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

  • Omar Abu ArqubEmail author
  • Mohammed Al-Smadi
  • Shaher Momani
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 303)


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.


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


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Copyright information

© Springer Nature Singapore Pte Ltd. 2019

Authors and Affiliations

  • Omar Abu Arqub
    • 1
    Email author
  • Mohammed Al-Smadi
    • 2
  • Shaher Momani
    • 1
    • 3
  1. 1.Department of Mathematics, Faculty of ScienceThe University of JordanAmmanJordan
  2. 2.Department of Applied ScienceAjloun College, Al-Balqa Applied UniversityAjlounJordan
  3. 3.Department of Mathematics and Sciences, College of Humanities and SciencesAjman UniversityAjmanUAE

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