Current telescopic surveys are aimed to identify meter-sized or larger asteroids before colliding with the Earth, as exemplified by successful observations of 2008 TC3 and recently, 2023 CX1. While a space-based infrared (IR) telescope monitoring system of the Earth’s environment would greatly benefit the scientific community, many extraterrestrial impacts occur in broad daylight. A sobering example of a destructive daylight event is the Chelyabinsk bolide. However, energetic daylight bolides are often elusive and require innovative techniques to reconstruct trajectories and orbits. The near-Earth population of bodies is dominated by remnants of fragmented objects, only a few meters in diameter or smaller, which have the benefit of being the main source of meteorite falls. Notable examples demonstrating the relevance of these events are the three impacting asteroids recovered as meteorites: 2008 TC (Almahata Sitta meteorite), 2018 LA (Motopi Pan), and 2023 CX1.

There is a growing interest in the study of potentially hazardous asteroids (PHAs), not only in space, but also during various phases upon their entry into the Earth’s atmosphere, to gain insight about their properties as well as infer their impact hazard capacity. The surprising entry of the Chelyabinsk bolide, which was not detected a priori, raised concerns about our monitoring capabilities from the ground. Moreover, there are casual reports of bright bolides or even meteorite falls occurring in broad daylight, but without any prior notice of the encounter, nor a recording of the fireball. Most small asteroids in the size range of one meter to about ten meters are of particular scientific interest because they produce meteorite falls. Depending on the entry geometry and the body’s physical properties, a significant part of the kinetic energy propagates towards the ground in the form of shock waves, which may subsequently produce coupled seismic waves, or even formation of impact craters. Consequently, infrasound and seismic detections of shock waves are essential for shedding more light on these events. However, quite frequently, detection data using various sensing modalities, including casual witness reports are scattered or incomplete, and only in very rare cases leveraged towards obtaining more clues about specific bolide events.

This Special Issue aims to collate original research and review articles focusing on research initiatives with the goal of advancing these research topics. We particularly welcome submissions on new observational techniques and fireball events without any a priori warning, as well as review articles discussing the current state of the art in this field.

Potential topics include but are not limited to the following:

• Theory and observations of fireball entries
• Infrasound and seismic observations of fireballs
• Fireball network studies and implications for meteorite-dropping bolides, particularly in broad daylight
• Detection of bolides from space: trajectory reconstruction comparison with ground-based data
• Consequences of impacts with space debris
• Break-up models of asteroids and comets in Earth’s atmosphere
• Bolides as a geohazard on Earth’s surface: shock waves and excavation of craters

Mathematical modeling is an active area of applied mathematics. At its beginning, engineers were the main practitioners of this area of mathematics, developing mathematical models for solving engineering problems in natural sciences.

Recently, mathematical modeling of social behavior, such as addictions, social disorders, or educational achievements, has emerged. A paramount issue in the study of social behavior is social contagion, not only by the influence of the Internet and social networking, but also by mimetic human behavior.

Analysis methods and models in social sciences are similar to those of nature sciences including engineering, with the only difference that instead of using principles of the nature one uses principles or theories from experts of such sciences.

In this special issue, we try to integrate models and methods not only in the scope of traditional natural sciences, but also in opening the scope to the social sciences framework. Also, theory and data-driven models, even in a synergy that gives rise to producing fertile, multidisciplinary, and hybrid models, are considered.

Potential topics include, but are not limited to:

• Deterministic differential equations: methods and models
• Random differential equations: methods and models
• Numeric and modelling analysis
• Optimization and evolutionary algorithms models and methods
• Educational models and methods
• Social and medical disorders modelling
• Social networks models and methods
• Engineering models and simulation
• Analysis modelling in economics
• Algebraic models and methods with applications
• Intelligent data analysis models and methods

The search for mathematical models and their analysis was born as a powerful tool for the investigations of nature and as an aid to solve the daily problems that life is offering us. On the other hand, the last decades have witnessed an increased interest in the growth and progress of the study of mathematical methods that are crucial in the quick advancement of the natural, physical, engineering, social, and life sciences.

The purpose of this special issue is to bring together papers covering a wide range of scientific interests concerned with differential equations emanating from mathematical models of classical fields, such as solids and fluid mechanics, or from some fields as biology, economics and finance, and sciences of life that, only in the second part of the last century, have begun to take into consideration the tools offered from mathematics. The mathematical techniques are not only devoted to the search for numerical and exact solutions of the underlying differential equations for general and special problems; they also include new theoretical developments suitable for novel applications of solving real-world problems.

The response to this special issue was beyond our expectation. We received 60 papers to be considered for publication. Those submissions, from different countries and continents, fall in different areas of the above-mentioned research fields. All papers submitted to this special issue went through a rigorous peer-refereeing process. Based on the reviewers reports, thirty-three original research articles have finally been accepted for publication. The contents embrace different qualitative and quantitative techniques, Lie symmetry techniques, numerical methods, stability analysis, and statistical methods to analyze different aspects that are concerned with differential equations arising in economics, finance, biology, physics, and fluid dynamics. It is certainly impossible to provide in this short editorial a more comprehensive description for all articles of this special issue. However, the guest editors believe that the results included reflect some recent trends in research and outline new ideas for future studies. In the following, we briefly describe the significance of the key contributions to our special session.

Lie symmetry methods are powerful tools to find conservation laws/first integrals, reductions, and exact solutions of differential equations. Some papers in our special issue have used Lie symmetry methods to study interesting problems like heat transfer in extended surfaces of different geometries, nonlinear Jaulent-Miodek equation, neutron transport equations in nonhomogeneous media, nonlinear fin equation, and general bond-option pricing equation of mathematical finance. Moreover, different approaches to construct first integrals for ordinary differential equations are compared, as well as symmetries, and their associated first integrals and double reduction of difference equations are also investigated.

Several authors study some interesting fluid mechanics models. Similarity solutions of MHD heat and mass transfer flow of the steady viscous incompressible fluid over a flat plate are derived and the ordinary differential equations obtained from similarity transformations are solved by a finite difference scheme known as the Keller Box method. The homotopy analysis method is employed to study the mixed convective heat transfer in an incompressible steady two-dimensional viscoelastic fluid flow over a wedge in the presence of buoyancy effects.

Stability analysis plays a vital role in understanding of the dynamic structure of complex dynamical systems. In our special issue several papers focus on stability analysis of biological systems and these models include Lotka-Voltera predator-prey system with time-delayed feedback, a nutrient-phytoplankton model with delay effect, a model of emerging infectious disease leading to amphibian decline, a nutrient-algae-zooplankton system with sinking of algae, a reaction-diffusion system modeling cancer network, Hodgkin-Huxley nonlinear model with constraint, malaria epidemic in Karonga district, stochastic cooperative predator-prey system with Beddington-DeAngelis functional response, a reaction-diffusion phytoplankton-zooplankton system with a double Allee effect on prey, and a stochastic delayed competitive system with impulsive toxicant input in a polluted environment. Dynamics of the oxygen, carbon dioxide, and water interaction across the insect is investigated. A traveling wave analysis is carried out for a tumour-immune interaction model with immunotherapy. Numerical simulations are also performed for few of these papers.

Papers collected in this special issue are also concerned with statistical methods which are important to study some tools essential for financial analysts. These papers include analysis of corporate bond valuation under an infinite dimensional compound Poisson framework, the pricing problem for convertible bonds with higher loan rate, convertible bonds via backward stochastic differential equations, the forward credit default swap with general random loss, and study of financial time series model by copula analysis.

Our special issue contains few papers in which different numerical techniques are employed. Haar wavelet method is used to find numerical solution of the system of Fredholm integral equations and the system of Volterra integral equations. The solution of a class of Lane-Emden equations is derived by the reproducing kernel method. Two nonstandard finite difference schemes to solve the regularized long wave equation are reported in our special issue. A new approach based on the effective combination of Lie symmetry method, homotopy perturbation method, finite element method, and simulation based error reduction techniques is presented and employed to transient nonlinear heat conduction problems. Numerical reduced variable optimization methods via implicit functional dependence are proposed and applied to nontrivail real-world problem. A method is developed for the approximate computation of the frequency-dependent magnetic and electric matrix Green’s functions in a rectangular parallelepiped with a perfect conducting boundary. Analytic formulation for the sound absorption of a panel absorber under the effects of microperforation, air pumping, and linear and nonlinear vibrations is reported.

We hope the papers published in this special issue will provide a useful guide to a large community of researchers and will give way to development of new innovative theories in the fields of natural, social, and life sciences.

The fractional and functional differential equations are two hot topics in mathematics, and they have multiple applications in various branches of science and engineering. The combination of these two types of differential equations leads to some interesting results from both theoretical and applied viewpoints.

The purpose of this special issue was to discuss some new algorithms and methods devoted to investigate the fractional and functional differential equations. Below we give a brief survey of the content of this special issue.

Existence theory of positive solutions for second-order singular differential equations with deviating arguments and its parameter dependence were reported. A parallel fast solution for Riesz space fractional reaction-diffusion equation was investigated. Positive solutions for a class of state dependent boundary value problems with fractional order differential operators were found. Step size restrictions for nonlinear stability properties of neutral delay differential equations and Green’s function and positive solutions for a second-order singular boundary value problem with integral boundary conditions and a delayed argument were investigated. The dynamics and synchronization of a fractional-order system with complex variables, the existence of solutions for Riemann-Liouville fractional boundary value problem, and the existence of almost periodic solutions for impulsive neutral functional differential equations were other topics of our special issue. Fractional differential equations with fractional impulsive and nonseparated boundary conditions and global well-posedness and longtime decay of fractional Navier-Stokes equations in Fourier-Besov spaces were reported. Parameter estimation for stochastic differential equations driven by mixed fractional Brownian motion, solutions of a nonlinear Erdelyi-Kober integral equation, and a novel kernel for RBF based neural networks are other subjects of the special issue. Numerical algorithm for the third-order partial differential equation with three-point boundary value problem, modelling the nonlinear wave motion within the scope of the fractional calculus, generalized Kudryashov method for time-fractional differential equations were presented. A novel approach for dealing with partial differential equations with mixed derivatives, the variational iteration transform method for fractional differential equations with local fractional derivative, an efficient collocation method for a class of boundary value problems arising in mathematical physics and geometry, and Cauchy problems for evolutionary pseudo differential equations over p-adic field were analyzed. New ultra spherical wavelets spectral solutions for fractional Riccati differential equations, nonlinear Gronwall-Bellman-Gamidov integral inequalities and their weakly singular analogues with applications, a new Legendre collocation method for solving atwo-dimensional fractional diffusion equation, computation of spectral parameter of discontinuous Dirac systems with a Gaussian multiplier, and an analytical study of fractional-order multiple chaotic FitzHugh-Nagumo neurons model using multistep generalized differential transform method were interesting reported topics. The small time asymptotics of SPDEs with reflection, the convergence of variational iteration method for solving singular partial differential equations with fractional order, the fourth-order compact difference scheme for the Riemann-Liouville and Riesz derivatives, the existence of solutions for fractional boundary value problem with nonlinear derivative dependence, and regularized fractional power parameters for image denoising based on convex solution of fractional heat equation were distinct topics of our special issue. Solving the Fokker-Planck equations on Cantor sets using local fractional decomposition method was also reported and an efficient series solution for fractional differential equations was analyzed. Two hybrid methods for solving two-dimensional linear time-fractional partial differential equations and the application of local fractional series expansion method to solve Klein-Gordon equations on Cantor sets are subjects investigated by the authors of our special issue. A pseudo-spectral algorithm for solving multipantograph delay systems on a semi-infinite interval using Legendre rational functions was developed and a modified groundwater flow model using the space time Riemann-Liouville fractional derivatives approximation was reported.

Dumitru Baleanu
Ali H. Bhrawy
Robert A. Van Gorder

In this special issue, we have solicited submissions from electrical engineers, control engineers, computer scientists, and mathematicians. After a rigorous peer review process, 18 papers have been selected that provide overviews, solutions, or early promises, to manage, analyse, and interpret dynamical behaviours of networked systems. These papers have covered both the theoretical and practical aspects of networked system with incomplete information in the broad areas of dynamical systems, mathematics, statistics, operational research, and engineering.

In this special issue, there is a survey paper on the recent advances of control and filtering problems for Takagi-Sugeno (T-S) fuzzy systems with network-induced phenomena. Specifically, in the paper entitled “Analysis, Filtering, and Control for Takagi-Sugeno Fuzzy Models in Networked Systems” by S. Zhang et al., the focus is to provide a timely review on some recent advances on the T-S fuzzy control and filtering problems with various network-induced phenomena. Because of the advantages in dealing with various nonlinear systems, the fuzzy logic theory has great success in industry applications. Among various kinds of models for fuzzy systems, the T-S fuzzy model is quite popular due to its convenient, simple dynamic structure and the capability of approximating any smooth nonlinear function to any specified accuracy within any compact set. This survey discusses a variety of T-S fuzzy control and filtering issues with network-induced phenomena in great detail firstly. Four network-induced phenomena (communication delays, packet dropouts, signal quantization, and randomly occurring uncertainties (ROUs)) are introduced. Both theories and techniques for dealing with the controller or filter design are systematically reviewed. Then, some latest results on T-S fuzzy control/filtering problems (bilinear T-S fuzzy model, event-based fuzzy control, fuzzy filtering with multiple network-induced phenomena, fuzzy filtering, fault detection, filtering with unknown membership functions, and nonfragile fuzzy filtering) for networked systems are surveyed and some challenging issues for future research are raised. Finally, some conclusions are drawn and several possible related research directions are pointed out.

In the past decades, the stability analysis of the networked systems has attracted much research attention. In the work entitled “Uniform Stability Analysis of Fractional-Order BAM Neural Networks with Delays in the Leakage Terms” by X. Yang et al., the uniform stability analysis is studied for a class of fractional-order BAM neural networks with delays in the leakage terms. By introducing a novel norm, several delay-dependent sufficient conditions are obtained to ensure the uniform stability of the proposed system by using inequality technique and analysis method. Moreover, sufficient conditions are established to guarantee the existence, uniqueness, and uniform stability of the equilibrium point. Three simulation examples are given to demonstrate the effectiveness of the obtained results. It should be pointed out that it is possible to extend the main results of this paper to other complex systems and establish novel stability conditions with less conservatism by using more up-to-date techniques. The -stability issue is discussed in “Global -Stability of Impulsive Complex-Valued Neural Networks with Leakage Delay and Mixed Delays” by X. Chen et al. for complex-valued neural networks (CVNNs) with leakage delay, discrete delay, and distributed delay under impulsive perturbations. The -stability is the concept for the purpose of unifying the exponential stability, power-rate stability, and log-stability of neural networks. CVNN is an extension of real-valued neural network which has been applied in physical systems dealing with electromagnetic, light, ultrasonic, and quantum waves. Based on the homeomorphism mapping principle of complex domain, a sufficient condition for the existence and uniqueness of the equilibrium point of the addressed CVNNs is proposed in terms of linear matrix inequality (LMI). By constructing appropriate Lyapunov-Krasovskii functionals and employing the free weighting matrix method, several delay-dependent criteria for checking the global -stability of the CVNNs are established in LMIs. As direct applications of these results, several criteria on the exponential stability, power-stability, and log-stability are obtained. In the paper entitled “A Switched Approach to Robust Stabilization of Multiple Coupled Networked Control Systems” by M. Yu et al., multiple coupled networked controlled systems (NCSs) with norm-bounded parameter uncertainties and multiple transmissions are considered. All the nodes in the proposed systems act over a limited bandwidth communication channel. The state information of every subsystem is split into different packets and there is only one packet of the subsystem that can be transmitted at a time. Based on the toking bus protocol, the nodes are arranged logically into a ring and transmit the corresponding packets in a prefixed circular order. Then, the proposed multiple NCSs can be modelled as periodic switched systems. Furthermore, the robust stabilization issue is dealt with by applying the switched system theory. State feedback controllers are constructed in terms of LMIs. A numerical example is given to show that the coupled NCSs considered can be effectively stabilized via the designed controller.

Control and fault estimation problems for stochastic systems have been of interest of many researchers during the past decades. In the paper entitled “Robust Control for a Class of Discrete Time-Delay Stochastic Systems with Randomly Occurring Nonlinearities” by Y. Wang et al., the robust problem is studied for a class of discrete time-delay stochastic systems with randomly occurring nonlinearities (RONs). It is assumed that all the system matrices contain the parameter uncertainties. The stochastic disturbances are both state- and control-dependent, and the RONs satisfy the sector boundedness conditions. The purpose of the problem proposed is to design a state feedback controller such that, for all admissible uncertainties, nonlinearities, and time-delays, the closed-loop system is robustly asymptotically stable in the mean square, and a prescribed disturbance rejection attenuation level is also guaranteed. By using the Lyapunov stability theory and stochastic analysis tools, a LMI approach is developed to derive sufficient conditions ensuring the existence of the desired controllers, where the conditions are dependent on the lower and upper bounds of the time-varying delays. The explicit parameterization of the desired controller gains is also given. The problem of control for network-based 2D systems with missing measurements is investigated in “ Control for Network-Based 2D Systems with Missing Measurements” by X. Bu et al. A state feedback controller is designed such that the closed-loop 2D stochastic system is mean-square asymptotic stability and has disturbance attenuation performance. A sufficient condition is derived in terms of LMIs technique, and formulas can be given for the control law design. The result is also extended to more general cases where the system matrices contain uncertain parameters. Numerical examples are also provided to show the effectiveness of proposed approach. In the work entitled “Krein Space-Based Fault Estimation for Discrete Time-Delay Systems” by X. Song and X. Yan, the finite-time fault estimation issue is investigated for linear time-delay systems where the delay appears in both state output and measurement output. Firstly, the design of finite horizon fault estimation is converted into a minimum problem of certain quadratic form. Then, a sufficient and necessary condition for the existence of the desired fault estimator is derived by employing the Krein-space theory. A solution of the desired fault estimator is obtained by recursively computing a partial difference Riccati equation which has the same dimension as the original systems. Therefore, solving a high dimension Riccati equation is avoided compared with the conventional augmented method. A numerical example is given to demonstrate the effectiveness of the approach.

Circulant type matrices have significant applications in network systems. In the work entitled “Equalities and Inequalities for Norms of Block Imaginary Circulant Operator Matrices” by X. Jiang and K. Hong, the block imaginary circulant operator matrices are studied. Firstly, by combining the special properties of block imaginary circulant operator matrix with unitarily invariant norm, several norm equalities are obtained. It should be pointed out that the norm in consideration is the weakly unitarily invariant norm. The usual operator norm and Schatten -norm are included. Then, several pinching type inequalities are presented by the triangle inequality and the invariance property of unitarily invariant norms. Furthermore, some special cases and examples are considered. Circulant and left circulant matrices with Fermat and Mersenne numbers are considered in “Exact Inverse Matrices of Fermat and Mersenne Circulant Matrix” by Y. Zheng and S. Shon. Moreover, the exact determinants and the inverse matrices of Fermat and Mersenne left circulant matrix are given. The nonsingularity of these special matrices is discussed. In the paper entitled “Norms and Spread of the Fibonacci and Lucas RSFMLR Circulant Matrices” by W. Xu and Z. Jiang the norms and spread of Fibonacci row skew first-minus-last right (RSFMLR) circulant matrices are investigated as well as the Lucas RSFMLR circulant matrices. Firstly, these two kinds of special matrices are defined. Then, the lower and upper bounds for the spectral norms of these matrices are proposed as well as the upper bounds for the spread of these matrices. Afterwards, some corollaries related to norms of Hadamard and Kronecker products of these matrices are obtained, respectively. The determinants and inverses of Tribonacci circulant type matrices are discussed in “Explicit Form of the Inverse Matrices of Tribonacci Circulant Type Matrices” by L. Liu and Z. Jian. The definition of Tribonacci circulant type matrices is given firstly. Then, the invertibility of Tribonacci circulant type matrices is studied. Based on constructing the transformation matrices, both the determinant and the inverse matrix are derived. Furthermore, by utilizing the relation between left circulant, -circulant matrices, and circulant matrix, the invertibility of Tribonacci left circulant and Tribonacci -circulant matrices is also studied. Finally, the determinants and inverse matrices of these matrices are presented, respectively. A future research direction is pointed out at last. In the paper entitled “Analysis of the Structured Perturbation for the BCSCB Linear System” by X. Tang and Z. Jian, the analysis problem associated with the BCSCB matrix is considered. The BCSCB matrix is an extension of the circulant matrix and skew circulant matrix. Firstly, the form of the BCSCB matrix is obtained based on the style spectral decomposition of the basic circulant matrix and the basic skew circulant matrix. Then, the structured perturbation analysis for BCSCB linear system is proposed, which includes the condition number and relative error of the BSCSB linear system. A new approach is presented to derive the minimal value of the perturbation bound, which is only related to the perturbation of the coefficient matrix and the vector. Simultaneously, the algorithm for the optimal backward perturbation bound is developed.

As is well known, the analysis of issues on network systems has important significance. In the paper entitled “On the Incidence Energy of Some Toroidal Lattices” by J.-B. Liu, et al., the closed-form formulae expressing the incidence energy of the 3.12.12 lattice and triangular kagomé lattice are derived as well as lattice. The calculations of the energy of graphs become a popular topic of research. However, it is not an easy task to deal with the problem of the asymptotic incidence energy of various lattices with the free boundary. By utilizing the applications of analysis approach with the help of software calculation, the explicit asymptotic values of the incidence energy in these lattices are derived simultaneously. This developed method can be used widely to handle the asymptotic behaviour of other lattices and can obtain some useful results simultaneously. For the purpose of studying the distribution of evolving networks, a kind of evolving network is proposed in “Asymptotic Degree Distribution of a Kind of Asymmetric Evolving Network” by Z. Li et al., where the model is a combination of preferential attachment model and uniform model. The distribution of the number of vertices with given degree is studied as well as the asymptotic degree distribution. It is shown that the proportional degree sequence obeys power law, exponential distribution, and other forms according to the relation of the degree and parameter . In the work entitled “Partial Synchronizability Characterized by Principal Quasi-Submatrices Corresponding to Clusters” by G. Zhang et al., a partial synchronization problem is studied in an oscillator network. In order to investigate the partial synchronization, the concept on a principal quasi-submatrix corresponding to the topology of a cluster is proposed. A novel criterion on partial synchronization is developed based on the analysis of principal quasi-submatrices corresponding to the clusters. The proposed criterion is not distinctly dependent on the intercluster couplings or the topology matrix of the whole network. If a network is composed of a large number of nodes, the enormous amount of calculation can be reduced by replacing the coupling matrix with several quasi-submatrices. Therefore, this criterion provides a novel index of partial synchronizability. It is shown that different types of partial synchronization occur in a star-global network when the coupling strength is increased. The proposed approach for partial synchronization might be applicable to the complex networks with networked induced phenomena. A backbone extraction heuristic with incomplete information (BEHwII) is investigated in “Extracting Backbones from Weighted Complex Networks with Incomplete Information” by L. Qian et al. The presence of the backbone is the signature or the abstraction of the nature of complex systems and can provide huge help for understanding them in more simplified forms. For the purpose of extracting backbones from large-scale weighted networks, a novel filter-based approach is presented which only needs incomplete information and then invokes the iteratively local search scheme for improving the efficiency. First, a strict filtering rule is designed to determine edges to be preserved or discarded. Then, a local search model is proposed to examine part of edges in an iterative way. Experimental results on four real-life networks demonstrate the advantage of BEHwII over the classic disparity filter method by either effectiveness or efficiency validity.

Recently, the application of networked systems has attracted a great deal of research interest. The data communication networks play an important role during the development of smart grid. Since the data communication network in smart gird is affected by plenty of decisive factors, different decision-making problems are presented according to the variable factors. In the paper entitled “Real-Time Pricing Decision Making for Retailer-Wholesaler in Smart Grid Based on Game Theory” by Y. Dai and Y. Gao, a novel game-theoretical decision-making scheme is investigated for electricity retailers and wholesaler in the smart grid with demand side management (DSM). The interaction between two retailers and their wholesaler is modelled by a two-stage dynamic game where the competition between two retailers is considered. According to the different action order between retailers and their wholesaler, two different game models are developed. The subgame perfect Nash equilibrium (SPE) for this game is determined through backward induction. It is shown that the wholesaler wants to decentralize certain management powers to the retailers through analysing the equilibrium revenues of the retailers for different situations. Imposing legal restrictions on the wholesaler’s discretionary policy suggests that the time-inconsistency problem is mitigated. The packing problem of unit equilateral triangles is investigated in “A New Quasi-Human Algorithm for Solving the Packing Problem of Unit Equilateral Triangles” by R. Wang et al. The packing problem of unit equilateral triangles offers broad prospects in different fields including the network resource optimization. This problem is nondeterministic polynomial (NP) hard and has the feature of continuity. A novel quasi-human algorithm for solving this problem is proposed according to the characteristic of the unit equilateral triangles and in the base of analysis of the general triangles packing problems. Time complexity analysis and the calculation results indicate that the proposed method is a polynomial time algorithm, which provides the possibility to solve the packing problem of arbitrary triangles.