Soft Numerical Computing in Uncertain Dynamic Systems

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. 

• Explores dynamic models, how time is fundamental to the structure of the model and data, and how a process unfolds
• Investigates the dynamic relationships between multiple components of a system in modeling using mathematical models and the concept of stability in uncertain environments
• Exposes readers to many soft numerical methods to simulate the solution function’s behavior

Researchers, professionals, and graduate students in computer science & engineering, bioinformatics, and electrical engineering

1. Introduction
   1.1. Importance of the subjects of the book
   1.2. Motivation
   1.3. The structure of the book
   2. Uncertain sets
   2.1. Introduction to aspects of uncertainty
   2.2. Uncertainty
   2.2.1. Distribution functions
   2.2.2. Uncertainty distribution functions
   2.2.3. Uncertain variable
   2.2.4. Uncertain set
   2.2.5. Function of uncertain set
   2.2.6. Membership function
   2.2.7. Interval parametric form
   2.2.8. Distance between uncertain sets
   2.2.9. Combined uncertain sets
   2.2.10. Level wise parametric format of a Combined uncertain set
   3. Soft computing with uncertain sets
   3.1. Introduction to soft computing
   3.2. Computing with uncertain sets
   3.2.1. Extension principle-based operations
   3.2.2. Computations with combined uncertain sets
   3.2.3. Computations with interval parametric form
   3.2.4. Difference between uncertain sets
   4. Continuous numerical solution of uncertain differential equations
   4.1. Introduction
   4.2. Uncertain differential equations
   4.2.1. First order uncertain differential equations
   4.2.1.1. Uncertain Taylor expansion methods
   4.2.1.2. Uncertain Homotopy perturbation method
   4.2.1.3. Uncertain Homotopy analysis perturbation method
   4.2.1.4. Uncertain differential transform method
   4.2.2. High order uncertain differential equation
   4.2.2.1. Uncertain Taylor expansion methods
   4.2.2.2. Uncertain Homotopy perturbation method
   4.2.2.3. Uncertain Homotopy analysis perturbation method
   4.2.2.4. Uncertain differential transform method
   5. Discrete numerical solution of uncertain differential equations
   5.1. Introduction
   5.2. First order uncertain differential equations
   5.2.1. Uncertain difference methods
   5.2.1.1. Uncertain Euler’s method
   5.2.1.2. Uncertain Runge-Kutta’s method ….
   5.2.2. Uncertain multi-step methods
   5.2.2.1. Uncertain explicit methods
   5.2.2.2. Uncertain implicit methods
   6. Numerical solution of uncertain fractional differential equations
   6.1. Introduction
   6.2. Uncertain fractional differential equations
   6.3. Numerical solution of uncertain fractional differential equations
   6.4. Convergence, Consistence and stability of the methods
   7. Numerical solution of uncertain partial differential equations
   7.1. Introduction
   7.2. Uncertain partial differential equations
   7.3. Numerical solutions of uncertain partial differential equations
   7.4. Convergence, Consistence and stability