Two numerical methods for solving a nonlinear system of integral equations of mixed Volterra-Fredholm type arising from a control problem related to leukemia


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.


Lorand Gabriel Parajdi
Department of Mathematics, West Virginia University, Morgantown, WV 26506, USA & Department of Mathematics, Babeş-Bolyai University, Cluj-Napoca, Romania

Flavius Pătrulescu
Department of Mathematics, Technical University of Cluj-Napoca, Cluj-Napoca, Romania;

Radu Precup
Faculty of Mathematics and Computer Science and Institute of Advanced Studies in Science and Technology, Babeş-Bolyai University, Cluj-Napoca 400084, Romania
Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy, Cluj-Napoca 400110, Romania

Ioan Ştefan Haplea
Department of Internal Medicine, Iuliu Haţieganu University of Medicine and Pharmacy, Cluj-Napoca, Romania


Control problem; myeloid leukemia; Volterra-Fredholm integral equation; numerical method.

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L.G. Parajdi, F. Pătrulescu, R. Precup, I. Şt. Haplea, Two numerical methods for solving a nonlinear system of integral equations of mixed Voltera-Fredholm type arising from a control problem related to leukemia, Journal of Applied Analysis & Computation,


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