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integral-differential-equations
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The Existence and Stability of Classical Solutions in the Neural Fields Equations
, M.Sc. Thesis Sharif University of Technology ; Fotouhi Firouzabadi, Morteza (Supervisor)
Abstract
In this thesis, first, the modeling method of neural fields is precisely presented. Then, Existence and Stability of different solutions of one dimensional neural fields like Standing Pulses, Traveling Waves and ... are investigated in three different models of neural fields. In order for proving Existence and Stability of the solutions the mathematical tools like Fourier transform and Evans function are applied. All the models which analysed in this thesis have the following Integro-Differential Equation form:
τ
∂u(x, t)
∂t
= −u(x, t) +
∫ +∞
−∞
w(x, y)f[u(y, t)]dy + I(x, t) + s(x, t)
and also in some models the parameters might be changed
τ
∂u(x, t)
∂t
= −u(x, t) +
∫ +∞
−∞
w(x, y)f[u(y, t)]dy + I(x, t) + s(x, t)
and also in some models the parameters might be changed
Asymptotic Analysis and Optimal Control of an Integro-Differential System Modeling Healthy and Cancer Cells Exposed to Chemotherapy
, M.Sc. Thesis Sharif University of Technology ; Hesaraki, Mahmoud (Supervisor)
Abstract
We consider a system of two coupled integro-differential equations modelling populations of healthy and cancer cells under chemotherapy. Both populations are structured by a phenotypic variables, representing their level of resistance to the treatment. we analyse the asymptotic behaviour of the model under constant infusion of drugs. By designing an appropriate Lyapunov function, we prove that both cell densities converge to Dirac masses. We then define an optimal control problem, by considering all possible infusion protocols and minimising the number of cancer cells over a prescribed time frame. We provide a quasi-optimal strategy and prove that it solves this problem for large final...