Polynomial Paradigms: Trends and applications in science and engineering

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