This paper investigates the population dynamics of a Fisher-KPP reaction-diffusion model with a nonlocal free boundary. Unlike classical Stefan-type problems, this model incorporates a nonlocal integral condition that accounts for long-range perception and memory effects in species movement. We develop an efficient numerical framework by first applying a front-fixing transformation to map the evolving habitat onto a fixed domain. Then, an Alternating Direction Explicit (ADE) scheme is constructed to solve the resulting nonlinear system.The long-term behavior of this model is characterized by a spreading-balancing-vanishing trichotomy. We propose a Physics-Informed Machine Learning (PIML) approach to predict the population's asymptotic behavior. By training an ensemble classifier on data generated by our numerical scheme and embedding theoretical mathematical features, such as kernel mass and critical survival ratios, into the model, we achieve a 94% accuracy rate in classifying the trichotomy. This hybrid framework enables the instantaneous generation of high-resolution bifurcation maps, providing a powerful tool for predicting species survival and range dynamics without the need for numerical simulations.

Free-boundary diffusive logistic model finds applications in diverse fields associated with population dynamics. These processes often possess stochastic characteristics and involve parameters with uncertainties. This study focuses on enhancing a two-dimensional diffusive logistic partial differential model with free boundary by incorporating randomness in the mean square sense, considering the conditions for well-posedness in the random case, which is crucial for the further analysis. Both unknown stochastic processes the solution and its moving front, and the parameters involved in the random problem as random variables, are constrained by a finite degree of randomness. To tackle this challenge, we propose a random level set method. Given the complexity of the problem, we employ alternating direction explicit methods for the interior solvers, to effectively address computational challenges. Since computing the mean and the standard deviation of both unknown stochastic processes are required, we combine the sample approach of the difference schemes together with Monte Carlo technique avoiding the storage accumulation of symbolic expressions of all the previous levels of the iteration process. Parallel computing is employed to enhance performance. A careful numerical analysis is performed in the mean square context to ensure stability, positivity, and boundedness. The set of presented examples illustrates these qualitative properties, assess numerical convergence and enables us to gain a deeper understanding of the system’s behavior attending to the geometry of the initial habitat. This approach provides valuable tools for analyzing and predicting spreading-vanishing dichotomy.

This paper deals with modelling the irregular immigration problem for the case of Spain. This problem is becoming a crucial humanitarian, social and political worldwide. From the scientific point of view is also challenging because the weaponization of human waves, in the sense that governments use the irregular immigration as a weapon to get political and economic advantages, from both parts the host country and the issuing countries. Geography plays its role and the case of Spain is particularly important. We construct a logistic type discrete dynamic mathematical model for irregular immigration for the case of Spain, starting from the data analysis of available institutions. Surprisingly interesting is the fact that host country attending to socio-political reasons implement regularization eventual measures to reduce the carrying capacity of the human behavior. The model is able to capture these government decisions when they occur, in order to forecast the amount of people involved in the phenomenon of irregular immigration.

In the domain of maritime security and safety, the surveillance and monitoring of vessels in ports are crucial for safeguarding against potential threats and ensuring the overall integrity of port operations. Current port surveillance methods, including radar systems, CCTV networks, and AIS, exhibit limitations, necessitating innovative solutions. The integration of drone-captured imagery holds promise, particularly for detecting non-cooperative vessels that may evade traditional surveillance methods. This paper presents VESSELimg dataset, a meticulously gathered and annotated collection of drone-captured images within port environments. This dataset serves as a benchmark for training and validating deep learning models, specifically tailored for automatic vessel detection, covering diverse vessel types. The implementation of a YOLO-based deep learning model for real-time inference demonstrates the practical applicability of the dataset, underscoring its potential for enhancing security and safety measures in port environments.

In this paper a discrete logistic–like dynamic model for the immigration arriving in Spain is constructed. The immigration population is split in two categories: regular and irregular. Taking into account economic, regulation, and political factors the model, is built in a vector form. For the sake of reliability the period of time is intentionally short, because the economic behavior and legal regulation are not stable.

A free boundary diffusive logistic model finds application in many different fields from biological invasion to wildfire propagation. However, many of these processes show a random nature and contain uncertainties in the parameters. In this paper we extend the diffusive logistic model with unknown moving front to the random scenario by assuming that the involved parameters have a finite degree of randomness. The resulting mathematical model becomes a random free boundary partial differential problem and it is addressed numerically combining the finite difference method with two approaches for the treatment of the moving front. Firstly, we propose a front-fixing transformation, reshaping the original random free boundary domain into a fixed deterministic one. A second approach is using the front-tracking method to capture the evolution of the moving front adapted to the random framework. Statistical moments of the approximating solution stochastic process and the stochastic moving boundary solution are calculated by the Monte Carlo technique. Qualitative numerical analysis establishes the stability and positivity conditions. Numerical examples are provided to compare both approaches, study the spreading-vanishing dichotomy, prove qualitative properties of the schemes and show the numerical convergence.

This paper introduces the exponential time differencing (ETD) technique as a numerical method to efficiently solve vulnerable American options pricing. We address several challenges, including removing cross-derivative terms through appropriate transformations, treating early-exercise opportunities using the penalty method, and substituting fixed boundary conditions with corresponding one-sided finite differences. The proposed method is shown to be both accurate and efficient through numerical experiments, which also compare the results with existing methods and analyze the numerical stability and convergence rate.

This paper presents a random discrete mathematical population model for immigration. This model incorporates not only rational factors, such as the economic gradient between destination and origin countries, geographical factors, and regulatory laws but also hidden intentional factors, such as the political interests of governments and the involvement of migrant smuggling by criminal organizations, which exploit immigration as a strategic tool. These non-rational factors are modeled as sudden, random arrival flow waves, represented by a Poisson distribution. The study’s time frame is short to ensure the reliability of economic forecasts for the coming years. Although the study focuses on Spain, the proposed approach is applicable to other geographic areas with appropriate data. The results obtained from this model can be applied to predict the national budget necessary for host countries to address this complex social phenomenon.

This paper deals with the construction of numerical solutions of moving boundary random problems where the uncertainty is limited to a finite degree of randomness in the mean square framework. Using a front fixing approach the problem is firstly transformed into a fixed boundary one. Then a random finite difference scheme for both the partial differential equation and the Stefan condition, allows the discretization. Since statistical moments of the approximate stochastic process solution are required, we combine the sample approach of the difference schemes together with Monte Carlo technique to perform manageable approximations of the expectation and variance of both the approximating stochastic process solution and the stochastic moving boundary solution. Qualitative and reliability properties such as positivity, monotonicity and stability in the mean square sense are treated. Feasibility of the proposed method is checked with illustrative examples of a melting problem and a binary metallic alloys problems.

In this paper, we propose a numerical method for American multi-asset options under jump-diffusion model based on the combination of the exponential time differencing (ETD) technique for the differential operator and Gauss-Hermite quadrature for the integral term. In order to simplify the computational stencil and improve characteristics of the ETD-scheme mixed derivative eliminating transformation is applied. The results are compared with recently proposed methods.

A two-dimensional free-boundary diffusive logistic model with radial symmetry is considered. This model is used in various fields to describe the dynamics of spreading in different media: fire propagation, spreading of population or biological invasions. Due to the radial symmetry, the free boundary can be treated by a front-fixing approach resulting in a fixed-domain non-linear problem, which is solved by an explicit finite difference method. Qualitative numerical analysis establishes the stability, positivity and monotonicity conditions. Special attention is paid to the spreading–vanishing dichotomy and a numerical algorithm for the spreading–vanishing boundary is proposed. Theoretical statements are illustrated by numerical tests.

In this paper a random discrete mathematical population model of immigration is constructed using not only rational factors as the gradient of economy level among the host and issuing countries, the geography, or the regulation laws, but also hidden intentional factors such as political interests of governments, or the business of smuggling of migrants by mafias, which use the immigration practically as a weapon. These non-rational factors are modeled as sudden random arrival flow waves represented by a Poisson distribution. The period of study is short in order to make reliable the economic forecast of the next years. Although the study is focused in Spain, the approach is exportable to other geographic areas by adapting the data. Results are applied to predict the necessary national budget of the host country to afford this social phenomenon.

This paper deals with the construction of a discrete dynamic population model addressed to estimate the expected size of the immigration population in a finite short period of time in Spain. By paying attention to a special subpopulation of interest, such as an irregular immigrant, unaccompanied minor immigrant and regular immigrant, a vector discrete population model is built after the discussion and introduction of proper hypotheses linked to economy, host and country of origin regulation policies, political interest and others. The model allows us to study the change of the results under variation of the parameters.

A new efficient numerical method is proposed for valuation of American option on zero‐coupon bond using Hull and White model. By applying the front‐fixing transformation suggested by Holmes and Yang, the original free boundary problem is transformed into a new fixed boundary partial differential equation (PDE) problem, where the optimal stopping boundary is one of the unknowns of the problem. The numerical finite difference scheme for the transformed problem is constructed. Stability and convergence rate is studied empirically. Numerical simulation of the computation of both the option price and the optimal stopping boundary are illustrated with examples and the comparison with the Hull and White tree method.

This paper deals with the search for reliable efficient finite difference methods for the numerical solution of random heterogeneous diffusion reaction models with a finite degree of randomness. Efficiency appeals to the computational challenge in the random framework that requires not only the approximating stochastic process solution but also its expectation and variance. After studying positivity and conditional random mean square stability, the computation of the expectation and variance of the approximating stochastic process is not performed directly but through using a set of sampling finite difference schemes coming out by taking realizations of the random scheme and using Monte Carlo technique. Thus, the storage accumulation of symbolic expressions collapsing the approach is avoided keeping reliability. Results are simulated and a procedure for the numerical computation is given.

In this paper, we consider random hyperbolic partial differential equation (PDE) problems following the mean square approach and Laplace transform technique. Randomness requires not only the computation of the approximating stochastic processes, but also its statistical moments. Hence, appropriate numerical methods should allow for the efficient computation of the expectation and variance. Here, we analyse different numerical methods around the inverse Laplace transform and its evaluation by using several integration techniques, including midpoint quadrature rule, Gauss–Laguerre quadrature and its extensions, and the Talbot algorithm. Simulations, numerical convergence, and computational process time with experiments are shown.

American options prices under jump–diffusion models are determined by a free boundary partial integro-differential equation (PIDE) problem. In this paper, we propose a front-fixing exponential time differencing (FF-ETD) method composed of several steps. First, the free boundary is included into equation by applying the front-fixing transformation. Second, the resulting nonlinear PIDE is semi-discretized, that leads to a system of ordinary differential equations (ODEs). Third, a numerical solution of the system is constructed by using exponential time differencing (ETD) method and matrix quadrature rules. Finally, numerical analysis is provided to establish empirical stability conditions on step sizes. Numerical results show the efficiency and competitiveness of the FF-ETD method.

This paper deals with the construction of numerical stable solutions of random mean square Fisher‐Kolmogorov‐Petrosky‐Piskunov (Fisher‐KPP) models with advection. The construction of the numerical scheme is performed in two stages. Firstly, a semidiscretization technique transforms the original continuous problem into a nonlinear inhomogeneous system of random differential equations. Then, by extending to the random framework, the ideas of the exponential time differencing method, a full vector discretization of the problem addresses to a random vector difference scheme. A sample approach of the random vector difference scheme, the use of properties of Metzler matrices and the logarithmic norm allow the proof of stability of the numerical solutions in the mean square sense. In spite of the computational complexity, the results are illustrated by comparing the results with a test problem where the exact solution is known.

The two‐phase Stefan problems with phase formation and depletion are special cases of moving boundary problems with interest in science and industry. In this work, we study a solidification problem, introducing a front‐fixing transformation. The resulting non‐linear partial differential system involves singularities, both at the beginning of the freezing process and when the depletion is complete, that are treated with special attention in the numerical modelling. The problem is decomposed in three stages, in which implicit and explicit finite difference schemes are used. Numerical analysis reveals qualitative properties of the numerical solution spatial monotonicity of both solid and liquid temperatures and the evolution of the solidification front. Numerical experiments illustrate the behaviour of the temperatures profiles with time, as well as the dynamics of the solidification front.

A numerical method for American options pricing on assets under the Heston stochastic volatility model is developed. A preliminary transformation is applied to remove the mixed derivative term avoiding known numerical draw-backs and reducing computational costs. Free boundary is treated by the penalty method. Transformed nonlinear partial differential equation is solved numerically by using the method of lines. For full discretization the exponential time differen-cing method is used. Numerical analysis establishes the stability and positivity of the proposed method. The numerical convergence behaviour and effectiveness are investigated in extensive numerical experiments.