mathematical, natural-science education and informatization. The reviewed collection of scientific work]. Samara: Samarskij filial MGPU, 2015. T. VI. Pp. 251—257.
[ 14] Kornilov V.S. Formirovanie fundamental'nyh znanij budushhih uchitelej informatiki i matematiki po funkcional'nomu analizupri obuchenii obratnym zadacham matematicheskoj fiziki [Formation of fundamental knowledge of future teachers of informatics and mathematics of the functional analysis when training in the inverse problems of mathematical physics]. Vestnik Moskovskogo gorodskogo pedagogicheskogo universiteta. Serija «Informatika i informatizacija obrazovanija» [Bulletin of the Moscow city pedagogical university. "Informatics and Informatization of Education" series]. 2015. No. 3 (33). Pp. 72—82.
[15] Kornilov VS. Realizacija nauchno-obrazovatel'nogopotenciala obuchenija studentov vuzov obratnym zadacham dlja differencial'nyh uravnenij [Realization of scientific and educational potential of training of students of higher education institutions in the inverse problems for the differential equations]. Kazanskij pedagogicheskij zhurnal [Kazan pedagogical journal]. 2016. No. 6. Pp. 55— 59.
[16] Kornilov V.S. Bazovyeponjatija informatiki v soderzhanii obuchenija obratnym zadacham dlja differencial'nyh uravnenij [Basic concepts of informatics of the content of training in the inverse problems for the differential equations]. Vestnik Rossijskogo universiteta druzhby narodov. Serija «Informatizacija obrazovanija» [Bulletin Peoples' Friendship University of Russia. "Education Informatization" series]. 2016. No. 1. Pp. 70—84.
[17] Kornilov V.S. Teorija i metodika obuchenija obratnym zadacham dlja differencial'nyh uravnenij [Theory and technique of training to the inverse problems for differential equations]: monografija. M.: Izd-vo «OntoPrint», 2017. 500 p.
[18] Kornilov V.S. Formirovanie fundamental'nyh znanij po matematicheskomu modelirovaniju pri obuchenii obratnym zadacham dlja differencial'nyh uravnenij [Formation of fundamental knowledge of mathematical modeling when training in the return tasks for the differential equations]. Vestnik Moskovskogo gorodskogo pedagogicheskogo universiteta. Serija «Informatika i informatizacija obrazovanija» [Bulletin ofthe Moscow city pedagogical university. "Informatics and Informatization of Education" series]. 2017. No. 1 (39). Pp. 92—99.
[19] Kraevskij V.V., Hutorskoj A.V. Osnovy obuchenija. Didaktika i metodika [Training bases. Didactics and technique]: ucheb. posobie. M.: Akademija, 2007. 352 p.
[20] Levchenko I.V, Kornilov V.S., Belikov VV. Rol' informatiki v podgotovke specialistov po prikladnoj matematike [Role of informatics in training of specialists on applied mathematics]. Vestnik Moskovskogo gorodskogo pedagogicheskogo universiteta. Serija «Informatika i informatizacija obrazovanija» [Bulletin ofthe Moscow city pedagogical university. "Informatics and Informatization of Education" series]. 2009. No. 2 (18). Pp. 108—112.
[21] Pidkasistyj P.I. Pedagogika [Pedagogics]. M.: Prosveshhenie, 1996. 602 p.
[22] Podlasyj I.P. Pedagogika: 100 voprosov — 100 otvetov [Pedagogics: 100 questions — 100 answers]: ucheb. posobie dlja vuzov. M.: VLADOS-press, 2004. 365 p.
[23] Romanov YG. Obratnye zadachi matematicheskoj fiziki [Inverse problems ofmathematical physics]: monografija. M.: Nauka, 1984. 264 p.
[24] Samarskij A.A., Vabishevich P.N. Chislennye metody reshenija obratnyh zadach matematicheskoj fiziki [Numerical methods of the solution of the inverse problems of mathematical physics]: monografija. M.: Editorial URSS, 2004. 480 p.
[25] Hutorskoj A.V Sovremennaja didaktika [Modern didactics]: ucheb. posobie. M.: Vysshaja shkola, 2007. 639 p.
История статьи:
Дата поступления в редакцию: 19 апреля 2017 Дата принятия к печати: 22 мая 2017
Для цитирования:
Корнилов В.С. Дидактический анализ учебных занятий по обратным задачам для дифференциальных уравнений // Вестник Российского университета дружбы народов. Серия «Информатизация образования». 2017. Т. 14. № 3. С. 348—356. DOI 10.22363/2312-86312017-14-3-348-356
Сведения об авторе:
Корнилов Виктор Семенович, доктор педагогических наук, кандидат физико-математических наук, профессор, заместитель заведующего кафедрой информатизации образования Московского городского педагогического университета. Контактная информация: е-mail: vs kornilov@mail.ru
