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 paper, 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.

Recently, a lot of attention has been paid to the field of research connected with the wireless sensor network and industrial internet of things. The solutions found by theorists are next used in practice in such area as smart industries, smart devices, smart home, smart transportation and the like. Therefore, there is a need to look for some new techniques for solving the problems described by means of the appropriate equations, including differential equations, integral equations and integro-differential equations. The object of interests of this paper is the method dedicated for solving some integro-differential equations with a retarded (delayed) argument. The proposed procedure is based on the Taylor differential transformation which enables to transform the given integro-differential equation into a respective system of algebraic (nonlinear, very often) equations. The described method is efficient and relatively simple to use, however a high degree of generality and complexity of problems, defined by means of the discussed equations, makes impossible to obtain a general form of their solution and enforces an individual approach to each equation, which, however, does not diminish the benefits associated with its use.

The solution of the one-phase Stefan problem is presented. A Stefan's task is first approximated with a system of ordinary differential equations. A comparison between the Adomian decomposition method and the fourth-order Runge–Kutta method for solving this system is then presented.

Ordinary differential equations (ODEs), and the systems of such equations, are used for describing many essential physical phenomena. Therefore, the ability to efficiently solve such tasks is important and desired. The goal of this paper is to compare three methods devoted to solving ODEs and their systems, with respect to the quality of obtained solutions, as well as the speed and reliability of working. These approaches are the classical and often applied Runge–Kutta method of order 4 (RK4), the method developed on the ground of the Taylor series, the differential transformation method (DTM), and the routine available in the Mathematica software (Mat).

Recovering energy from waste is a positive element in the operation of a waste incineration plant. Hazardous waste is a very diverse group, including in terms of its fuel properties. Carrying out the thermal process in this case is associated with the difficulty in maintaining stable conditions. This may translate into the efficiency of energy recovery from waste. The article presents a tool supporting the work of hazardous waste incineration plant operators, the aim of which is to select waste for a batch of input material in a manner that ensures process stability and efficient energy recovery. The tool is an application in which the bee algorithm is implemented. It selects the optimal solution to the problem, in accordance with the assumed parameters. The application tests in laboratory conditions were satisfactory and indicated compliance with the assumptions and stability of the solution.

The history of variational calculus dates back to the late 17th century when Johann Bernoulli presented his famous problem concerning the brachistochrone curve. Since then, variational calculus has developed intensively as many problems in physics and engineering are described by equations from this branch of mathematical analysis. This paper presents two non-classical, distinct methods for solving such problems. The first method is based on the differential transform method (DTM), which seeks an analytical solution in the form of a certain functional series. The second method, on the other hand, is based on the physics-informed neural network (PINN), where artificial intelligence in the form of a neural network is used to solve the differential equation. In addition to describing both methods, this paper also presents numerical examples along with a comparison of the obtained results.Comparingthe two methods, DTM produced marginally more accurate results than PINNs. While PINNs exhibited slightly higher errors, their performance remained commendable. The key strengths of neural networks are their adaptability and ease of implementation. Both approaches discussed in the article are effective for addressing the examined problems.