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 Front Matter Copyright Dedication Preface Introduction Introduction Introduction to uncertain dynamic systems History Structure of the book References Uncertain sets Short introduction to this chapter Textual short outline Measures Measurable space Examples Uncertain sets and variables Examples Zigzag uncertain variable Experimental uncertain variables Membership function Fuzzy numbers and their properties Definition of a fuzzy number Level-wise form of a fuzzy number Definition of a fuzzy number in level-wise form Definition of a fuzzy number in level-wise form A singleton fuzzy number Definition of a fuzzy number in parametric form Nonlinear fuzzy number Trapezoidal fuzzy number Triangular fuzzy number Operations on level-wise form of fuzzy numbers Summation Example Multiplication Difference Hukuhara difference Example Example Generalized Hukuhara difference The level-wise form of generalized difference Some properties of gH-difference Example Partial ordering Some properties of partial ordering Absolute value of a fuzzy number Some properties of partial ordering in gH-difference Approximately generalized Hukuhara difference Some properties of g-difference Example Example Generalized division Some properties of division Examples Approximately generalized division Example Piece-wise membership function Some properties of addition and scalar product on fuzzy numbers Definition-singleton fuzzy number Advanced uncertainties and their properties Pseudo-octagonal sets Z-process Definition-Z-process Example Example Computations on Z-numbers Summation of two Z-numbers Difference of two Z-numbers Multiplication of two Z-numbers Division of two Z-numbers Level-wise form of a Z-number High membership degree does have high reliability Definition-Level-wise form of a standard Z-number Summation in level-wise form Scalar multiplication in level-wise form Hukuhara difference in level-wise form Generalized Hukuhara difference in level-wise form Some properties of generalized Hukuhara References Further reading Soft computing with uncertain sets Introduction Expected value Distance of two fuzzy numbers P-distance Hausdorff Distance Limit of fuzzy number valued functions Definition-Fuzzy set valued function Definition-Fuzzy number valued function Definition-The limit of Fuzzy number valued function Theorem-Limit of summation of functions Theorem-Limit of difference of functions Theorem-Limit of multiplication Other properties of limit Fuzzy Riemann integral operator Some properties of fuzzy Riemann integral Differential operator Definition-gH-differentiability Example Example Definition-gH-differentiability in level-wise form Definition-Switching points of gH-differentiability Example Proposition-Summation in gH-differentiability Proposition-Difference in gH-differentiability Proposition-Production in gH-differentiability Proposition-Composition of gH-differentiability Proposition-Minimum and maximum Definition-Continuous fuzzy number valued function Proposition Proposition-Cauchy's fuzzy mean value theorem Corollary-Fuzzy mean value theorem Proposition-Increasing and decreasing function Proposition-Integral of gH-differentiability Proposition-Switching points in integration High order differentiability Extended integral relation Part-by-part integration Taylor expansion Example gH-partial differentiability Example Another simple example Level-wise form of gH-partial differentiability Switching point in gH-partial differentiability Example Higher order of gH-partial differentiability Integral relation in gH-partial differentiability Multivariate fuzzy chain rule in gH-partial differentiability The fuzzy Laplace transform operator Example Definition-Absolutely convergence First translation theorem Second translation theorem Laplace transform on the derivative Derivative theorem High order derivation theorem Fuzzy improper integral Definition-Uniform convergence Theorem-Interchanging integrals Theorem-Integral and derivative Fourier transform operator Definition-Fuzzy Fourier transform Example-Fuzzy Fourier transform Definition-Fuzzy inverse Fourier transform Theorem-Existence Theorem-Linearity property Theorem-Fourier transform of gH-derivative References Continuous numerical solutions of uncertain differential equations Introduction Uncertain differential equations Definition-Uncertain process as a canonical Liu process Definition-Liu integral of an uncertain process Theorem-Chain rule Theorem-Integration by parts Definition-Uncertain differential equation Remark Fuzzy differential equations Theorem-Existence and uniqueness Fuzzy differential equations-Variation of constants Theorem-Existence of the solution Length function Definition-Length function Nonlinear property Theorem-Nonlinear property of fuzzy functions Remark Remark-Differentiability and length Theorem-Nonlinear property of fuzzy functions Theorem-Derivative of integral equation The length function-Fuzzy differential equations Fuzzy differential equations-Laplace transform Fuzzy differential equations-Second order Fuzzy differential equations-Variational iteration method Fuzzy differential equations-Legendre differential equation Definition-Power series with fuzzy coefficients Some properties of fuzzy series Fuzzy calculated operations Fuzzy power series method for solving Legendre's equation Linear systems of fuzzy differential equations Homogeneous fuzzy linear differential systems Nonhomogeneous fuzzy linear differential systems Reduction of a second order fuzzy differential equations to a system of first order equations Z-differential equations References Discrete numerical solutions of uncertain differential equations Introduction Fuzzy Euler method Analysis of the fuzzy Euler method Local truncation error and consistency Global truncation error and convergence Theorem-Convergence Stability Fuzzy modified Euler method Analysis of the fuzzy modified Euler method Local truncation error and consistency Global truncation error and convergence Theorem-Convergence Stability of the modified fuzzy Euler method Fuzzy Euler method for fuzzy hybrid differential equations Fuzzy Euler method for fuzzy impulsive differential equations Error analysis Stability Fuzzy predictor and corrector methods Definition-Fuzzy explicit method Definition-Fuzzy implicit method Fuzzy explicit three steps method Fuzzy implicit two steps method Fuzzy predictor and corrector three steps methods Numerical solution of fuzzy nth-order differential equations References Further reading Numerical solutions of uncertain fractional differential equations Introduction Fuzzy Riemann-Liouville Derivative-Fuzzy RL Derivative Note-Combination Property Level-Wise form of Fuzzy Riemann-Liouville Integral Operators The RL Fractional Integral Operator The Fuzzy Riemann-Liouville Derivative Operators Fuzzy Caputo Fractional Derivative Caputo gH-Differentiability Caputo-Katugampola gH-Fractional Derivative Fuzzy Fractional Differential Equations-Caputo-Katugampola Derivative Definition-Fuzzy Fractional Differential Equations Existence and Uniqueness of the Solution Theorem-Existence and Uniqueness in Real Fractional Differential Equation Theorem-Existence and Uniqueness in Fuzzy Fractional Differential Equation Some Properties of the Mittag-Leffler Function Fuzzy Generalized TaylorS Expansion Fuzzy Fractional Euler Method References Numerical solutions of uncertain partial differential equations Introduction Partial ordering Continuity Minimum and maximum Production in partial gH-differentiability Fuzzy integrating factor The fuzzy heat equation Theorem-Fuzzy maximum principle Theorem-Existence Analytical solution of the fuzzy heat equation The fundamental solution of the fuzzy heat equation Fuzzy Fourier transform Fuzzy inverse Fourier transform Fourier transform of gH-derivative Fuzzy finite difference method for solving the fuzzy Poisson's equation Theorem-Uniqueness Error analysis References Index A B C D E F G H I L M N P R S T U V W Z __Soft Numerical Computing in Uncertain Dynamic Systems__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.