Derivation of a mathematical model of oscillation processes of spatially loaded rods with account for temperature is considered in the paper. On the basis of the problem posed, using the generalized Hamilton-Ostrogradsky principle, the variations in kinetic, potential energy and work of external forces were calculated, and a nonlinear mathematical model of spatially loaded rods was derived, taking into account temperature. Based on the mathematical model, the resolving equations of spatially loaded rods with generalized natural initial and boundary conditions were derived.
In this paper, we study the conditions of global solvability and unsolvability in time of solutions to the nonlinear diffusion problem based on self-similar analysis. We constructed various self-similar solutions of the nonlinear diffusion problem in the slow diffusion case. We established critical exponents of the Fujita type and critical exponents for the global existence of the solution. Using the asymptotic formulas as the initial approximation for the iterative process, numerical calculations are performed.
In the present paper, we study the boundary value problem for a mixed parabolic-hyperbolic equation with fractional derivative and degeneration in time. Using the method of separation of variables, we prove theorems on the uniqueness and existence for the classical solution of this problem.
In the paper, we give the upper bounds for moments of branching processes in a varying environment starting with a random number of particles.
We explore an application from the author’s work in neuroscience. A code used to investigate neural development modeled 100 neurons with all-to-all excitatory connectivity. We used a simple ordinary differential equation system to model each neuron, and this 100-neuron model was used to produce a paper published in the Journal of Neurophysiology. Later a colleague used our code to continue this work, and found he could not reproduce our results. This lead us to thoroughly investigate this code and we discovered that it offered many different ways to thwart reproducibility that could be explained by round-off error arising from floating-point arithmetic.Numerical reproducibility is considered a task that directly follows from the determinism in computations. However, reproducibility has become an intense concern and and issue for research. In fact, the author developed an series of international workshops on numerical reproducibility and computational reproducibility that is now a regular offering at the annual international supercomputing conferences. We will show how this particular code provides a lack of reproducibility from the following three mechanisms: (i) the introduction of floating-point errors in an inner product; (ii) introduction of floating-point errors at each an increasing number of time steps during temporal refinement (ii); and (iii) differences in the output of library mathematical functions at the level of round-off error. This code’s sensitivity makes it a very powerful tool to explore many different manifestations of numerical reproducibility. However, this code is by no means exceptional, as in neuroscience these types of models are used extensively to gain insights on the functioning of the nervous system. In addition, these types of models are widely used in many other fields of study as they just nonlinear evolution equations.
