New Numerical Scheme with Newton Polynomial

New Numerical Scheme with Newton Polynomial

Theory, Methods, and Applications

First published on June 10, 2021

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  • Authors: Abdon Atangana, Seda Igret Araz
  • eBook ISBN: 9780323858021
  • Paperback ISBN: 9780323854481

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Description

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.

Key Features

  • 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

Readership

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

Table of Contents

  • 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

Product details

  • No. of pages: 460
  • Language: English
  • Copyright: © Academic Press 2021
  • Published: June 10, 2021
  • Imprint: Academic Press
  • eBook ISBN: 9780323858021
  • Paperback ISBN: 9780323854481

About the Authors

Abdon Atangana

Dr. Atangana is Academic Head of Department and Professor of Applied Mathematics at the University of the Free State, Bloemfontein, Republic of South Africa. He obtained his honours and master’s degrees from the Department of Applied Mathematics at the UFS with distinction. He obtained his PhD in applied mathematics from the Institute for Groundwater Studies. He serves as an editor for 20 international journals and lead guest editor in 10 journals and is also a reviewer of more than 200 international accredited journals. His research interests are methods and applications of partial and ordinary differential equations, fractional differential equations, perturbation methods, asymptotic methods, iterative methods, and groundwater modelling. Prof Atangana is the founder of fractional calculus with non-local and non-singular kernels popular in applied mathematics today. Since 2013, he has published in 250 international accredited journals of applied mathematics, applied physics, geo-hydrology and biomathematics. He is also the single author of two books in Academic Press Elsevier and a co-author of a book published in springer and author of more than 20 chapters in books. He has graduated 7 PhD and 20 masters students, and 6 postdoc fellows.

Affiliations and Expertise

Academic Head of Department and Professor of Applied Mathematics, University of the Free State, Bloemfontein, South Africa

Seda Igret Araz

Dr. Seda Igret Araz is an Assistant Professor of Mathematics at Siirt University, Siirt, Turkey. She obtained her master’s and PhD at Ataturk University, Turkey. She is the author of more than 15 papers published in top-tier journals. She has acted as guest editor on some special issues in Q1 journals.

Affiliations and Expertise

Assistant Professor of Mathematics, Department of Mathematics, Siirt University, Siirt, Turkey

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