### Applications of an extended $({G}^{\text{'}}/G)$-expansion method to find exact solutions of nonlinear PDEs in mathematical physics.

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Hematologic disorders such as the myelodysplastic syndromes (MDS) are discussed. The lingering controversies related to various diseases are highlighted. A simple biomathematical model of bone marrow - peripheral blood dynamics in the normal state is proposed and used to investigate cell behavior in normal hematopoiesis from a mathematical viewpoint. Analysis of the steady state and properties of the model are used to make postulations about the...

The time-ordered exponential of a time-dependent matrix $\U0001d5a0\left(t\right)$ is defined as the function of $\U0001d5a0\left(t\right)$ that solves the first-order system of coupled linear differential equations with non-constant coefficients encoded in $\U0001d5a0\left(t\right)$. The authors have recently proposed the first Lanczos-like algorithm capable of evaluating this function. This algorithm relies on inverses of time-dependent functions with respect to a non-commutative convolution-like product, denoted by $*$. Yet, the existence of such inverses, crucial to...

The existence of two continuous solutions for a nonlinear singular elliptic equation with natural growth in the gradient is proved for the Dirichlet problem in the unit ball centered at the origin. The first continuous solution is positive and maximal; it is obtained via the regularization method. The second continuous solution is zero at the origin, and follows by considering the corresponding radial ODE and by sub-sup solutions method.

A new method to solve stationary one-dimensional Schroedinger equation is investigated. Solutions are described by means of representation of circles with multiple winding number. The results are demonstrated using the well-known analytical solutions of the Schroedinger equation.