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Introduction to Numerical Methods
Published in Bilal M. Ayyub, Richard H. Mccuen, Numerical Analysis for Engineers, 2015
Bilal M. Ayyub, Richard H. Mccuen
Many operations in mathematical analysis can be identified as belonging to one of two types: analytical or numerical. For example, when we need to solve a second-order polynomial aX2 + bX + c = 0, for the values of X that satisfy the equality, it is common to use the closed-form solution X=(-b±b2-4ac)/2a. This is an analytical solution. Alternatively, we could assume some value for the solution (X), and through a systematic method of trial and error, we could obtain essentially the same solution as the analytical approach. However, this numerical approach is much more tedious, so when an analytical solution exists, it is usually the preferred method. For many types of problems, such as solving a fifth-order polynomial, a closed-form or analytical solution does not exist. Then the iterative, or numerical, approach must be used.
Introduction
Published in Vladimir A. Dobrushkin, Applied Differential Equations with Boundary Value Problems, 2017
The independent discovery of the calculus by I. Newton and G. Leibniz was immediately followed by its intensive application in mathematics, physics, and engineering. Since the late seventeenth century, differential equations have been of fundamental importance in the study, development, and application of mathematical analysis. Differential equations and their solutions play one of the central roles in the modeling of real-life phenomena.
Introduction
Published in Vladimir A. Dobrushkin, Applied Differential Equations, 2022
The independent discovery of the calculus by I. Newton and G. Leibniz was immediately followed by its intensive application in mathematics, physics, and engineering. Since the late seventeenth century, differential equations have been of fundamental importance in the study, development, and application of mathematical analysis. Differential equations and their solutions play one of the central roles in the modeling of real-life phenomena.
Accelerating flow for engine oil base fluid with graphene oxide and molybdenum disulfide nanoparticles: modified fractional simulations
Published in Waves in Random and Complex Media, 2023
Ali Raza, Sami Ullah Khan, Muhammad Yasir, Sumera Dero
Fractional calculus is a branch of mathematical analysis that deals with generalizations of differentiation and integration to non-integer orders. It extends the concepts of derivatives and integrals to fractional or non-integer orders, allowing for the analysis of functions with fractional or non-integer degrees of differentiability. Fractional calculus has found applications in various fields, including physics, engineering, finance, signal processing, and control theory. It provides powerful mathematical tools for modeling and understanding complex systems that exhibit fractional order behavior, offering insights and solutions that are not achievable with traditional integer-order calculus. In fractional calculus, various mathematical algorithms have been developed. The Caputo–Fabrizio fractional derivative is a specific type of fractional derivative operator introduced by Caputo and Fabrizio in 2013. It is used in the field of fractional calculus to generalize the concept of differentiation to non-integer orders. The Caputo–Fabrizio fractional derivative provides a way to analyze and model phenomena that exhibit fractional order dynamics, such as fractional diffusion and anomalous transport processes. It is one of several fractional derivative operators used in fractional calculus, each with its own properties and applications [31–34].
A multidimensional approach to training mathematics students at a university: improving the efficiency through the unity of social, psychological and pedagogical aspects
Published in International Journal of Mathematical Education in Science and Technology, 2018
Elena Kuznetsova, Marina Matytcina
We conducted an internet survey of first-year students of applied mathematics majors from 27 Russian regional universities. In Russia, there are about 100 regions and in almost each of them there is a university that trains industrial mathematicians for regional enterprises. Education in Russian schools is regulated by the Federal State Educational Standard. The final certifying examination is the Unified State Exam (the USE) in a number of academic disciplines. Examination in mathematics is compulsory. The USE serves as both school finals and university entrance examinations. Currently, the USE in Mathematics contains 19 tasks on algebra, geometry, fundamentals of mathematical analysis and probability theory. The tasks are arranged in order of increasing complexity: from the simplest tasks to tasks, which requires the use of non-standard approaches and informal reasoning. USE results are estimated on a 100-point scale, which is also converted into a 5-point scale traditional in Russia.
Inverse nodal problem for a conformable fractional diffusion operator
Published in Inverse Problems in Science and Engineering, 2021
The fractional derivative based on 1695 is widely used in applied mathematics and mathematical analysis. Since then, many researchers have developed different types of fractional derivative (see [29–32]). In 2014, Khalil et al. introduced the definition of conformable fractional derivative (see [33]). In 2015, the basic properties and main results of this derivative were given by Abdeljawad and Atangana et al. [34,35]. This derivative arises in various fields such as quantum mechanics, dynamical systems, time scale problems, diffusions, conservation of mass, etc. (see [36–39]).