115,499 research outputs found
INTEGRATION OF INTERDISCIPLINARY SCIENTIFIC KNOWLEDGE IN TEACHING INVERSE PROBLEMS FOR DIFFERENTIAL EQUATIONS
Problem and goal. In the process of teaching students inverse problems for differential equations, one of the important goals is to form students ‘ fundamental knowledge in the field of inverse problems, in the field of applied and computational mathematics; to develop mathematical creativity that allows students after graduation, working in research institutions, to successfully solve a variety of complex mathematical problems in the implementation of practical applied research. Methodology. In the process of teaching students inverse problems for differential equations, the system of humanitarian-oriented training sessions is designed, the methods of rational reasoning are used, an individual approach to learning is implemented. Results. Humanitarian-oriented training sessions on inverse problems for differential equations are aimed at creating situations that require students, according to the results of solving the inverse problem, to make logical conclusions of applied and humanitarian character, to overcome moral contradictions, to make a reasonable choice of the right position in society. The skills and experience gained in the course of training in the application of rational reasoning in the study of inverse problems for differential equations allow students not only to effectively investigate applied problems, but also to form fundamental knowledge in the field of applied mathematics. Individual approach in teaching inverse problems for differential equations acts as a didactic principle of training, education and development of students, taking into account the personal characteristics of students, the level of intellectual development, cognitive interests and other factors that affect the success of learning. Conclusion. Humanitarian-oriented training sessions on inverse problems for differential equations, methods of rational reasoning, individual approach to learning allows the students to form a system of fundamental knowledge in inverse problems for partial differential equations, integrating multidisciplinary scientific knowledge, to identify humanitarian and scientific-educational potential of such learning, to justify the positive contribution of teaching inverse problems for differential equations in humanization and fundamentalization of mathematical education
Direct and Inverse Computational Methods for Electromagnetic Scattering in Biological Diagnostics
Scattering theory has had a major roll in twentieth century mathematical
physics. Mathematical modeling and algorithms of direct,- and inverse
electromagnetic scattering formulation due to biological tissues are
investigated. The algorithms are used for a model based illustration technique
within the microwave range. A number of methods is given to solve the inverse
electromagnetic scattering problem in which the nonlinear and ill-posed nature
of the problem are acknowledged.Comment: 61 pages, 5 figure
New integral transform: Shehu transform a generalization of Sumudu and Laplace transform for solving differential equations
In this paper, we introduce a Laplace-type integral transform called the
Shehu transform which is a generalization of the Laplace and the Sumudu
integral transforms for solving differential equations in the time domain. The
proposed integral transform is successfully derived from the classical Fourier
integral transform and is applied to both ordinary and partial differential
equations to show its simplicity, efficiency, and the high accuracy
Inverse heat conduction problems by using particular solutions
Based on the method of fundamental solutions, we develop in this paper a new computational method to solve two-dimensional transient heat conduction inverse problems. The main idea is to use particular solutions as radial basis functions (PSRBF) for approximation of the solutions to the inverse heat conduction problems. The heat conduction equations are first analyzed in the Laplace transformed domain and the Durbin inversion method is then used to determine the solutions in the time domain. Least-square and singular value decomposition (SVD) techniques are adopted to solve the ill-conditioned linear system of algebraic equations obtained from the proposed PSRBF method. To demonstrate the effectiveness and simplicity of this approach, several numerical examples are given with satisfactory accuracy and stability.Peer reviewe
An inverse Sturm-Liouville problem with a fractional derivative
In this paper, we numerically investigate an inverse problem of recovering
the potential term in a fractional Sturm-Liouville problem from one spectrum.
The qualitative behaviors of the eigenvalues and eigenfunctions are discussed,
and numerical reconstructions of the potential with a Newton method from finite
spectral data are presented. Surprisingly, it allows very satisfactory
reconstructions for both smooth and discontinuous potentials, provided that the
order of fractional derivative is sufficiently away from 2.Comment: 16 pages, 6 figures, accepted for publication in Journal of
Computational Physic
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