Soft Numerical Algorithm with Convergence Analysis for Time-Fractional Partial IDEs Constrained by Neumann Conditions

Omar Abu Arqub; Mohammed Al-Smadi; Shaher Momani

doi:10.1007/978-981-15-0430-3_7

Soft Numerical Algorithm with Convergence Analysis for Time-Fractional Partial IDEs Constrained by Neumann Conditions

Omar Abu Arqub
, Mohammed Al-Smadi
, Shaher Momani

Nonlinear Dynamic Research Centre (NDRC)

University of Jordan
Al-Balqa Applied University

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

1 Scopus citations

Abstract

Some scientific pieces of research are governed by classes of partial integro-differential equations (PIDEs) of fractional order that are leading to novel challenges in simulation and optimization. In this chapter, a soft numerical algorithm is proposed and analyzed to fitted analytical solutions of PIDEs with appropriate initial and Neumann conditions in Sobolev space. Meanwhile, the solutions are represented in series form with strictly computable components. By truncating n-term approximation of the analytical solution, the solution methodology is discussed for both linear and nonlinear problems based on the nonhomogeneous term. Analysis of convergence and smoothness are given under certain assumptions to show the theoretical structures of the method. Dynamic features of the approximate solutions are studied through an illustrated example. The yield of numerical results indicates the accuracy, clarity, and effectiveness of the proposed algorithm as well as provide a proper methodology in handling such fractional issues.

Original language	English
Title of host publication	Fractional Calculus - ICFDA 2018
Editors	Praveen Agarwal, Praveen Agarwal, Praveen Agarwal, Dumitru Baleanu, YangQuan Chen, Shaher Momani, José António Tenreiro Machado
Publisher	Springer
Pages	107-119
Number of pages	13
ISBN (Print)	9789811504297
DOIs	https://doi.org/10.1007/978-981-15-0430-3_7
State	Published - 2019
Event	International Conference on Fractional Differentiation and its Applications, ICFDA 2018 - Amman, Jordan Duration: 16 Jul 2018 → 18 Jul 2018

Publication series

Name	Springer Proceedings in Mathematics and Statistics
Volume	303
ISSN (Print)	2194-1009
ISSN (Electronic)	2194-1017

Conference

Conference	International Conference on Fractional Differentiation and its Applications, ICFDA 2018
Country/Territory	Jordan
City	Amman
Period	16/07/18 → 18/07/18

Keywords

Fractional derivatives
Fredholm and Volterra operators
Partial integro-differential equations
Reproducing kernel algorithm

Access to Document

10.1007/978-981-15-0430-3_7

Cite this

Arqub, O. A., Al-Smadi, M., & Momani, S. (2019). Soft Numerical Algorithm with Convergence Analysis for Time-Fractional Partial IDEs Constrained by Neumann Conditions. In P. Agarwal, P. Agarwal, P. Agarwal, D. Baleanu, Y. Chen, S. Momani, & J. A. T. Machado (Eds.), Fractional Calculus - ICFDA 2018 (pp. 107-119). (Springer Proceedings in Mathematics and Statistics; Vol. 303). Springer. https://doi.org/10.1007/978-981-15-0430-3_7

Arqub, Omar Abu ; Al-Smadi, Mohammed ; Momani, Shaher. / Soft Numerical Algorithm with Convergence Analysis for Time-Fractional Partial IDEs Constrained by Neumann Conditions. Fractional Calculus - ICFDA 2018. editor / Praveen Agarwal ; Praveen Agarwal ; Praveen Agarwal ; Dumitru Baleanu ; YangQuan Chen ; Shaher Momani ; José António Tenreiro Machado. Springer, 2019. pp. 107-119 (Springer Proceedings in Mathematics and Statistics).

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title = "Soft Numerical Algorithm with Convergence Analysis for Time-Fractional Partial IDEs Constrained by Neumann Conditions",

abstract = "Some scientific pieces of research are governed by classes of partial integro-differential equations (PIDEs) of fractional order that are leading to novel challenges in simulation and optimization. In this chapter, a soft numerical algorithm is proposed and analyzed to fitted analytical solutions of PIDEs with appropriate initial and Neumann conditions in Sobolev space. Meanwhile, the solutions are represented in series form with strictly computable components. By truncating n-term approximation of the analytical solution, the solution methodology is discussed for both linear and nonlinear problems based on the nonhomogeneous term. Analysis of convergence and smoothness are given under certain assumptions to show the theoretical structures of the method. Dynamic features of the approximate solutions are studied through an illustrated example. The yield of numerical results indicates the accuracy, clarity, and effectiveness of the proposed algorithm as well as provide a proper methodology in handling such fractional issues.",

keywords = "Fractional derivatives, Fredholm and Volterra operators, Partial integro-differential equations, Reproducing kernel algorithm",

author = "Arqub, \{Omar Abu\} and Mohammed Al-Smadi and Shaher Momani",

note = "Publisher Copyright: {\textcopyright} 2019, Springer Nature Singapore Pte Ltd.; International Conference on Fractional Differentiation and its Applications, ICFDA 2018 ; Conference date: 16-07-2018 Through 18-07-2018",

year = "2019",

doi = "10.1007/978-981-15-0430-3\_7",

language = "English",

isbn = "9789811504297",

series = "Springer Proceedings in Mathematics and Statistics",

publisher = "Springer",

pages = "107--119",

editor = "Praveen Agarwal and Praveen Agarwal and Praveen Agarwal and Dumitru Baleanu and YangQuan Chen and Shaher Momani and Machado, \{Jos{\'e} Ant{\'o}nio Tenreiro\}",

booktitle = "Fractional Calculus - ICFDA 2018",

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Arqub, OA, Al-Smadi, M & Momani, S 2019, Soft Numerical Algorithm with Convergence Analysis for Time-Fractional Partial IDEs Constrained by Neumann Conditions. in P Agarwal, P Agarwal, P Agarwal, D Baleanu, Y Chen, S Momani & JAT Machado (eds), Fractional Calculus - ICFDA 2018. Springer Proceedings in Mathematics and Statistics, vol. 303, Springer, pp. 107-119, International Conference on Fractional Differentiation and its Applications, ICFDA 2018, Amman, Jordan, 16/07/18. https://doi.org/10.1007/978-981-15-0430-3_7

Soft Numerical Algorithm with Convergence Analysis for Time-Fractional Partial IDEs Constrained by Neumann Conditions. / Arqub, Omar Abu; Al-Smadi, Mohammed; Momani, Shaher.
Fractional Calculus - ICFDA 2018. ed. / Praveen Agarwal; Praveen Agarwal; Praveen Agarwal; Dumitru Baleanu; YangQuan Chen; Shaher Momani; José António Tenreiro Machado. Springer, 2019. p. 107-119 (Springer Proceedings in Mathematics and Statistics; Vol. 303).

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

TY - GEN

T1 - Soft Numerical Algorithm with Convergence Analysis for Time-Fractional Partial IDEs Constrained by Neumann Conditions

AU - Arqub, Omar Abu

AU - Al-Smadi, Mohammed

AU - Momani, Shaher

PY - 2019

Y1 - 2019

N2 - Some scientific pieces of research are governed by classes of partial integro-differential equations (PIDEs) of fractional order that are leading to novel challenges in simulation and optimization. In this chapter, a soft numerical algorithm is proposed and analyzed to fitted analytical solutions of PIDEs with appropriate initial and Neumann conditions in Sobolev space. Meanwhile, the solutions are represented in series form with strictly computable components. By truncating n-term approximation of the analytical solution, the solution methodology is discussed for both linear and nonlinear problems based on the nonhomogeneous term. Analysis of convergence and smoothness are given under certain assumptions to show the theoretical structures of the method. Dynamic features of the approximate solutions are studied through an illustrated example. The yield of numerical results indicates the accuracy, clarity, and effectiveness of the proposed algorithm as well as provide a proper methodology in handling such fractional issues.

AB - Some scientific pieces of research are governed by classes of partial integro-differential equations (PIDEs) of fractional order that are leading to novel challenges in simulation and optimization. In this chapter, a soft numerical algorithm is proposed and analyzed to fitted analytical solutions of PIDEs with appropriate initial and Neumann conditions in Sobolev space. Meanwhile, the solutions are represented in series form with strictly computable components. By truncating n-term approximation of the analytical solution, the solution methodology is discussed for both linear and nonlinear problems based on the nonhomogeneous term. Analysis of convergence and smoothness are given under certain assumptions to show the theoretical structures of the method. Dynamic features of the approximate solutions are studied through an illustrated example. The yield of numerical results indicates the accuracy, clarity, and effectiveness of the proposed algorithm as well as provide a proper methodology in handling such fractional issues.

KW - Fractional derivatives

KW - Fredholm and Volterra operators

KW - Partial integro-differential equations

KW - Reproducing kernel algorithm

UR - https://www.scopus.com/pages/publications/85076758198

U2 - 10.1007/978-981-15-0430-3_7

DO - 10.1007/978-981-15-0430-3_7

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AN - SCOPUS:85076758198

SN - 9789811504297

T3 - Springer Proceedings in Mathematics and Statistics

SP - 107

EP - 119

BT - Fractional Calculus - ICFDA 2018

A2 - Agarwal, Praveen

A2 - Baleanu, Dumitru

A2 - Chen, YangQuan

A2 - Momani, Shaher

A2 - Machado, José António Tenreiro

PB - Springer

T2 - International Conference on Fractional Differentiation and its Applications, ICFDA 2018

Y2 - 16 July 2018 through 18 July 2018

ER -

Arqub OA, Al-Smadi M, Momani S. Soft Numerical Algorithm with Convergence Analysis for Time-Fractional Partial IDEs Constrained by Neumann Conditions. In Agarwal P, Agarwal P, Agarwal P, Baleanu D, Chen Y, Momani S, Machado JAT, editors, Fractional Calculus - ICFDA 2018. Springer. 2019. p. 107-119. (Springer Proceedings in Mathematics and Statistics). doi: 10.1007/978-981-15-0430-3_7

Soft Numerical Algorithm with Convergence Analysis for Time-Fractional Partial IDEs Constrained by Neumann Conditions

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