In this paper we obtain asymptotic estimates of eigenfunctions for regular Sturm-Liouville problems having the eigenparameter in the boundary condition without smoothness conditions on the potential. haskız coşkun and ayşe kabataşThe aim of this paper is to obtained asymptotic approximations for the eigenfunctions of (1)-(3) including the above mentioned cases (a 4 = 0, a 4 = 0) where q(t) is integrable on [a, b]. We use similar approach in [4], [7] to derive these approximations by making use of solutions of (1) defined with the initial conditions and asymptotic approximations of the eigenvalues of (1)-(3) obtained in [2].Acknowledgements. The first author is grateful to Professor Bernard J. Harris for introducing her to the field. Both authors record their gratitude to the referee for his (or her) careful reading of the paper.

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In this paper we obtain Green's function for a regular Sturm-Liouville problem having the eigenparameter in all boundary conditions in which the left one is in quadratic form. We assume no smoothness condition on the potential.

Background: Cancer is one of the most complex phenomena in biology and medicine. Extensive attempts have been made to work around this complexity. In this study, we try to take a selective approach; not modeling each particular facet in detail but rather only the pertinent and essential parts of the tumor system are simulated and followed by optimization, revealing specific traits. This leads us to a pellucid personalized model which is noteworthy as it closely approximates existing experimental results.

Purpose: Cancer is one of the most complex phenomena in biology and medicine. Extensive attempts have been made to work around this complexity. In this study, we try to take a selective approach; not modeling each particular facet in detail but rather only the pertinent and essential parts of the tumor system are simulated and followed by optimization, revealing specific traits. This leads us to a pellucid personalized model which is noteworthy as it closely approximates existing experimental results.Methods: In the present study, a hybrid modeling approach which consists of cellular automata for discrete cell state representation and diffusion equations to calculate distribution of relevant substances in the tumor microenvironment is favored. Moreover, naive Bayesian decision making with weighted stochastic equations and a Bayesian network to model the temporal order of mutations is presented. The model is personalized according to the evidence using Markov Chain Monte Carlo. To validate the tumor model, a data set belonging to the A549 cell line is used. The data represents the growth of a tumor for 30 days. We optimize the coefficients of the stochastic decision-making equations using the first half of the timeline.Results: Simulation results of the developed model are promising with their low error margin (all correlation coefficients are over 0.8 under different microenvironment conditions) and simulated growth data is in line with laboratory results (r=0.97, p<0.01).Conclusions: Our approach of using simulated annealing for parameter estimation and the subsequent validation of the prediction with invitro tumor growth data are, to our knowledge, is novel.