طرح عددی جدید با چندجمله‌ای نیوتن : نظریه، روش‌ها و کاربردها

__New Numerical Scheme with Newton Polynomial: Theory, Methods, and Applications__ provides a detailed discussion on the underpinnings of the theory, methods and real-world applications of this numerical scheme. The book's authors explore how this efficient and accurate numerical scheme is useful for solving partial and ordinary differential equations, as well as systems of ordinary and partial differential equations with different types of integral operators. Content coverage includes the foundational layers of polynomial interpretation, Lagrange interpolation, and Newton interpolation, followed by new schemes for fractional calculus. Final sections include six chapters on the application of numerical scheme to a range of real-world applications. Over the last several decades, many techniques have been suggested to model real-world problems across science, technology and engineering. New analytical methods have been suggested in order to provide exact solutions to real-world problems. Many real-world problems, however, cannot be solved using analytical methods. To handle these problems, researchers need to rely on numerical methods, hence the release of this important resource on the topic at hand. Front Cover New Numerical Scheme With Newton Polynomial Copyright Contents Preface Acknowledgments List of symbols 1 Polynomial interpolation 1.1 Some interpolation polynomials 1.1.1 Bernstein polynomial 1.1.2 The Newton polynomial interpolation 1.1.3 Hermite interpolation 1.1.4 Cubic polynomial 1.1.5 B-spline polynomial 1.1.6 Legendre polynomial 1.1.7 Chebyshev polynomial 1.1.8 Lagrange–Sylvester interpolation 2 Two-steps Lagrange polynomial interpolation: numerical scheme 2.1 Classical differential equation 2.1.1 Numerical illustrations 2.2 Fractal differential equation 2.2.1 Numerical illustrations 2.3 Differential equation with the Caputo–Fabrizio operator 2.3.1 Error analysis with exponential kernel 2.3.2 Numerical illustrations 2.4 Differential equation with the Caputo fractional operator 2.4.1 Error analysis with power-law kernel 2.4.2 Numerical illustrations 2.5 Differential equation with the Atangana–Baleanu operator 2.5.1 Error analysis with the Mittag-Leffler kernel 2.5.2 Numerical illustrations 2.6 Differential equation with fractal–fractional with power-law kernel 2.6.1 Error analysis with the Caputo fractal–fractional derivative 2.6.2 Numerical illustrations 2.7 Differential equation with fractal–fractional derivative with exponential decay kernel 2.7.1 Error analysis with the Caputo–Fabrizio fractal–fractional derivative 2.7.2 Numerical illustrations 2.8 Differential equation with fractal–fractional derivative with the Mittag-Leffler kernel 2.8.1 Error analysis with the Atangana–Baleanu fractal–fractional derivative 2.8.2 Numerical illustrations 2.9 Differential equation with fractal–fractional with variable order with exponential decay kernel 2.9.1 Error analysis with fractal–fractional derivative with variable order with exponential decay kernel 2.9.2 Numerical illustrations 2.10 Differential equation with fractal–fractional derivative with variable order with the Mittag-Leffler kernel 2.10.1 Error analysis with fractal–fractional derivative with variable order with Mittag-Leffler kernel 2.10.2 Numerical illustrations 2.11 Differential equation with fractal–fractional derivative with variable order with power-law kernel 2.11.1 Error analysis with fractal–fractional derivative with variable order with power-law kernel 2.11.2 Numerical illustrations 3 Newton interpolation: introduction of the scheme for classical calculus 3.1 Error analysis with classical derivative 3.2 Numerical illustrations 4 Numerical method for fractal differential equations 4.1 Error analysis with fractal derivative 4.2 Numerical illustrations 5 Numerical method for a fractional differential equation with Caputo–Fabrizio derivative 5.1 Error analysis with Caputo–Fabrizio fractional derivative 5.2 Numerical illustrations 6 Numerical method for a fractional differential equation with power-law kernel 6.1 Error analysis with Caputo fractional derivative 6.2 Numerical illustrations 7 Numerical method for a fractional differential equation with the generalized Mittag-Leffler kernel 7.1 Error analysis with the Atangana–Baleanu fractional derivative 7.2 Numerical illustrations 8 Numerical method for a fractal–fractional ordinary differential equation with exponential decay kernel 8.1 Predictor–corrector method for fractal–fractional derivative with the exponential decay kernel 8.2 Error analysis with the Caputo–Fabrizio fractal–fractional derivative 8.3 Numerical illustrations 9 Numerical method for a fractal–fractional ordinary differential equation with power law kernel 9.1 Predictor–corrector method for fractal–fractional derivative with power law kernel 9.2 Error analysis with Caputo fractal–fractional derivative 9.3 Numerical illustrations 10 Numerical method for a fractal–fractional ordinary differential equation with Mittag-Leffler kernel 10.1 Predictor–corrector method for fractal–fractional derivative with the generalized Mittag-Leffler kernel 10.2 Error analysis with the Atangana–Baleanu fractal–fractional derivative 10.3 Numerical illustrations 11 Numerical method for a fractal–fractional ordinary differential equation with variable order with exponential decay kernel 11.1 Numerical illustrations 12 Numerical method for a fractal–fractional ordinary differential equation with variable order with power-law kernel 12.1 Numerical illustrations 13 Numerical method for a fractal–fractional ordinary differential equation with variable order with the generalized Mittag-Leffler kernel 13.1 Numerical illustrations 14 Numerical scheme for partial differential equations with integer and non-integer order 14.1 Numerical scheme with classical derivative 14.1.1 Numerical illustrations 14.2 Numerical scheme with fractal derivative 14.2.1 Numerical illustrations 14.3 Numerical scheme with the Atangana–Baleanu fractional operator 14.3.1 Numerical illustrations 14.4 Numerical scheme with the Caputo fractional operator 14.4.1 Numerical illustrations 14.5 Numerical scheme with the Caputo–Fabrizio fractional operator 14.5.1 Numerical illustration 14.6 Numerical scheme with the Atangana–Baleanu fractal–fractional operator 14.7 Numerical scheme with the Caputo fractal–fractional operator 14.8 Numerical scheme for Caputo–Fabrizio fractal–fractional operator 14.9 New scheme with fractal–fractional with variable order with exponential decay kernel 14.10 New scheme with fractal–fractional with variable order with the Mittag-Leffler kernel 14.11 New scheme with fractal–fractional with variable order with power-law kernel 15 Application to linear ordinary differential equations 15.1 Linear ordinary differential equations with integer and non-integer orders 15.1.1 A non-homogeneous linear differential equation 15.1.2 Non-homogeneous linear differential equation with the Atangana–Baleanu derivative 15.1.3 Non-homogeneous linear differential equation with the Caputo derivative 15.1.4 Non-homogeneous linear differential equation with fractal–fractional with the exponential law 15.1.5 Non-homogeneous linear differential equation with fractal–fractional derivative with the Mittag-Leffler kernel 16 Application to non-linear ordinary differential equations 16.1 Non-linear ordinary differential equations with integer and non-integer orders 16.1.1 Non-homogeneous nonlinear differential equation with classical derivative 16.1.2 Non-homogeneous non-linear differential equation with Caputo–Fabrizio derivative 16.1.3 Non-homogeneous non-linear differential equation with fractal derivative 16.1.4 Non-homogeneous non-linear differential equation with the fractal–fractional derivative with exponential law 16.1.5 Non-homogeneous non-linear differential equation with fractal–fractional with variable order with power law 17 Application to linear partial differential equations 17.1 Linear partial differential equations with integer and non-integer orders 17.1.1 Linear partial differential equation with the classical derivative 17.1.2 Linear partial differential equation with the fractal derivative 17.1.3 Linear partial differential equation with the Caputo fractional derivative 17.1.4 Linear partial differential equation with the fractal–fractional derivative with the exponential law 17.1.5 Linear partial differential equation with the fractal–fractional with variable order with the Mittag-Leffler kernel 18 Application to non-linear partial differential equations 18.1 Non-linear partial differential equations with integer and non-integer orders 18.1.1 Non-linear partial differential equation with the classical derivative 18.1.2 Non-linear partial differential equation with the fractal derivative 18.1.3 Non-linear partial differential equation with the Caputo fractional derivative 18.1.4 Non-linear partial differential equation with the fractal–fractional with exponential law 18.1.5 Non-linear partial differential equation with the fractal–fractional with variable order with the Mittag-Leffler kernel 19 Application to a system of ordinary differential equations 19.1 System of ordinary differential equations with integer and non-integer orders 19.1.1 A hybrid attractor with the classical derivative 19.1.2 Shaw oscillator with the Caputo fractional derivative 19.1.3 Dengue model with the Atangana–Baleanu fractional derivative 19.1.4 HIV model with fractal–fractional derivative with power law 19.1.5 Ebola model with fractal–fractional derivative with variable order with the exponential law 20 Application to system of non-linear partial differential equations 20.1 System of non-linear partial differential equations 20.1.1 System of non-linear partial differential equations with the classical derivative 20.1.2 System of non-linear partial differential equations with the Atangana–Baleanu derivative 20.1.3 System of non-linear partial differential equations with the Caputo fractional derivative 20.1.4 System of non-linear partial differential equations with the fractal–fractional with the Mittag-Leffler kernel 20.1.5 System of non-linear partial differential equations with fractal–fractional with the power law A Appendix AS_Method_for_Chaotic_with_AB_Fractal-Fractional.m AS_Method_for_Chaotic_with_AB_Fractal-Fractional_with_Variable_Order.m AS_Method_for_Chaotic_with_AB_Fractional.m AS_Method_for_Chaotic_with_Caputo_Fractal-Fractional_with_Variable_Order.m AS_Method_for_Chaotic_with_Caputo_Fractional.m AS_Method_for_Chaotic_with_CF_Fractal-Fractional_with_Variable_Order.m AS_Method_for_Chaotic_with_CF_Fractional.m AS_Method_for_Differential_Equation_with_AB_Fractal-Fractional.m AS_Method_for_Differential_Equation_with_AB_Fractal-Fractional_with_Variable_Order.m AS_Method_for_Differential_Equation_with_AB_Fractional.m AS_Method_for_Differential_Equation_with_Caputo_Fractal-Fractional.m AS_Method_for_Differential_Equation_with_Caputo_Fractional.m AS_Method_for_Differential_Equation_with_Caputo_Fractal-Fractional_with_Variable_Order.m AS_Method_for_Differential_Equation_with_CF_Fractal-Fractional.m AS_Method_for_Differential_Equation_with_CF_Fractional.m AS_Method_for_Differential_Equation_with_CF_Fractal-Fractional_with_Variable_Order.m AS_Method_for_Differential_Equation_with_Classical.m AS_Method_for_Differential_Equation_with_Fractal.m AT_Method_for_Chaotic_with_AB_Fractal-Fractional.m AT_Method_for_Chaotic_with_AB_Fractal-Fractional_with_Variable_Order.m AT_Method_for_Chaotic_with_AB_Fractional.m AT_Method_for_Chaotic_with_Caputo_Fractal-Fractional_with_Variable_Order.m AT_Method_for_Chaotic_with_Caputo_Fractal-Fractional.m AT_Method_for_Chaotic_with_Caputo_Fractional.m AT_Method_for_Chaotic_with_CF_Fractal-Fractional.m AT_Method_for_Chaotic_with_CF_Fractal-Fractional_with_Variable_Order.m AT_Method_for_Chaotic_with_CF_Fractional.m AT_Method_for_Differential_Equation_with_AB_Fractal-Fractional.m AT_Method_for_Differential_Equation_with_AB_Fractal-Fractional_with_Variable_Order.m AT_Method_for_Differential_Equation_with_AB_Fractional.m AT_Method_for_Differential_Equation_with_Caputo_Fractal-Fractional_with_Variable_Order.m AT_Method_for_Differential_Equation_with_Caputo_Fractal-Fractional.m AT_Method_for_Differential_Equation_with_Caputo_Fractional.m AT_Method_for_Differential_Equation_with_CF_Fractal-Fractional.m AT_Method_for_Differential_Equation_with_CF_Fractal-Fractional_with_Variable_Order.m AT_Method_for_Differential_Equation_with_CF_Fractional.m AT_Method_for_Differential_Equation_with_Classical.m AT_Method_for_Differential_Equation_with_Fractal.m A.1 Supplementary material References Index Back Cover