New Numerical Scheme with Newton Polynomial

Theory, Methods, and Applications

1st Edition - June 10, 2021
Imprint: Academic Press
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.

Cover image
Title page
Table of Contents
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

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

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