New Numerical Scheme with Newton Polynomial

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.

• Offers an overview of the field of numerical analysis and modeling real-world problems
• Provides a deeper understanding and comparison of Adams-Bashforth and Newton polynomial numerical methods
• Presents applications of local fractional calculus to a range of real-world problems
• Explores new scheme for fractal functions and investigates numerical scheme for partial differential equations with integer and non-integer order
• Includes codes and examples in MATLAB in all relevant chapters

Graduate students and researchers in mathematics (pure and applied), engineering, physics, economics

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 Lagrange 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 Caputo-Fabrizio Operator
2.3.1 Error Analysis with Exponential Kernel
2.3.2 Numerical Illustrations
2.4 Differential Equation with Caputo Fractional Operator
2.4.1 Error Analysis with Power-Law Kernel
2.4.2 Numerical Illustrations
2.5 Differential Equation with Atangana-Baleanu Operator
2.5.1 Error Analysis with 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 Caputo Fractal-Fractional Derivative
2.6.2 Numerical Illustrations
2.7 Differential Equation with Fractal-Fractional with Exponential Decay Kernel
2.7.1 Error Analysis with Caputo-Fabrizio Fractal-Fractional Derivative
2.7.2 Numerical Illustrations
2.8 Differential Equation with Fractal-Fractional with Mittag-Leffler Kernel
2.8.1 Error Analysis with 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 with Variable Order with Exponential Decay Kernel
2.9.2 Numerical Illustrations
2.10 Differential Equation with Fractal-Fractional with Variable Order with Mittag-Leffler Kernel
2.10.1 Error Analysis with Fractal-Fractional with Variable Order with Mittag-Leffler Kernel
2.10.2 Numerical Illustrations
2.11 Differential Equation with Fractal-Fractional with Variable Order with Power-Law Kernel
2.11.1 Error Analysis with Fractal-Fractional with Variable Order with Power-Law Kernel
2.11.2 Numerical Illustrations

3 Newton Interpolation: Introduction to New Scheme for Classical Calculus
3.1 Error Analysis with Classical Derivative
3.2 Numerical Illustrations

4 New Scheme for Fractal Calculus
4.1 Error Analysis with Fractal Derivative
4.2 Numerical Illustrations

5 New Scheme for Fractional Calculus with Exponential Decay Kernel
5.1 Error Analysis with Caputo-Fabrizio Fractional Derivative
5.2 Numerical Illustrations

6 New Scheme for Fractional Calculus with Power-Law Kernel
6.1 Error Analysis with Caputo Fractional Derivative
6.2 Numerical Illustrations

7 New scheme for fractional calculus with the generalized Mittag-Leffler kernel
7.1 Error Analysis with Atangana-Baleanu fractional derivative
7.2 Numerical Illustrations

8 New scheme for fractal-fractional with exponential decay kernel
8.1 Predictor-corrector method for fractal-fractional with the exponential decay kernel
8.2 Error Analysis with Caputo-Fabrizio fractal-fractional derivative
8.3 Numerical Illustrations

9 New scheme for fractal-fractional with power law kernel
9.1 Predictor-corrector method for fractal-fractional with power law kernel
9.2 Error Analysis with Caputo fractal-fractional derivative
9.3 Numerical Examples

10 New Scheme for Fractal-Fractional with The Generalized Mittag-Leffler Kernel
10.1 Predictor-Corrector Method for Fractal-Fractional with The Generalized Mittag-Leffler Kern
10.2 Error Analysis with Atangana-Baleanu Fractal-Fractional Derivative
10.3 Numerical Illustrations

11 New Scheme with Fractal-Fractional with Variable Order with Exponential Decay Kernel
11.1 Numerical Illustrations

12 New Scheme with Fractal-Fractional with Variable Order with Power-Law Kernel
12.1 Numerical Illustrations

13 New Scheme with Fractal-Fractional with Variable Order with 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 Atangana-Baleanu Fractional Operator
14.3.1 Numerical Illustrations
14.4 Numerical Scheme with Caputo Fractional Operator
14.4.1 Numerical Illustrations
14.5 Numerical scheme with Caputo-Fabrizio fractional operator
14.5.1 Numerical Illustration
14.6 Numerical Scheme with Atangana-Baleanu Fractal-Fractional Operator
14.7 Numerical Scheme with Caputo Fractal-Fractional Operator
14.8 Numerical Scheme Caputo-Fabrizio Fractal-Fractional Operator
14.9 New Scheme with Fractal-Fractional with Variable Order with Exponential Decay Kernel
14.10New Scheme with Fractal-Fractional with Variable Order with Mittag-Leffler Kernel
14.11New Scheme with Fractal-Fractional with Variable Order with Power-Law Kernel

15 Application to Linear Ordinary Differential Equations

16 Application to Nonlinear Ordinary Differential Equations

17 Application to Linear Partial Differential Equations

18 Application to Nonlinear Partial Differential Equations

19 Application to System of Ordinary Differential Equations

20 Application to System of Nonlinear Partial Differential Equations