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Table of Contents
Intro
Preface
Acknowledgement
Editor biography
S Chakraverty
List of contributors
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
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
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
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.
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
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
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
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
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
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
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.
Chapter 12 Polynomials in hybrid artificial intelligence.
Preface
Acknowledgement
Editor biography
S Chakraverty
List of contributors
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
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
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
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.
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
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
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
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
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
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
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.
Chapter 12 Polynomials in hybrid artificial intelligence.