New Numerical Scheme with Newton Polynomial: Theory, Methods, and Applications

By Abdon Atangana and Seda İğret Araz

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

• Language: English
• Publisher: Academic Press
• Release date: Jun 10, 2021
• ISBN: 9780323858021

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.

Book preview

Front Cover for New Numerical Scheme With Newton Polynomial

New Numerical Scheme With Newton Polynomial

Theory, Methods, and Applications

First edition

Abdon Atangana

University of the Free State, Bloemfontein, South Africa

Seda İğret Araz

Department of Mathematics, Siirt University, Siirt, Turkey

publogo

Table of Contents

Cover image

Title page

Copyright

Preface

Acknowledgments

List of symbols

1: Polynomial interpolation

Abstract

1.1. Some interpolation polynomials

References

2: Two-steps Lagrange polynomial interpolation: numerical scheme

Abstract

2.1. Classical differential equation

2.2. Fractal differential equation

2.3. Differential equation with the Caputo–Fabrizio operator

2.4. Differential equation with the Caputo fractional operator

2.5. Differential equation with the Atangana–Baleanu operator

2.6. Differential equation with fractal–fractional with power-law kernel

2.7. Differential equation with fractal–fractional derivative with exponential decay kernel

2.8. Differential equation with fractal–fractional derivative with the Mittag-Leffler kernel

2.9. Differential equation with fractal–fractional with variable order with exponential decay kernel

2.10. Differential equation with fractal–fractional derivative with variable order with the Mittag-Leffler kernel

2.11. Differential equation with fractal–fractional derivative with variable order with power-law kernel

References

3: Newton interpolation: introduction of the scheme for classical calculus

Abstract

3.1. Error analysis with classical derivative

3.2. Numerical illustrations

References

4: Numerical method for fractal differential equations

Abstract

4.1. Error analysis with fractal derivative

4.2. Numerical illustrations

References

5: Numerical method for a fractional differential equation with Caputo–Fabrizio derivative

Abstract

5.1. Error analysis with Caputo–Fabrizio fractional derivative

5.2. Numerical illustrations

References

6: Numerical method for a fractional differential equation with power-law kernel

Abstract

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

Abstract

7.1. Error analysis with the Atangana–Baleanu fractional derivative

7.2. Numerical illustrations

References

8: Numerical method for a fractal–fractional ordinary differential equation with exponential decay kernel

Abstract

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

References

9: Numerical method for a fractal–fractional ordinary differential equation with power law kernel

Abstract

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

References

10: Numerical method for a fractal–fractional ordinary differential equation with Mittag-Leffler kernel

Abstract

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

References

11: Numerical method for a fractal–fractional ordinary differential equation with variable order with exponential decay kernel

Abstract

11.1. Numerical illustrations

References

12: Numerical method for a fractal–fractional ordinary differential equation with variable order with power-law kernel

Abstract

12.1. Numerical illustrations

References

13: Numerical method for a fractal–fractional ordinary differential equation with variable order with the generalized Mittag-Leffler kernel

Abstract

13.1. Numerical illustrations

References

14: Numerical scheme for partial differential equations with integer and non-integer order

Abstract

14.1. Numerical scheme with classical derivative

14.2. Numerical scheme with fractal derivative

14.3. Numerical scheme with the Atangana–Baleanu fractional operator

14.4. Numerical scheme with the Caputo fractional operator

14.5. Numerical scheme with the Caputo–Fabrizio fractional operator

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

Abstract

15.1. Linear ordinary differential equations with integer and non-integer orders

16: Application to non-linear ordinary differential equations

Abstract

16.1. Non-linear ordinary differential equations with integer and non-integer orders

17: Application to linear partial differential equations

Abstract

17.1. Linear partial differential equations with integer and non-integer orders

18: Application to non-linear partial differential equations

Abstract

18.1. Non-linear partial differential equations with integer and non-integer orders

19: Application to a system of ordinary differential equations

Abstract

19.1. System of ordinary differential equations with integer and non-integer orders

20: Application to system of non-linear partial differential equations

Abstract

20.1. System of non-linear partial differential equations

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

References

Index

Copyright

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Preface

... and the Lord God granted mankind power to control the world within which they live. In particular he granted humans the knowledge to understand using mathematics the nature within which they live. The assignment can only be well achieved if humans follow three important steps including, a correct observation of physical problems, a deeper analysis of these observations and finally an accurate prediction in space and time of such observations. The first step thus requires an adequate, efficient and accurate methodology of data collection and elimination of possible uncertainties. The second, which is fundamental, is managing the data-free-uncertainties of a complex observed fact in order to gain a better understanding of it. The genesis of this second step can be traced back to before Aristotle (385–322 B.C). Many forefathers of modern science endorsed this step as a format concept, for instance Alhazen, Galieo Galilei, and Descartes; also it has been credited to Newton who used it as a real-world technique of physical discovery. The last step requires first a systematic conversion of real world problems into a mathematical formulation, solving the mathematical models and fitting the obtained solution with the data-free uncertainties. If the solution of the mathematical model is in good agreement with data-free uncertainties, an accurate and reliable prediction could be performed. We should mention that these mathematical models are constructed using the concept of differentiation and integration. Due to the complexity of nature, the conversion from observed facts to mathematical models usually ends up with highly nonlinear ordinary and partial differential equations that could not be solved using known analytical methods; therefore numerical schemes are needed to at least provide an approximate solution of the problem. The literature counts several suggested numerical schemes, some very efficient and user friendly, with of course limitations and disadvantages besides the advantages. Some of these numerical schemes are based on polynomial interpolation, for instance, the associated Adams–Basforth approach is based on the Lagrange method, and we have the spline polynomial, and many others that will not be listed here. Very recently Atangana and Seda suggested an alternative numerical scheme based on the Newton polynomial. The introduction of the method met a warm welcome and was revealed to be an efficient and accurate numerical scheme able to solve fractional ordinary differential equations and fractional systems of ordinary with different types of fractional integral operators. This book is therefore devoted to the detailed discussions underpinning the theory, methods and applications of this alternative numerical scheme.

Acknowledgments

The fear of the LORD is the beginning of wisdom; all who follow His precepts have a rich understanding. His praise endures forever!

We could not have written this book without God King of heaven, who made it possible for us to write this book, through his mighty grace, divine love, and unending supply of his divine protection, love and knowledge. We thank him for his grace and everyday breath.

We thank our respective families for their support, and for their patience for allowing us to finish this book.

List of symbols

: The vector space of polynomials of degree at most m

: Lagrange basis polynomials

: Normalization function

: Normalization function

: Fractional derivative

: Fractional integral

: Fractal derivative in Chen's sense

: Fractional integral in Chen's sense

: Caputo–Fabrizio fractional derivative

: Caputo–Fabrizio fractional integral

: Caputo fractional derivative

: Caputo fractional integral

: Atangana–Baleanu fractional derivative in Riemann–Liouville's sense

: Atangana–Baleanu fractional integral in Riemann-Liouville's sense

: Atangana–Baleanu fractional derivative in Caputo's sense

: Atangana–Baleanu fractional integral in Caputo's sense

: Caputo–Fabrizio fractal–fractional derivative

: Caputo–Fabrizio fractal–fractional integral

: Caputo fractal–fractional derivative

: Caputo fractal–fractional integral

: Atangana–Baleanu fractal–fractional derivative

: Atangana–Baleanu fractal–fractional integral

: Fractal–fractional derivative with variable order with exponential decay kernel

: Fractal–fractional integral with variable order with exponential decay kernel

: Fractal–fractional derivative with variable order with power-law kernel

: Fractal–fractional integral with variable order with power-law kernel

: Fractal–fractional derivative with variable order with Mittag-Leffler kernel

: Fractal–fractional integral with variable order with Mittag-Leffler kernel

: Error function

: Hilbert space consisting of the elements having generalized derivative of first order

1: Polynomial interpolation

Abstract

Numerical methods are powerful mathematical methods used to solve complex ordinary differential equations. They sometime rely on polynomial interpolation technique, which is viewed as interpolation of a given set of collected data by a polynomial of lowest possible degree able to fit through the points of the given dataset. There exist many polynomial interpolations in the available literature, in this chapter, we present some well-known polynomial interpolation and their properties. In particular, we present in this chapter the following polynomial including: Berstein polynomial, Newton polynomial interpolation, Hermite interpolation, cubic polynomial, B-spline polynomial, Legendre polynomial, Chebyschev polynomial, Lagrange-Sylvester interpolation.

Keywords

Polynomial interpolation; error analysis; numerical analysis

One of the greatest achievements in the framework of numerical analysis is perhaps the introduction of approximating a giving nonlinear function using a polynomial interpolation. In this framework, a polynomial interpretation can be viewed as the interpolation of a given collected data set by the polynomial of lowest possible degree, which is able to fit the points of the data [1–10]. Such polynomials can be used to approximate complex curves that could be difficult to be evaluated, a clear example has been supplied in the literature, the shapes of the letters in typography given a few points. We have to point out the wider applicability of this technique [1,2,10]. One of the relevant applications is the evaluation of the natural logarithm and trigonometric functions, however, many other important applications can be found in many already published, books, papers and even in Wikipedia. The process of implementing this algorithm follows these steps: Select a few given data points, create a lookup table, then interpolate between those data points [1,2,10]. It is important to recognize the efficiency and the accuracy of these results, as they are significantly faster computations. The polynomial interpolation have been documented to also constitute the basis for algorithms in numerical quadrature and numerical ordinary differential equations, partial differential equations and secure multi party computation and secret sharing schemes. The polynomial interpolation is essential to engage sub-quadratic multiplication and squares like Toom–Cook multiplication and Karatsuba multiplication, where an interpolation via points on a polynomial like expression defines the product yields the product itself [1–10].

Definition 1.1

If we assume a set of given data points, where no two are the same, one is looking for a polynomial P which is suitable with degree c at most n with the property

(1.1)

The well-established unisolvence theorem stipulates that such a polynomial P exists and more importantly is unique. The proof can be achieved thanks to the Vandermonde matrix. A detailed proof can be found in many text books and also some already published papers, however, to make it