New Numerical Scheme with Newton Polynomial

Theory, Methods, and Applications

1st Edition - June 10, 2021
Authors: Abdon Atangana, Seda İğret Araz
Language: English
Paperback ISBN:
9 7 8 - 0 - 3 2 3 - 8 5 4 4 8 - 1
eBook ISBN:
9 7 8 - 0 - 3 2 3 - 8 5 8 0 2 - 1

New Numerical Scheme with Newton Polynomial: Theory, Methods, and Applications provides a detailed discussion on the underpinnings of the theory, methods and real-world applicati… Read more

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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.

1 Polynomial Interpolation1.1 Some Interpolation Polynomials1.1.1 Bernstein Polynomial1.1.2 The Newton Polynomial Interpolation1.1.3 Hermite Interpolation1.1.4 Cubic Polynomial1.1.5 B-spline Polynomial1.1.6 Legendre Polynomial1.1.7 Chebyshev Polynomial1.1.8 Lagrange-Sylvester interpolation

2 Lagrange Interpolation: Numerical Scheme2.1 Classical Differential Equation2.1.1 Numerical Illustrations2.2 Fractal Differential Equation2.2.1 Numerical Illustrations2.3 Differential Equation with Caputo-Fabrizio Operator2.3.1 Error Analysis with Exponential Kernel2.3.2 Numerical Illustrations2.4 Differential Equation with Caputo Fractional Operator2.4.1 Error Analysis with Power-Law Kernel2.4.2 Numerical Illustrations2.5 Differential Equation with Atangana-Baleanu Operator2.5.1 Error Analysis with Mittag-Leffler Kernel2.5.2 Numerical Illustrations2.6 Differential Equation with Fractal-Fractional with Power-Law Kernel2.6.1 Error Analysis with Caputo Fractal-Fractional Derivative2.6.2 Numerical Illustrations2.7 Differential Equation with Fractal-Fractional with Exponential Decay Kernel2.7.1 Error Analysis with Caputo-Fabrizio Fractal-Fractional Derivative2.7.2 Numerical Illustrations2.8 Differential Equation with Fractal-Fractional with Mittag-Leffler Kernel2.8.1 Error Analysis with Atangana-Baleanu fractal-fractional derivative2.8.2 Numerical Illustrations2.9 Differential equation with Fractal-Fractional with Variable Order with Exponential Decay Kernel2.9.1 Error Analysis with Fractal-Fractional with Variable Order with Exponential Decay Kernel2.9.2 Numerical Illustrations2.10 Differential Equation with Fractal-Fractional with Variable Order with Mittag-Leffler Kernel2.10.1 Error Analysis with Fractal-Fractional with Variable Order with Mittag-Leffler Kernel2.10.2 Numerical Illustrations2.11 Differential Equation with Fractal-Fractional with Variable Order with Power-Law Kernel2.11.1 Error Analysis with Fractal-Fractional with Variable Order with Power-Law Kernel2.11.2 Numerical Illustrations

3 Newton Interpolation: Introduction to New Scheme for Classical Calculus3.1 Error Analysis with Classical Derivative3.2 Numerical Illustrations

4 New Scheme for Fractal Calculus4.1 Error Analysis with Fractal Derivative4.2 Numerical Illustrations

5 New Scheme for Fractional Calculus with Exponential Decay Kernel5.1 Error Analysis with Caputo-Fabrizio Fractional Derivative5.2 Numerical Illustrations

6 New Scheme for Fractional Calculus with Power-Law Kernel6.1 Error Analysis with Caputo Fractional Derivative6.2 Numerical Illustrations

7 New scheme for fractional calculus with the generalized Mittag-Leffler kernel7.1 Error Analysis with Atangana-Baleanu fractional derivative7.2 Numerical Illustrations

8 New scheme for fractal-fractional with exponential decay kernel8.1 Predictor-corrector method for fractal-fractional with the exponential decay kernel8.2 Error Analysis with Caputo-Fabrizio fractal-fractional derivative8.3 Numerical Illustrations

9 New scheme for fractal-fractional with power law kernel9.1 Predictor-corrector method for fractal-fractional with power law kernel9.2 Error Analysis with Caputo fractal-fractional derivative9.3 Numerical Examples

10 New Scheme for Fractal-Fractional with The Generalized Mittag-Leffler Kernel10.1 Predictor-Corrector Method for Fractal-Fractional with The Generalized Mittag-Leffler Kern10.2 Error Analysis with Atangana-Baleanu Fractal-Fractional Derivative10.3 Numerical Illustrations

11 New Scheme with Fractal-Fractional with Variable Order with Exponential Decay Kernel11.1 Numerical Illustrations

12 New Scheme with Fractal-Fractional with Variable Order with Power-Law Kernel12.1 Numerical Illustrations

13 New Scheme with Fractal-Fractional with Variable Order with Mittag-Leffler Kernel13.1 Numerical Illustrations

14 Numerical Scheme for Partial Differential Equations with Integer and Non-integer Order14.1 Numerical Scheme with Classical Derivative14.1.1 Numerical Illustrations14.2 Numerical Scheme with Fractal Derivative14.2.1 Numerical Illustrations14.3 Numerical Scheme with Atangana-Baleanu Fractional Operator14.3.1 Numerical Illustrations14.4 Numerical Scheme with Caputo Fractional Operator14.4.1 Numerical Illustrations14.5 Numerical scheme with Caputo-Fabrizio fractional operator14.5.1 Numerical Illustration14.6 Numerical Scheme with Atangana-Baleanu Fractal-Fractional Operator14.7 Numerical Scheme with Caputo Fractal-Fractional Operator14.8 Numerical Scheme Caputo-Fabrizio Fractal-Fractional Operator14.9 New Scheme with Fractal-Fractional with Variable Order with Exponential Decay Kernel14.10New Scheme with Fractal-Fractional with Variable Order with Mittag-Leffler Kernel14.11New 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

Abdon Atangana

Dr. Abdon 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 was included in the 2019 (Maths), 2020 (Cross-field) and the 2021 (Maths) Clarivate Web of Science lists of the World's top 1% scientists, and he was awarded The World Academy of Sciences (TWAS) inaugural Mohammed A. Hamdan award for contributions to science in developing countries. In 2018 Dr. Atangana was elected as a member of the African Academy of Sciences and in 2021 a member of The World Academy of Sciences. He also ranked number one in the world in mathematics, number 186 in the world in all fields, and number one in Africa in all fields, according to the Stanford University list of top 2% scientists in the world. He was one of the first recipients of the Obada Award in 2018. Dr. Atangana published a paper that was ranked by Clarivate in 2017 as the most cited mathematics paper in the world. Dr. Atangana serves as an editor for 20 international journals, lead guest editor for 10 journals, and is also a reviewer of more than 200 international accredited journals. His research interests include methods and applications of partial and ordinary differential equations, fractional differential equations, perturbation methods, asymptotic methods, iterative methods, and groundwater modelling. Dr. Atangana is a pioneer in research on fractional calculus with non-local and non-singular kernels popular in applied mathematics today. He is the author of numerous books, including Integral Transforms and Engineering: Theory, Methods, and Applications, Taylor and Francis/CRC Press; Numerical Methods for Fractal-Fractional Differential Equations and Engineering: Simulations and Modeling, Taylor and Francis/CRC Press; Numerical Methods for Fractional Differentiation, Springer; Fractional Stochastic Differential Equations, Springer; Fractional Order Analysis, Wiley; Applications of Fractional Calculus to Modeling in Dynamics and Chaos, Taylor and Francis/CRC Press; Fractional Operators with Constant and Variable Order with Application to Geo-hydrology, Elsevier/Academic Press; Derivative with a New Parameter: Theory, Methods, and Applications, Elsevier/Academic Press; and New Numerical Scheme with Newton Polynomial: Theory, Methods, and Applications, Elsevier/Academic Press; among others.

Affiliations and expertise

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

Seda İğret Araz

Dr. Seda İğret 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 special issues in Q1 journals. Dr. Araz is the co-author with Dr. Atangana of New Numerical Scheme with Newton Polynomial: Theory, Methods, and Applications, Elsevier/Academic Press; and Fractional Stochastic Differential Equations, Springer

Affiliations and expertise

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