The ordinary differential equation of second order is being used in many engineering disciplines and sciences to model many real-life problems. These problems are mostly uncertain, vague and incomplete and thus they require some more advanced tool for modelling. Neutrosophic logic becomes the solution of all these kind of uncertain problems as it describe the conditions of uncertainty which occurs during the process of modelling on the basis of grade of membership of truth values, indeterminacy values and falsity values, that means it consider all the uncertain parameters on the basis of these degrees. In this research paper, we have considered the ordinary differential equation of second order with neutrosophic numbers as initial conditions of spring mass system is solved using Sumudu transform method which has advantage of unit preserving property over the well established Laplace Transform method. The solution obtained at various computational points by this method is shown in the form of table. Furthermore, the results obtained at different (α, β, γ)-cut and time values are also depicted graphically and are verified analytically by de-fuzzifying the data.
Neutrosophic Logic is a tool based on non-standard analysis to represent mathematical model of uncertainty, vagueness, ambiguity, incompleteness, and inconsistency. In Neutrosophic set, indeterminacy is quantified explicitly whereas the truth membership, indeterminacy membership, and falsity membership are independent. This plays a vital role in many situations whenwehandle inconsistent and incomplete information. In modeling problems, differential equations have major applications in the field of science and engineering and the study of differential equationwith uncertainty is one of emerging field of research. In this paper, the differential equations in neutrosophic environment are explored, also the solution of second-order linear differential equation with trapezoidal neutrosophic numbers as boundary conditions is discussed. Furthermore, the numerical example is given to demonstrate the solution with different values of (α, β, γ )-cut of trapezoidal neutrosophic number.
“Neutrosophic Sets and Systems” has been created for publications on advanced studies in neutrosophy, neutrosophic set, neutrosophic logic, neutrosophic probability, neutrosophic statistics that started in 1995 and their applications in any field, such as the neutrosophic structures developed in algebra, geometry, topology, etc.
The present book entitled “ A Solution for Ordinary Differential Equations- Solving Techniques and Applications” has been written so as to cover the syllabi of mathematics of various semesters of all the branches of engineering and for under graduate and post graduate students of most of the universities in our country.
In this paper the Neutrosophic ordinary differential equation of first order via neutrosophic numbers is epitomized. We also intend to define the neutrosophic numbers and their (α, β, γ)-cut. Finally a numerical example is given to demonstrate its practicality and effectiveness of the differential equation involving neutrosophic numbers.
This book is an invaluable resource for applied researchers to find the analytical solution of differential equations describing the dynamical system with less computational effort and time. It describes the basic concepts of the differential transform method and solution of various real-world problems described by simple to complicated differential equations. It provides a computational technique that is not only conceptually simple and easy to use but also readily adaptable for computer coding. Different chapters of the book deal with the basic differential equations involved in the physical phenomena as well as a complicated system of differential equations described by the mathematical model. The book offers comprehensive coverage of the most essential topics, including Basic concepts and fundamental properties of the proposed technique with proof The solution of linear, nonlinear, homogeneous, and nonhomogeneous ordinary differential equations (ODEs) and partial differential equations (PDEs) The initial and boundary value problems Real-world ODE and PDE problems are also discussed Applications of Differential Transform to Real World Problems is primarily aimed at undergraduates, graduates, and researchers studying differential equations. Scientists dealing with complicated differential equations or systems of differential equations will also find this book useful.
The world abounds with introductory texts on ordinary differential equations and rightly so in view of the large number of students taking a course in this subject. However, for some time now there is a growing need for a junior-senior level book on the more advanced topics of differential equations. In fact the number of engineering and science students requiring a second course in these topics has been increasing. This book is an outgrowth of such courses taught by us in the last ten years at Worcester Polytechnic Institute. The book attempts to blend mathematical theory with nontrivial applications from varipus disciplines. It does not contain lengthy proofs of mathemati~al theorems as this would be inappropriate for its intended audience. Nevertheless, in each case we motivated these theorems and their practical use through examples and in some cases an "intuitive proof" is included. In view of this approach the book could be used also by aspiring mathematicians who wish to obtain an overview of the more advanced aspects of differential equations and an insight into some of its applications. We have included a wide range of topics in order to afford the instructor the flexibility in designing such a course according to the needs of the students. Therefore, this book contains more than enough material for a one semester course.
A liinear second order differential equation may be considered as a 2 X 2 system of first order equations. The question is whether the solutions of this system can be written in the form exp Omega is a 2 X2 matrix. A motivation for the problem is given, based on the question of defining "lump constants" for an inhomogenous layer. Conditions necessary for the existence of Omega are given for a variety of circumstances.
Fuzzy sets have long been employed to handle imprecise and uncertain information in the real world, but their limitations in dealing with incomplete and inconsistent data led to the emergence of neutrosophic sets. In this thought-provoking book, titled Data-Driven Modelling with Fuzzy Sets: A Neutrosophic Perspective, the authors delve into the theories and extensive applications of neutrosophic sets, ranging from neutrosophic graphs to single-valued trapezoidal neutrosophic sets and their practical implications in knowledge management, including student learning assessment, academic performance evaluation, and technical article screening. This comprehensive resource is intended to benefit mathematicians, physicists, computer experts, engineers, scholars, practitioners, and students seeking to deepen their understanding of neutrosophic sets and their practical applications in diverse fields. This book comprises 11 chapters that provide a thorough examination of neutrosophic set theory and its extensions. Each chapter presents valuable insights into various aspects of data-driven modeling with neutrosophic sets and explores their applications in different domains. The book covers a wide range of topics. The specific topics covered in the book include neutrosophic submodules, applications of neutrosophic sets, solutions to differential equations with neutrosophic uncertainty, cardinalities of neutrosophic sets, neutrosophic cylindrical coordinates, applications to graphs and climatic analysis, neutrosophic differential equation approaches to growth models, neutrosophic aggregation operators for decision making, and similarity measures for Fermatean neutrosophic sets. The diverse contributions from experts in the field, coupled with the constructive feedback from reviewers, ensure the book's high quality and relevance. This book presents a qualitative assessment of big data in the education sector using linguistic quadripartitioned single-valued neutrosophic soft sets showcases application of n-cylindrical fuzzy neutrosophic sets in education using neutrosophic affinity degree and neutrosophic similarity index covers scientific evaluation of student academic performance using single-valued neutrosophic Markov chain illustrates multi-granulation single-valued neutrosophic probabilistic rough sets for teamwork assessment examines estimation of distribution algorithms based on multiple-attribute group decision-making to evaluate teaching quality With its wealth of knowledge, this book aims to inspire further research and innovation in the field of neutrosophic sets and their extensions, providing a valuable resource for scholars, practitioners, and students alike.
