Polynomial Paradigms: Trends and applications in science and engineering

Description

Polynomials play an important role in developing numerical and analytical methods to solve various practical problems of physics, mathematics, engineering and industry. This research and reference text reports and reviews recent developments and applications of different polynomials in numerical and analytical/semi-analytical methods for solving a variety of science and engineering problems. It contains contributions from leading experts in areas such as basic theory and concepts of polynomials, mathematical modelling, mathematical physics, engineering, high-order numerical methods for differential, integral and integro-differential equations, artificial intelligence, fuzzy and interval based models and beyond. This book would be useful for graduates and researchers of various sciences and engineering fields. PRELIMS.pdf Preface Acknowledgement Editor biography S Chakraverty List of contributors CH001.pdf Chapter 1 Formulas for the sums of the series of reciprocals of the cubic polynomials with integer roots, at least one zero 1.1 Introduction 1.2 Integer roots of the special reduced cubic polynomial 1.3 The case of triple zero root 1.4 The case of double zero root 1.4.1 The case of double zero root and one negative integer root 1.4.2 The case of double zero root and one positive integer root 1.4.3 The case of double zero root and one arbitrary integer root 1.5 The case of one zero root 1.5.1 The case of two different negative integer roots and one zero root 1.5.2 The case of one negative, one zero and one positive integer root 1.5.3 The case of two different positive integer roots and one zero root 1.5.4 The case of one zero root and two arbitrary integer roots 1.5.5 Three special cases of one zero and two integer roots 1.6 Some approximate values for seven types of the sums 1.7 Conclusion References CH002.pdf Chapter 2 Polynomials for meshless methods in finding solutions in gradient elasticity problems 2.1 Introduction 2.2 Strain gradient plate model 2.2.1 Kinematics 2.2.2 Constitutive equation 2.2.3 Equilibrium equations 2.3 Mesh free approach 2.3.1 RBF 2.3.2 Discretized equations 2.4 Numerical analyses and verification 2.4.1 Isotropic plates 2.4.2 Composite plates 2.5 Dynamic analysis 2.6 Conclusions References CH003.pdf Chapter 3 Numerical solution of fractal-fractional variable orders differential equations using two-step and three-step Newton and Lagrange interpolation polynomials 3.1 Introduction 3.2 Definitions and notations 3.3 Algorithm for fractal-fractional variable-order differential equation based on Newton interpolation polynomial via power-law type kernel 3.3.1 Implementation of two-step Newton interpolation polynomial 3.3.2 Implementation of three-step Newton interpolation polynomial 3.4 Algorithm for fractal-fractional variable-order differential equation based on Newton interpolation polynomials via Mittag-Leffler type kernel 3.4.1 Implementation of two-step Newton interpolation polynomial 3.4.2 Implementation of three-step Newton interpolation polynomial 3.5 Algorithm for fractal-fractional variable-order differential equation based on Lagrange polynomial interpolation via power-law type kernel 3.5.1 Implementation of two-step Lagrange interpolation polynomial 3.5.2 Implementation of three-step Lagrange interpolation polynomial 3.6 Algorithm for fractal-fractional variable-order differential equation based on Lagrange polynomial interpolation via Mittag-Leffler type kernel 3.6.1 Implementation of two-step Lagrange polynomial interpolation 3.6.2 Implementation of three-step Lagrange polynomial interpolation 3.7 Numerical examples 3.8 Conclusion Conflicts of interest Acknowledgement References CH004.pdf Chapter 4 Polynomial-based numerical methods for singularly perturbed differential equation on layer-adapted meshes Symbols 4.1 Introduction 4.2 Derivative bounds and solution decomposition 4.3 The discrete problem 4.3.1 Construction of non-uniform meshes 4.3.2 Cubic spline-based numerical method 4.3.3 Spline-based hybrid scheme 4.4 Error analysis 4.4.1 Error estimate on S-mesh 4.4.2 Error estimate on B–S mesh 4.5 Numerical results 4.6 Conclusion References CH005.pdf Chapter 5 Modelling the impact of preventive and treatment-based control interventions on the transmission dynamics of Leptospirosis disease 5.1 Introduction 5.2 Model formulation 5.2.1 Notations and meanings 5.3 Model qualitative analysis 5.3.1 Existence and positivity of solutions 5.3.2 Existence and stability of equilibrium points 5.3.3 Existence of disease-free equilibrium point 5.3.4 Reproduction number 5.3.5 Stability of disease-free equilibrium point 5.3.6 Existence of endemic equilibrium point 5.3.7 Global stability 5.4 Bifurcation analysis for the Leptospirosis model 5.5 Sensitivity and elasticity analysis of the parameters in the model 5.6 Determining intervention strategies for Leptospirosis diseases 5.7 Stochastic model of the transmission dynamics of Leptospirosis 5.7.1 Positivity of the solution for the SDE model 5.8 Homotopy analysis approach 5.9 Solution of the Leptospirosis model by HAM 5.10 Numerical results and discussions 5.11 Numerical simulations 5.11.1 Numerical simulations on the time evolution of the human and rodents population 5.11.2 Numerical simulation on the effect of pesticides and rodent control 5.11.3 Numerical simulations on the effect of treatment on the exposed and infected human population 5.11.4 Numerical simulations on the effect of variation of transmissibility rate on the susceptible human population 5.11.5 Numerical simulations on the variation of infected human and rodents against the force of infection (FOI) 5.11.6 Numerical simulation of the reproduction number against some important model parameters 5.11.7 Numerical simulation of the SDE model 5.12 Conclusion References CH006.pdf Chapter 6 Polynomials based semi-analytical methods for the solutions of fractional order Volterra-Fredholm integro differential equations Symbols 6.1 Introduction 6.2 Some definitions and properties 6.3 Model problem 6.4 Methodology 6.4.1 Adomian decomposition method (ADM) 6.4.2 ADM based on Chebyshev polynomials (ADM-CP) 6.4.3 ADM based on Bernstein polynomials (ADM-BP) 6.5 Analysis of the proposed methods 6.5.1 Existence and uniqueness of the solution 6.5.2 Error bound 6.6 Numerical experiments 6.7 Conclusion References CH007.pdf Chapter 7 Comparing different polynomials-based shape functions in the Rayleigh–Ritz method for investigating dynamical characteristics of nanobeam 7.1 Introduction 7.2 Preliminaries 7.2.1 Chebyshev polynomials 7.2.2 Legendre polynomials 7.2.3 Hermite polynomials 7.3 Governing equations of motion for the proposed model 7.4 Solution procedures 7.4.1 Application of different polynomials in Rayleigh–Ritz method 7.5 Numerical results and discussions 7.5.1 Validation 7.5.2 Convergence 7.5.3 Comparisons of shape functions with respect to convergence 7.6 Conclusion References CH008.pdf Chapter 8 Application of polynomial functions in analyzing anti-plane wave profiles in a functionally graded piezoelectric–viscoelastic–poroelastic structure with buffer layer 8.1 Introduction 8.2 Statement and geometry of the problem 8.3 Constitutive and governing equations 8.3.1 For PV layer 8.3.2 For PP half-space 8.3.3 For the buffer layer and air medium 8.3.4 For the coated film 8.4 Boundary conditions 8.5 Solution procedure involved 8.5.1 For PV layer 8.5.2 For the buffer layer 8.5.3 For PP half-space 8.5.4 For air medium 8.6 Dispersion relation 8.7 Special cases pertaining to this study 8.7.1 Case 1—validation with the work of [32] 8.7.2 Case 2—validation with the work of [42] 8.8 Numerical discussion 8.8.1 Effect of guiding layer width 8.8.2 Effect of sandwiched FG-buffer layer 8.8.3 Electromechanical coupling parameter (K2) 8.8.4 Effect of mass loading sensitivity 8.8.5 Attenuation of anti-plane wave 8.9 Conclusions Appendix A References CH009.pdf Chapter 9 Vibration analysis of single-link robotic manipulator by polynomial based Galerkin method in uncertain environment 9.1 Introduction 9.2 Preliminaries 9.2.1 Fuzzy number 9.2.2 Gaussian fuzzy number 9.2.3 Fuzzy Arithmetic 9.3 Mathematical modelling of single-link manipulator 9.4 Application of the Galerkin method in the present model 9.5 Proposed model in fuzzy environment using Gaussian fuzzy number 9.6 Results and discussions 9.7 Conclusion References CH010.pdf Chapter 10 Solving Type-2 Fuzzy Differential Equations Using Collocation Method with Type-2 Fuzzy Polynomials 10.1 Introduction 10.2 Preliminaries 10.2.1 Type-1 fuzzy numbers 10.2.2 Parametric form of fuzzy number 10.2.3 Type-2 fuzzy set 10.2.4 Vertical slice of type-2 fuzzy set 10.2.5 r1-plane of type-2 fuzzy set 10.2.6 FOU of a type-2 fuzzy set 10.2.7 LMF and UMF of a type-2 fuzzy set 10.2.8 Principle set of A˜ 10.2.9 r2-cut of r1-plane 10.2.10 Triangular perfect quasi type-2 fuzzy numbers 10.2.11 Type-2 fuzzy functions 10.2.12 Hukuhara differential of type-2 fuzzy numbers (H2 differential) 10.3 Proposed method 10.4 Numerical examples 10.5 Results and discussions 10.6 Conclusions Acknowledgments References CH011.pdf Chapter 11 Shannon entropy determination for the elastic Euler–Bernoulli beam via random polynomials and stochastic finite difference method 11.1 Introduction 11.2 Problem statement 11.3 Probabilistic response with polynomial bases 11.4 Computational implementation 11.5 Numerical experiments 11.6 Concluding remarks Acknowledgment References CH012.pdf Chapter 12 Polynomials in hybrid artificial intelligence 12.1 Artificial intelligence 12.1.1 Introduction 12.1.2 Abilities 12.1.3 Advantages and disadvantages 12.1.4 Symbolic and non-symbolic artificial intelligence 12.2 Hybrid artificial intelligence 12.2.1 Introduction 12.2.2 Why we need hybrid AI? 12.2.3 Abilities of hybrid AI 12.2.4 Applications of hybrid AI 12.3 Polynomials in hybrid artificial intelligence 12.3.1 Introduction of some polynomials such as regression, Rss, etc 12.3.2 Approximation of polynomials 12.3.3 Homogenious and estimation of error 12.4 Polynomials’ applications References CH013.pdf Chapter 13 Comparative study of Chebyshev and Legendre polynomial-based neural models for approximating multidimensional poverty for an Indian State 13.1 Introduction 13.2 Preliminaries 13.2.1 Chebyshev neural network (ChNN) 13.2.2 Legendre neural network (LeNN) 13.3 Methodology 13.3.1 Dataset construction 13.3.2 Data preprocessing module 13.3.3 Mechanism of MLNN model 13.3.4 Mechanism of FLNN model 13.4 Results and discussion 13.5 Conclusion Acknowledgements References CH014.pdf Chapter 14 Polynomial based model for solving unconstrained optimization problem with smoothing parameters 14.1 Introduction 14.2 Preliminaries of support vector machine (SVM) 14.2.1 Direction index matrix [6] 14.2.2 Analysis of polynomial function 14.2.3 Broyden–Fletcher–Goldfarb–Shanno (BFGS) algorithm for PSSVM [6] 14.3 Result and discussion 14.4 Conclusion Acknowledgement References CH015.pdf Chapter 15 Interval root finding and interval polynomials: methods and applications in science and engineering 15.1 Introduction 15.2 On sets, relations, and structures: setting the stage 15.3 The ring of polynomials and real polynomial arithmetic 15.3.1 Monomials, polynomials, and polynomial equations 15.3.2 The algebra of real polynomials 15.4 Beyond ordinary polynomials: generalizations of polynomials 15.4.1 Formal power series: infinite polynomials 15.4.2 Trigonometric and cylindrical polynomials 15.4.3 Polynomials over semirings 15.5 Two more generalizations of polynomials: n-adic polynomials and S-polynomials 15.5.1 Generalized n-adic polynomials: polynomials of polynomials 15.5.2 Polynomials over S-semirings and n-adic S-polynomials 15.6 Systems of generalized n-adic polynomial equations and inequalities 15.7 Interval arithmetic and interval polynomials 15.7.1 The algebra of real closed intervals 15.7.2 Interval functions and interval polynomials: guaranteed interval enclosures of families of generalized real polynomials 15.7.3 The interval subdivision method: more refined enclosures of families of generalized real polynomials 15.8 Taylor models: sharper interval enclosures with infinite polynomials 15.9 Interval root finding: guaranteed interval enclosures of roots of real polynomials 15.9.1 Interval Newton–Raphson method for root enclosures 15.9.2 Interval branch-and-prune method for root enclosures 15.9.3 A more refined interval branch-and-prune method 15.10 Concluding remarks Supplementary materials Acknowledgement References