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Abstract
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We develop the fractional model of stem cell population dynamics with state-dependent and time- dependent delays. In this model, the stem cell division rate and self-renewal rate are controlled by an external signal, depending on the effects of the environment’s heterogeneity. We quantify a relationship between the fractional derivative order, which shows the effects of the environment’s heterogeneity and stem cell division and stem cell self-renewal rate. We consider a general form of the fractional neutral delay differential equations with state-dependent and time-dependent delay to study this relationship. First, we show the solution’s existence and uniqueness using the fixed point theorem on the Banach space. We define a completely continuous operator on the non-empty closed convex set to use the fixed point theorem on the Banach space and show this operator has a uniquely defined fixed point. Also, we proof the Ulam–Hyers stability to make sure a close exact solution could be reached using the numerical approximation. Then, we develop a new numerical method based on Jacobi polynomials for solving the fractional neutral delay differential equations with state-dependent and time-dependent delay. We use the least-squares approximation of the candidate function to reduce the solution of fractional neutral de- lay differential equations to a set of algebraic equations and compare the results obtained by using differ- ent collocation points. We evaluate the accuracy of the numerical method, theoretically and numerically. We have used the numerical method to evaluate the fractional model’s behavior of stem cell population dynamics and quantify the relationship between the effects of the environment’s heterogeneity and the rate of stem cell division and stem cell self-renewal
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