Many engineering problems are described using differential equations. The work presents the use of certain bio-inspired heuristic algorithms for solving ordinary differential equations, the form of which may be non-linear, and the equation itself does not have to be presented in explicit form.

Based on sensor networks within a dynamic and real-time environment, a novel parking-lot navigation method is proposed based on diffusion equation and Poisson equation to achieve convenient and efficient navigation process with edge computing mind to aid information query and navigation. From the perspective of theoretical proof, is presented parallel method mainly for ordinary differential equations (ODEs) by partitioning the time domain. In this article, our new method is combined by parallelization for linear heat equations. The model problem is decoupled into several sub-problems in space-time sub-domains. We prove the super-linear convergence when the time interval is bounded. Numerical experiments testify our theoretical analysis. Simultaneously, the proposed method holds the lower constraint condition and some improper navigation routes can be updated. Mathematical analysis and simulations show that the method is accurately and efficiently enabled to solve typical sensor network configuration information navigation problem.

In this article, Polar Bear Optimization Algorithm (PBO) is parallelized to solve the problem of computed tomography (CT) with incomplete data. It is vary hard to model correctly 2D and 3D objects by using CT scanners when information is incomplete. Our approach is to use PBO to reduce recovery time and simplify specificity of the phenomenon. Results from our research show that proposed approach is enabling fast and accurate reconstruction of objects modeled in projection space.

The paper is devoted to discussion of the minimal cycles of the so called Kaprekar's transformations and some of its generalizations.The considered transformations are the self-maps of the sets of natural numbers possessing n digits in their decimal expansions.In the paper there are introduced several new characteristics of such maps, among others, the ones connected with the Sharkovsky's theorem and with the Erdős-Szekeres theorem concerning the monotonic subsequences.Because of the size the study is divided into two parts.Part I includes the considerations of strictly theoretical nature resulting from the definition of Kaprekar's transformations.We find here all the minimal orbits of Kaprekar's transformations Tn, for n = 3, ..., 7.Moreover, we define many different generalizations of the Kaprekar's transformations and we discuss their minimal orbits for the selected cases.In Part II (ibidem), which is a continuation of the current paper, the theoretical discussion will be supported by the numerical observations.For example, we notice there that each fixed point, familiar to us, of any Kaprekar's transformation generates an infinite sequence of fixed points of the other Kaprekar's transformations.The observed facts concern also several generalizations of the Kaprekar's transformations defined in Part I.