Soft Numerical Computing in Uncertain Dynamic Systems

1st Edition - August 19, 2020
Authors: Tofigh Allahviranloo, Witold Pedrycz
Language: English
Paperback ISBN:
9 7 8 - 0 - 1 2 - 8 2 2 8 5 5 - 5
eBook ISBN:
9 7 8 - 0 - 1 2 - 8 2 2 9 9 4 - 1

Soft Numerical Computing in Uncertain Dynamic Systems is intended for system specialists interested in dynamic systems that operate at different time scales. The book discusses… Read more

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Soft Numerical Computing in Uncertain Dynamic Systems is intended for system specialists interested in dynamic systems that operate at different time scales. The book discusses several types of errors and their propagation, covering numerical methods—including convergence and consistence properties and characteristics—and proving of related theorems within the setting of soft computing. Several types of uncertainty representation like interval, fuzzy, type 2 fuzzy, granular, and combined uncertain sets are discussed in detail. The book can be used by engineering students in control and finite element fields, as well as all engineering, applied mathematics, economics, and computer science students.

One of the important topics in applied science is dynamic systems and their applications. The authors develop these models and deliver solutions with the aid of numerical methods. Since they are inherently uncertain, soft computations are of high relevance here. This is the reason behind investigating soft numerical computing in dynamic systems. If these systems are involved with complex-uncertain data, they will be more practical and important. Real-life problems work with this type of data and most of them cannot be solved exactly and easily—sometimes they are impossible to solve.

Clearly, all the numerical methods need to consider error of approximation. Other important applied topics involving uncertain dynamic systems include image processing and pattern recognition, which can benefit from uncertain dynamic systems as well. In fact, the main objective is to determine the coefficients of a matrix that acts as the frame in the image. One of the effective methods exhibiting high accuracy is to use finite differences to fill the cells of the matrix.

1. Introduction1.1. Importance of the subjects of the book1.2. Motivation1.3. The structure of the book2. Uncertain sets2.1. Introduction to aspects of uncertainty2.2. Uncertainty2.2.1. Distribution functions2.2.2. Uncertainty distribution functions2.2.3. Uncertain variable2.2.4. Uncertain set2.2.5. Function of uncertain set2.2.6. Membership function2.2.7. Interval parametric form2.2.8. Distance between uncertain sets2.2.9. Combined uncertain sets2.2.10. Level wise parametric format of a Combined uncertain set3. Soft computing with uncertain sets 3.1. Introduction to soft computing3.2. Computing with uncertain sets3.2.1. Extension principle-based operations3.2.2. Computations with combined uncertain sets3.2.3. Computations with interval parametric form3.2.4. Difference between uncertain sets4. Continuous numerical solution of uncertain differential equations4.1. Introduction 4.2. Uncertain differential equations4.2.1. First order uncertain differential equations4.2.1.1. Uncertain Taylor expansion methods4.2.1.2. Uncertain Homotopy perturbation method4.2.1.3. Uncertain Homotopy analysis perturbation method4.2.1.4. Uncertain differential transform method4.2.2. High order uncertain differential equation4.2.2.1. Uncertain Taylor expansion methods4.2.2.2. Uncertain Homotopy perturbation method4.2.2.3. Uncertain Homotopy analysis perturbation method 4.2.2.4. Uncertain differential transform method5. Discrete numerical solution of uncertain differential equations5.1. Introduction5.2. First order uncertain differential equations5.2.1. Uncertain difference methods5.2.1.1. Uncertain Euler’s method5.2.1.2. Uncertain Runge-Kutta’s method ….5.2.2. Uncertain multi-step methods5.2.2.1. Uncertain explicit methods5.2.2.2. Uncertain implicit methods6. Numerical solution of uncertain fractional differential equations6.1. Introduction6.2. Uncertain fractional differential equations6.3. Numerical solution of uncertain fractional differential equations6.4. Convergence, Consistence and stability of the methods7. Numerical solution of uncertain partial differential equations7.1. Introduction7.2. Uncertain partial differential equations7.3. Numerical solutions of uncertain partial differential equations7.4. Convergence, Consistence and stability

Tofigh Allahviranloo

Tofigh Allahviranloo is a full professor of applied mathematics at Istinye University, Turkey. As a trained mathematician and computer scientist, Prof. Allahviranloo has developed a passion for multi- and interdisciplinary research. He is not only deeply involved in fundamental research in fuzzy applied mathematics, especially fuzzy differential equations, but he also aims at innovative applications in the applied biological sciences. He is the author of several books and many papers published by Elsevier and Springer. He actively serves the research community, as Editor-in-Chief of the International J. of Industrial Mathematics, and Associate Editor or editorial board member of several other journals, including Information Sciences, Fuzzy Sets and Systems, Journal of Intelligent and Fuzzy Systems, Iranian J. of Fuzzy Systems and Mathematical Sciences.

Affiliations and expertise

Full Professor, Istinye University, Istanbul, Turkey

Witold Pedrycz

Dr. Witold Pedrycz (IEEE Fellow, 1998) is Professor and Canada Research Chair (CRC) in computational intelligence in the Department of Electrical and Computer Engineering, University of Alberta, Edmonton, Canada. In 2012 he was elected a fellow of the Royal Society of Canada. His main research directions involve computational intelligence, fuzzy modeling and granular computing, knowledge discovery and data science, pattern recognition, data science, knowledge-based neural networks, and control engineering. He is also an author of 18 research monographs and edited volumes covering various aspects of computational intelligence, data mining, and software engineering. Dr. Pedrycz is vigorously involved in editorial activities. He is the editor-in-chief of Information Sciences, editor-in-chief of WIREs Data Mining and Knowledge Discovery, and co-editor-in-chief of International Journal of Granular Computing, and Journal of Data Information and Management. He serves on the advisory board of IEEE Transactions on Fuzzy Systems.

Affiliations and expertise

Professor, Department of Electrical and Computer Engineering, University of Alberta, Canada

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Soft Numerical Computing in Uncertain Dynamic Systems

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Tofigh Allahviranloo

Witold Pedrycz