Defense of the dissertation of Taishiyeva Aigul Galymzhanovna for the degree of Doctor of Philosophy (PhD) in the educational program «8D06105 - Mathematical and Computer Modeling»
L.N. Gumilyov Eurasian National University, a dissertation defense for the degree of Doctor of Philosophy (PhD) by Taishiyeva Aigul Galymzhanovna on the topic «On some nonlinear physical models with derivatives of integer and fractional order» to the educational program «8D06105 – Mathematical and Computer Modeling».
The dissertation was carried out at the Department of «Mathematical and computer modeling» of L.N. Gumilyov Eurasian National University.
The language of defense is kazakh.
Official reviewers:
Kanguzhin Baltabek – Doctor of Physical and Mathematical Sciences, Professor of the Department of Mathematics at Al-Farabi Kazakh National University (Almaty, Republic of Kazakhstan);
Ramazanov Murat – Doctor of Physical and Mathematical Sciences, Professor of the Department of Mathematical Analysis and Differential Equations of the E.A. Buketov Karaganda University (Karaganda, Republic of Kazakhstan).
Temporary members of the Dissertation Council:
Jenaliyev Muvasharkhan – Doctor of Physical and Mathematical Sciences, Professor, Institute of Mathematics and Mathematical Modeling (Almaty, Republic of Kazakhstan);
Shakenov Kanat – Doctor of Physical and Mathematical Sciences, Professor of the Department of Computational Sciences and Statistics of the Al-Farabi Kazakh National University (Almaty, Republic of Kazakhstan);
Sakabekov Auzhan – Doctor of Physical and Mathematical Sciences, Professor of the Department of Higher Mathematics and Modeling of the Satbayev Kazakh National Technical University (Almaty, Republic of Kazakhstan);
Sultanov Murat – International Kazakh-Turkish University named after Khoja Ahmed Yasawi, Professor of the Department of Mathematics, Candidate of Physical and Mathematical Sciences (Turkestan, Republic of Kazakhstan).
Scientific consultants:
Nugmanova Gulgassyl – Candidate of Physical and Mathematical Sciences, Associate Professor of the Department of Mathematical and Computer Modeling at L.N. Gumilev ENU (Astana, Republic of Kazakhstan);
Valchev Tikhomir – PhD, Professor at the Institute of Mathematics and Computer Science of the Bulgarian Academy of Sciences (Sofia, Republic of Bulgaria).
The defense will take place on August 28, 2024, at 02:00 PM in the Dissertation Council for the training direction «8D061 – Information and communication technologies» in the educational program «8D06105 – Mathematical and Computer Modeling» of L.N. Gumilyov Eurasian National University. The defense meeting is planned to be held offline and online.
Link: http://surl.li/iumigy
Address: Astana, st. Kazhymukan, 13, auditorium No. 205 of educational building No. 3.
Abstract (English): TAISHIYEVA AIGUL GALIMZHANOVNA «On some nonlinear physical models with derivatives of integer and fractional order» dissertation for the degree of Doctor of Philosophy (PhD) in the educational program «8D06105 – Mathematical and computer modeling». Novelty of the research topic: solving partial differential equations is one of the most important mathematical tools for describing physical phenomena in various scientific fields. However, studies carried out using the classical derivative model cannot show the propagation behavior of the end of ordered long waves. Therefore, the scientific works prove that the use of fractional derivative operators in modern science research to solve such problems leads to significant results. The fact that interest in objects described by fractional order differential equations today is not decreasing, but rather increasing, is due, first of all, to their numerous applications in various fields of physics and mathematics. When modeling dynamic processes of a fractional or fractal nature, it is often necessary to solve not a direct, but an inverse problem, i.e. find the original function in which the fractional order derivative is used in this model. New possibilities appear in mathematical and theoretical physics when the order α of a differential operator D_x^α or integral operator I_x^α becomes an arbitrary parameter. Moreover, most of the known properties of integer-order differentiation D_x^n do not satisfy the fractional differentiation operators D_x^α. For example, the rule for differentiation of integer order, the rule for differentiation of complex functions, and properties obvious for the first-order derivative D_x do not hold for the operators D_x^α, where0<α<1. However, these rules and properties have analogues and are determined by very complex relationships. In recent years, the study of analytical and numerical solutions of fractional derivatives of mathematical models reflecting known phenomena has become a hot topic in science and technology. Objective of the thesis: Description of some natural phenomena, which cannot be demonstrated by research results obtained via classical derivative models, using solutions of nonlinear differential equations with fractional order derivatives. Research tasks: - find soliton and soliton-like solutions of the generalized Burgers equation with fractional order derivatives, create their computer models demonstrating the behavior of the end propagation of ordered long waves; - find analytical solutions of the generalized Schrödinger equation with a self-consistent potential and create their computer models; - explore the relationship between integrable models that describe physical processes in various media using the example of the Camassa-Holm equation and spin system such as Heisenberg ferromagnet; - find solutions to the Camassa-Holm equation with fractional order derivatives, illustrate their computer models and conduct a comparative analysis with the solutions of this equation and the solutions obtained in the case of an integer order derivative. Objects of study: nonlinear differential equations with derivatives of integer and fractional order, which are models of natural and physical processes in various media: the ordered long-wave Burgers equation, the Schrödinger-type equation with a self-consistent potential, the Camassa-Holm equation, and the generalized Heisenberg ferromagnet model. Research methods: when studying nonlinear fractional differential equations, the definitions and properties of the beta-fractional derivative and the M-reduced fractional derivative are used. Solutions to nonlinear differential equations of fractional order are found using the new method of auxiliary equations and the Kudryashov method. The construction and analysis of computer images of the obtained solutions is carried out using the Maple application software package. Provisions for defense: - application of the properties of the beta derivative to the solution of the ordered long-wavelength nonlinear Burgers equation; - solutions of the ordered long-wave Burgers equation with time derivative obtained using a new method of auxiliary equations and computer models of the propagation behavior of the end of ordered long waves; - application of the properties of the reduced M-fractional derivative to the solution of the generalized Schrödinger equation with a consistent potential; - solutions of the generalized Schrödinger equation with self-consistent potential with a reduced M-fractional derivative with respect to time-spatial variables, obtained by the Kudryashov method and their computer models; - gauge and algebraic-geometric equivalence between the Camassa-Holm equation and the spin system with integer derivatives; - solutions of the Camassa-Holm equation with fractional derivatives and their computer models, comparative analysis between solutions of the Camassa-Holm equation with integer and fractional derivatives and main conclusions. Theoretical and practical significance. The results obtained on the topic of the dissertation work can be used for an in-depth study of the equations of mathematical physics of the fractional derivative. The methods supplemented and expanded in this work are available for studying and solving other models of mathematical physics. Reliability and validity of the research results. The reliability and validity of scientific results is confirmed by publications in international peer-reviewed journals included in the Web of Science/Scopus databases, as well as citations from foreign scientists. The article, published in the journal «Fractal and Fractional», contains 17 citations, without self-citation. Scientometric indicators of doctoral student A. G. Taishiyeva: H-index – 1 in the Web of Science database; h-index – 2 in the Scopus database. Approbation of the obtained results. The research results obtained from the dissertation work were reported at the following scientific conferences: 1. 10th International Conference on Mathematical Modeling in Physical Sciences (Greece), 2021; 2. Ufa Basic Mathematical School (Russia, Ufa с.), 2019; 3. International conference «Modern problems of mathematics and computer science», dedicated to the 10th anniversary of the Department of Mathematical and Computer Modeling (Astana c.), 2024; 4. XVIII International Conference of Students and Young Scientists «Science and Education – 2024» (Astana c.), 2024. Also, discussed at scientific seminars of the departments «Mathematical and Computer Modeling» and «General and Theoretical Physics». Publications on the topic of the dissertation. As part of the research on the topic of the dissertation, 6 works were published: 1. The Sensitive Visualization and Generalized Fractional Solitons’ Construction for Regularized Long-Wave Governing Model // Fractal and Fractional. – 2023. – №7. – Р. 136 (IF 3.577, WoS CC: Q1, Scopus CS: 86). 2. Effect of truncated M-fractional derivative on the new exact solitons to the Shynaray-IIA equation and stability analysis // Results in Physics. – 2024. - №57. - р. 107422 (IF 5.51, WoS CC: Q2, Scopus CS: 89). 3. On equivalence of one spin system and two-component Camassa-Holm equation // Ufa Mathematical Journal. – 2020. – Vol. 12, №2. – P. 50-55 (IF 0.34, WoS CC: Q3, Scopus CS: 34). 4. Geometric Flows of Curves, Two-Component Camassa-Holm Equation and Generalized Heisenberg Ferromagnet Equation // Journal of Physics: Conference Series. – 2021. - №2090. - р. 012068 (Scopus CS:18). 5. On the equivalence of one spin system and the two-component Сamassa-Holm formation // Collection of abstracts of the international scientific conference «Ufa Basic Mathematical School» (Ufa, 2019). 6. Бөлшек туындылы Камасса-Холм теңдеуі және оның шешімдері туралы // International scientific conference «Modern problems of mathematics and computer science» (L. N. Gumilyov ENU, Astana, 2024). Structure of the dissertation. The dissertation includes a section of definitions and notations, introduction, 4 main chapters, conclusion and bibliographies. The volume of work is 86 pages, including 16 images. In the first part, the beta and M-reduced fractional derivative operators are applied to the ordered long-wavelength Burgers equation. This fractional time derivative equation is initially converted to a fractional ordinary differential equation using wave transformation. A new auxiliary equation approach was used to find soliton solutions. As a result, based on the ratio of constraints to the parameters of the auxiliary equation, light, periodic, combined-periodic, rational, and soliton solutions were found. Graphical representations of the obtained results are shown by selecting appropriate parametric values and predicting whether the fractional order parameter is responsible for controlling the propagation behavior of local waves. In the second part, solutions to the nonlinear Schrödinger equation with a consistent potential with a reduced M-fractional derivative are found using the Kudryashov method, and their three-dimensional and two-dimensional computer models are compiled. In the third section, the gauge and algebraic-geometric connection between the fractional-order Camassa-Holm equation and the generalized Heisenberg model was established, and an exact solution of the latter was found. In the fourth part, a soliton solution of the Camassa-Holm equation with a fractional derivative is obtained and computer images are created. A comparative analysis of solutions to the equation under consideration, found in two cases - integer and fractional derivatives, is carried out. Internal unity of the dissertation work The results obtained within the framework of the dissertation topic are presented according to the principle «from simple to complex» depending on the structure of the research objects. The equation studied in the first chapter is a generalization of the classical Burgers equation. Likewise, the Schrödinger-type equation and the Camassa-Holm equation discussed in chapters two, three, and four cover the ordered long-wavelength Burgers equation studied in chapter one. Structure of the dissertation.