Article no.9

Two Numerical Methods for Solving a Nonlinear System of Integral Equations of Mixed Volterra-Fredholm Type Arising from a Control Problem Related to LeukemiaResearch Paper, July 12, 2023 / Lorand Gabriel Parajdi

Published in Journal of Applied Analysis & Computation (JAAC) 13(4), 1797-1812, DOI: 10.11948/20220197,  2023.

  The full paper is available on the Journal of Applied Analysis & Computation (JAAC) website:

Authors: Lorand Gabriel Parajdi1,2, Flavius Pătrulescu3, Radu Precup4,5 and Ioan Ștefan Haplea6

1 Department of Mathematics, West Virginia University, Morgantown, WV, USA 2 Department of Mathematics, “Babeş–Bolyai” University, Cluj-Napoca, Romania3 Department of Mathematics, Technical University of Cluj-Napoca, Cluj-Napoca, Romania 4 Faculty of Mathematics and Computer Science and Institute of Advanced Studies in Science and Technology, “Babeș-Bolyai” University, Cluj-Napoca, Romania5 “Tiberiu Popoviciu” Institute of Numerical Analysis, Romanian Academy, Cluj-Napoca, Romania6 Department of Internal Medicine, “Iuliu Hațieganu” University of Medicine and Pharmacy, Cluj-Napoca, Romania

Abstract: The aim of this paper is to present two algorithms for numerical solving of a fixed final state control problem in connection with the leukemia treatment strategy. In the absence of the controllability condition, our model leads to a nonlinear integral system of Volterra type to whom explicit iterative techniques apply and converge. Once using the controllability condition, the control variable is expressed in terms of the state variables and the integral system changes to a mixed Volterra-Fredholm type one making direct iterative techniques inoperative. However, two paths can be followed. One consists in an iterative procedure where at each step the control variable is calculated using the approximate values of the state variables from the previous step. The other looks for the numerical value of the control variable by using the bisection method. Numerical simulations, error analysis and biological interpretation are given.

Subject Classification: 45G15, 34H05, 65R20, 92C50, 93B05.

Keywords: control problem; myeloid leukemia; Volterra-Fredholm integral equation; numerical method.

Cite As: Parajdi LG, Pătrulescu F, Precup R, Haplea IȘ. Two numerical methods for solving a nonlinear system of integral equations of mixed Volterra-Fredholm type arising from a control problem related to leukemia. Journal of Applied Analysis & Computation. 2023; 13(4):1797-1812.

This work was supported by the project “The Development of Advanced and Applicative Research Competencies in the Logic of STEAM + Health” (POCU/993/6/13/153310), a project co-financed by the European Social Fund through the Romanian Operational Programme ‘Human Capital’, 2014-2020. L.G. Parajdi is the only author of this paper who benefited from financial support through this project, in his position as a postdoctoral researcher at Babeș-Bolyai University. For more details about the project, please visit the project website: