Time Space Research Articles

Online Three-Dimensional Fuzzy Reinforcement Learning Modeling for Nonlinear Distributed Parameter Systems

Distributed parameter systems (DPSs) frequently appear in industrial manufacturing processes, with complex characteristics such as time–space coupling, nonlinearity, infinite dimension, uncertainty and so on, which is full of challenges to the modeling of the system. At present, most DPS modeling methods are offline. When the internal parameters or external environment of DPS change, the offline model is incapable of accurately representing the dynamic attributes of the real system. Establishing an online model for DPS that accurately reflects the real-time dynamics of the system is very important. In this paper, the idea of reinforcement learning is creatively integrated into the three-dimensional (3D) fuzzy model and a reinforcement learning-based 3D fuzzy modeling method is proposed. The agent improves the strategy by continuously interacting with the environment, so that the 3D fuzzy model can adaptively establish the online model from scratch. Specifically, this paper combines the deterministic strategy gradient reinforcement learning algorithm based on an actor critic framework with a 3D fuzzy system. The actor function and critic function are represented by two 3D fuzzy systems and the critic function and actor function are updated alternately. The critic function uses a TD (0) target and is updated via the semi-gradient method; the actor function is updated by using the chain derivation rule on the behavior value function and the actor function is the established DPS online model. Since DPS modeling is a continuous problem, this paper proposes a TD (0) target based on average reward, which can effectively realize online modeling. The suggested methodology is implemented on a three-zone rapid thermal chemical vapor deposition reactor system and the simulation results demonstrate the efficacy of the methodology.

Humanitarian Architecture in Service of Children: Designing an Orphanage for Vulnerable Children in Yaoundé

This research explores the impact of humanitarian architecture in designing an orphanage for vulnerable children in Yaoundé, Cameroon. In a context where millions of children are affected by social, economic, and political crises, particularly in Sub- Saharan Africa, humanitarian architecture emerges as a viable response to ensure their protection and well-being. The literature review highlights the key principles of humanitarian architecture, such as user-centered design, the use of local and sustainable materials, and the creation of resilient spaces in times of crisis. Through case studies, including projects in Burkina Faso and Mali, this research examines how architectural approaches can be integrated into educational and care facilities to improve the living environment of vulnerable children. The work concludes with design proposals for an orphanage in Yaoundé, focusing on adaptation to the local climate, material sustainability, and community participation, aiming to strengthen the resilience and autonomy of children.

Transformation of informal elder care practices and mobilities during the pandemic

ABSTRACT The COVID-19 pandemic revealed that when formal care services closed down, informal care burden was unequally transferred to women. Women across the world became part of “caregiver pools”. Using textual archive material collected by the Finnish Literature Society, we analyse the transformation of informal elder care practices in Finland. We adopt a care mobility and care process perspective to analyse how informal elder care was transformed due to social distancing measures that restricted ageing individuals and their family members’ local and translocal mobilities. We find that over 70-year-olds had to adapt their daily mobilities according to risk assessment. Caregiving provided by mostly female family members was replaced by digital care at a distance, which reinforced existing inequalities in care. We argue that the pandemic simultaneously brought about both a time–space compression and what we call a time–space expansion that affected the everyday practices and mobilities of care.

Time-dependent electromagnetic scattering from dispersive materials

Abstract This paper studies time-dependent electromagnetic scattering from obstacles that are described by dispersive material laws. We consider the numerical treatment of a scattering problem in which a dispersive material law, for a causal and passive homogeneous material, determines the wave–material interaction in the scatterer. The resulting problem is nonlocal in time inside the scatterer and is posed on an unbounded domain. Well-posedness of the scattering problem is shown using a formulation that is fully given on the surface of the scatterer via a time-dependent boundary integral equation. Discretizing this equation by convolution quadrature in time and boundary elements in space yields a provably stable and convergent method that is fully parallel in time and space. Under regularity assumptions on the exact solution we derive error bounds with explicit convergence rates in time and space. Numerical experiments illustrate the theoretical results and show the effectiveness of the method.

Rapidly Screening the Correlation between the Rotational Mobility and the Hydrogen Bonding Strength of Confined Water.

Automated Deuterium Relaxation-Ordered SpectroscopY in solution (ADROSYS), an automated two-dimensional deuterium NMR methodology, discriminates between D2O populations (as well as deuterium-labeled alcohol groups) whose properties differ as a result of being confined inside nanoscale volumes. In this contribution, a proof-of-principle demonstration on reverse micelles (RMs) yields the insight that as the length scale of the confinement decreases from several nanometers down to less than a nanometer, the position of the signal peak migrates through the two-dimensional (2D) spectrum, tracing out a distinctive path in the 2D space (of relaxation time vs chemical shift). The signals typically follow a relatively gentle linear path for water confined on the scale of several nanometers, before curving once the surfactants confine the water molecules to length scales smaller than 1-2 nm. The qualitative shape of this path, especially in the regime of strong confinement, can change with different choices of surfactants, i.e., a different choice of chemistry at the edges of the confining environment. An important facet of this research was to demonstrate the relatively wide applicability of these techniques by showing that both: (1) Standard modern NMR instrumentation is capable of deploying an automated measurement, even though the choice of a deuterium nucleus is nonstandard and frequently requires companion proton spectra in order to reference the chemical shifts; and (2) well-established (though underutilized) modern techniques can process the resulting signal even though it involves the somewhat unusual combination of chemical shifts along one dimension and a distribution of relaxation times along another dimension. In addition to demonstrating that this technique can be deployed across many samples of interest, detailed facts pertaining to the broadening or shifting of resulting signals upon inclusion of various guest molecules are also discussed.

Analytical modelling and analysis of the meander-line coil EMATs

The meander-line coil electromagnetic acoustic transducer (EMAT) is widely used in the field of ultrasonic nondestructive testing due to its convenience to generate specific mode of guided waves. Some design methods of the meander-coil EMATs are developed in the frequency-wavenumber domain while others in the time–space domain. In this paper, a unified theoretical framework is developed by proposing an analytical model from the system perspective. Signal transfers between different physical fields in EMAT excitation, wave propagation and EMAT reception are represented as linear time–space-invariant systems. Taking the Rayleigh wave EMAT detection as an example, the analytical model for the transfer functions of these systems is established. The analytical model is experimentally verified by different Rayleigh wave detection techniques: the conventional EMAT, the spatial pulse compression (SPC) EMAT, temporal-spatial pulse compression (TSPC) EMAT and detection cases employing the same receiving EMAT. From the system perspective, the received signal of EMAT is interpreted as the response of the filter system to the input signal. It is found that the meander-coil EMAT can be regarded as the frequency domain expression during the detection. And the frequency domain expression plays different roles in different techniques.

Macroscopic State-Level Analysis of Pavement Roughness Using Time–Space Econometric Modeling Methods

This paper used pavement condition data collected by the Federal Highway Administration (FHWA) between 2001 and 2006 aggregated by U.S. states to identify macroscopic factors affecting pavement roughness in time and space. To account for prior pavement conditions and preservation expenditure over time, time autocorrelation parameters were introduced in a spatial modeling scheme that accounted for spatial autocorrelation and heterogeneity. The proposed framework accommodates data aggregation in network-level pavement deterioration models. Because pavement roughness across different roadway classes is anticipated to be affected by different explanatory parameters, separate time–space models are estimated for nine roadway classes (rural interstate roads, rural collectors, urban minor arterials, urban principal arterials, and other freeways). The best model specifications revealed that different time–space models were appropriate for pavement performance modeling across the different roadway classes. Factors that were found to affect state-level pavement roughness in time and space included preservation expenditure, predominant soil type, and predominant climatic conditions. The results have the potential to assist governmental agencies in planning effectively for pavement preservation programs at a macroscopic level.

Instabilities and Pattern Formation in Epidemic Spread Induced by Nonlinear Saturation Effects and Ornstein–Uhlenbeck Noise

Abstract We analytically study the emergence of instabilities and the consequent steady-state pattern formation in a stochastic partial differential equation (PDE) based, compartmental model of spatiotemporal epidemic spread. The model is characterized by: (1) strongly nonlinear forces representing the infection transmission mechanism and (2) random environmental forces represented by the Ornstein–Uhlenbeck (O–U) stochastic process which better approximates real-world uncertainties. Employing second-order perturbation analysis and computing the local Lyapunov exponent, we find the emergence of diffusion-induced instabilities and analyze the effects of O–U noise on these instabilities. We obtain a range of values of the diffusion coefficient and correlation time in parameter space that support the onset of instabilities. Notably, the stability and pattern formation results depend critically on the correlation time of the O–U stochastic process; specifically, we obtain lower values of steady-state infection density for higher correlation times. Also, for lower correlation times the results approach those obtained in the white noise case. The analytical results are valid for lower-order correlation times. In summary, the results provide insights into the onset of noise-induced, and Turing-type instabilities in a stochastic PDE epidemic model in the presence of strongly nonlinear deterministic infection forces and stochastic environmental forces represented by Ornstein–Uhlenbeck noise.

Unconditional superconvergence analysis of a new energy stable nonconforming BDF2 mixed finite element method for BBM–Burgers equation

The focus of this paper is to establish a new energy stable 2-step backward differentiation formula (BDF2) fully-discrete mixed finite element method (MFEM) for the Benjamin–Bona–Mahony–Burgers (BBM–Burgers) equation with the nonconforming rectangular EQ1rot element and zero-order Raviart–Thomas (R–T) element (EQ1rot/Q10×Q01). Based on the energy stable property, the existence and uniqueness of the numerical solution are proved by the Brouwer fixed point theorem. Subsequently, with the assistance of some typical properties of this element pair and the interpolation post-processing approach, the unconditional superclose and superconvergence results with O(h2+τ2) are obtained directly without any constraints between the spatial partition parameter h and the time step τ. It is worthy to mention that the method and analysis presented herein are very different from the time–space splitting approach utilized in the previous studies and simplify the implement. Finally, two numerical experiments are executed to validate the theoretical analysis.

Redefinition for Fundamental Characteristics of Waterway Traffic Stream Considering Vessel Displacement

As the basis for characterizing traffic conditions in waterways, fundamental characteristics of waterway traffic streams are of great practical significance in ensuring traffic safety and improving navigation efficiency. In the study of the fundamental characteristics of waterway traffic flow, although some scholars consider the length or area of the vessel, few scholars take the displacement of the vessel into account and make light of the influence of the three-dimensional size of the vessel. This paper proposes a method for defining the fundamental characteristics of a waterway traffic stream considering vessel size. This method defines fundamental characteristics in terms of vessel displacement and quantifies flow, density, and speed based on vessel trajectories in time–space regions. This study selects a unidirectional channel in the south trough waters of Shanghai Harbor for a case study and draws the fundamental diagram of the waterway traffic stream while considering vessel number and displacement. The comparison result shows that the definition considering vessel displacement can more accurately reflect the actual traffic condition of the selected channel. Finally, based on the flow–density subgraph of the waterway traffic stream (measured by vessel displacement), this paper constructs a traffic stream model and derives critical parameters. The definition proposed in this study effectively characterizes how vessels occupy the time–space resources of waterways, revealing the inherent mechanisms governing waterway traffic stream and, thus, enhancing accuracy in describing fundamental relationships among waterway stream characteristics. The outcome of this research underlies how the waterway traffic stream is measured, operated, and managed to ensure safety and productivity.

Development of a virtual teaching module for advanced semiconductor fabrication and its learning effectiveness analysis

AbstractSemiconductor fabrication is the process of manufacturing semiconductor devices, typically integrated circuits (ICs) such as microprocessors and memory. This involves transferring circuit diagrams onto a silicon wafer using photomasks and photoresists. After a series of fabrication processes, ICs are created on the wafer surface and then diced into individual chips, which are packaged and tested with quality control procedures to become the final products. Virtual reality (VR) simulates imaginary experiences or environments difficult to achieve in the real world through human senses and immersive equipment, allowing users to interact in a virtual 3D space in real time, making it well suited for applications in science education and industrial training. This study transforms the essential knowledge of advanced semiconductor manufacturing processes into an easily understandable virtual teaching module, thereby creating educational resources for high school and college students. The objective is to enhance their scientific and technological literacy, yielding substantial benefits for the general public. This study utilized VR technology to simplify and clarify the knowledge about the semiconductor manufacturing process, making it more engaging for learners. The virtual teaching module's learning content includes an overview of wafer preparation, semiconductor fabrication, chip packaging, and IC testing. Users can interact with the virtual teaching module and conduct virtual experiments to enhance their understanding by trial and error. Experimental results show that it can improve students' learning achievement and learning motivation. Therefore, the virtual teaching module is suitable for high‐school students and the general public to understand semiconductor technology and its applications. The effectiveness of the virtual teaching module is heavily dependent on the availability and quality of VR hardware and software. Limited access to advanced VR equipment or technical issues could have affected the learning experience, thereby influencing the learning outcomes.

High-Accuracy Positivity-Preserving Finite Difference Approximations of the Chemotaxis Model for Tumor Invasion.

Numerical simulation of the complex evolution process for tumor invasion plays an extremely important role in-depth exploring the bio-taxis phenomena of tumor growth and metastasis. In view of the fact that low-accuracy numerical methods often have large errors and low resolution, very refined grids have to be used if we want to get high-resolution simulating results, which leads to a great deal of computational cost. In this paper, we are committed to developing a class of high-accuracy positivity-preserving finite difference methods to solve the chemotaxis model for tumor invasion. First, two unconditionally stable implicit compact difference schemes for solving the model are proposed; second, the local truncation errors of the new schemes are analyzed, which show that they have second-order accuracy in time and fourth-order accuracy in space; third, based on the proposed schemes, the high-accuracy numerical integration idea of binary functions is employed to structure a linear compact weighting formula that guarantees fourth-order accuracy and nonnegative, and then a positivity-preserving and time-marching algorithm is established; and finally, the accuracy, stability, and positivity-preserving of the proposed methods are verified by several numerical experiments, and the evolution phenomena of tumor invasion over time are numerically simulated and analyzed.

A boundary-type algorithm based on the discrete ordinates for neutron transport equation

The spatial distribution of neutron flux in the core of a nuclear reactor plays a crucial role in ensuring nuclear safety. It can be determined by solving the neutron transport equation, where computational resource consumption is gradually receiving more attention, especially in problems with multiple reflective boundary conditions. It is common practice to assume initial boundary values in one angular direction to obtain a global solution, and then use reflective boundary conditions to solve for the other directions, iterating until convergence. However, this approach has the drawback of requiring the global solution, and each iteration necessitates storing the neutron flux values, consuming time and storage space. In order to address this issue, this paper proposes a precise and efficient boundary-type method, the Half Boundary Method (HBM). This method establishes relationships between adjacent node values through integration and mathematical deduction, ultimately obtaining the relationship among any node and the boundary values in any direction. As a result, iteration is only required for boundary values, reducing the iteration amount and thus saving time and storage space. Using the discrete ordinate (SN) method for angular discretization and HBM for spatial discretization, this paper solved the steady-state neutron transport equation for two-dimensional systems modeled with Cartesian geometry. The method is validated using several benchmark problems, including the 2D ISSA benchmark, the 2D mono-group k-eigenvalue problem and BWR rod bundle test problem. All of these problems demonstrate good results and effectively reduced computational time.

Numerical simulation of an effective transform mechanism with convergence analysis of the fractional diffusion-wave equations

In the current study, we solve two very important mathematical models, such as the time fractional-order space-fractional telegraph and diffusion-wave equations using a reliable technique called the Adomian decomposition natural method (ADNM), which combines Adomian decomposition and natural transform. The diffusion wave equation describes the flood wave propagation, which is used in solving overland and open channel flow problems. For this reason, it is critical to fully understand and effectively solve the diffusion wave equations. Because telegraph equations are crucial for modeling and developing voltage or frequency transmission, they are widely used in physics and engineering. Furthermore, the designing process is greatly impacted by the uncertainty in the system parameters. For nonlinear ordinary differential equations based on the theorem of Banach fixed point, we provide existence and uniqueness theorem proofs. The present approach has been successfully used to explore exact solutions for time fractional-order and space fractional-order applications. The results show how effective and valuable the ADNM. This paper presents a methodology that will be used in future work to address similar nonlinear problems related to fractional calculus.

A new higher-order super compact finite difference scheme to study three-dimensional non-Newtonian flows

This work introduces a new higher-order accurate super compact (HOSC) finite difference scheme for solving complex unsteady three-dimensional (3D) non-Newtonian fluid flow problems. As per the author's knowledge, the proposed scheme is the first ever developed higher-order compact finite difference scheme to solve 3D non-Newtonian flow problems. The proposed scheme is fourth-order accurate in space and second-order accurate in time, utilizing only seven adjacent grid points at the (n+1)th time level for the finite difference discretization. A time-marching methodology is employed with pressure calculated via a pressure-correction strategy based on the modified artificial compressibility method. Using the power-law viscosity model, we tackle the benchmark problem of a 3D lid-driven cavity, systematically analyzing the varied rheological behavior of shear-thinning (n = 0.5), shear-thickening (n = 1.5), and Newtonian (n = 1.0) fluids across different Reynolds numbers (Re=1,50,100,200). Both Newtonian and non-Newtonian results are carefully investigated in terms of streamlines, velocity variation, pressure distributions, and viscosity contours, and the computed results are validated with the existing benchmark results. The findings demonstrate excellent agreement with the existing results. It is found that for shear-thinning fluid (n = 0.5), u velocity is higher near the top moving wall than the case of Newtonian (n = 1.0) and shear-thickening fluid (n = 1.5) for all Re values. This extensive analysis, using the new HOSC scheme, not only increases our understanding of non-Newtonian fluid behavior but also provides a robust foundation for future research and practical applications.

Implementation of a peer-delivered opioid overdose response initiative in New York City emergency departments: Insight from multi-stakeholder qualitative interviews

BackgroundEmergency departments (EDs) are critical touchpoints for overdose prevention efforts. In New York City (NYC), the Health Department's Relay initiative dispatches trained peer “Wellness Advocates” (WAs) to engage with patients in EDs after an overdose and for up to 90 days subsequently. Interest in peer-delivered interventions for patients at risk for overdose has grown nationally, but few studies have explored challenges and opportunities related to implementing such interventions in EDs. MethodsWe conducted in-depth interviews with Relay WAs, ED patients, and ED providers across 4 diverse NYC EDs. Sampling was purposeful and continued until theoretical saturation was reached. Interviews followed a semi-structured interview guide based on key domains from the Consolidated Framework for Implementation Research (CFIR). Interviews were conducted by telephone or web conferencing; audio recordings were professionally transcribed. The study utilized rapid qualitative analysis using template summaries and summary matrices followed by line-by-line coding conducted independently by 3 researchers, then discussed and harmonized at group coding meetings. Coding was both inductive (using an a priori code list based on CFIR domains and study goals) and deductive (new codes allowed to emerge from transcripts). Dedoose software was used for data organization. ResultsWe conducted 32 in-depth interviews (10 WAs, 12 patients, 10 ED providers). Four overarching themes emerged: 1) EDs are characterized by multiple competing demands (e.g., related to provider time and physical space), underscoring the utility of Relay and leading to some practical challenges for its delivery; 2) There is a strong role distinction of WAs as peers with lived experience; 3) ED providers value Relay, even though they have a limited understanding of its full scope and outcomes; 4) While the role of structural factors (e.g., homelessness and unstable housing) is recognized, responsibility is often placed on patients for controlling their own success. ConclusionsWe identified four themes that shed new light on the implementation of peer-based overdose prevention programs in EDs. Our findings highlight unique ED inner and outer setting factors that may impact program implementation and effectiveness. The findings provide actionable information to inform implementation of similar programs nationally.

Reducing neural network complexity via optimization algorithms for fault diagnosis in renewable energy systems

This article discusses the challenges involved in diagnosing faults in renewable energy systems. A significant challenge is the high computational demands of the artificial intelligence algorithms needed for grid-connected photovoltaic (GCPV) and wind energy conversion (WEC) systems in fault diagnosis. To address this issue, several methods are proposed to reduce computation time, minimize memory requirements, and improve efficiency. First, optimization algorithms such as the Salp Swarm Algorithm (SSA), Genetic Algorithm (GA), and Particle Swarm Optimization (PSO), combined with machine learning classifiers for feature selection, are suggested to reduce computational time and memory space requirements. This approach notably decreases computation time but has a limited impact on memory space. Secondly, further reduction in memory space requirements is recommended by using the variogram method for data reduction. This method can leverage the outputs of preceding algorithms to optimize renewable energy systems, making their operations cost-effective and efficient. Finally, the outputs of the variogram are used to train neural network (NN), recurrent neural network (RNN), and long short-term memory (LSTM) classifiers to differentiate between various modes of operation in GCPV and WEC systems. Experimental results demonstrate the effectiveness and robustness of the proposed methods, showing that memory space can be reduced by 1.3 to 5.6 times and CPU time by 1.1 to 11 times while maintaining accuracy and improving efficiency compared to conventional algorithms.

Adaptive mesh extended cubic B-spline method for singularly perturbed delay Sobolev problems

The purpose of this paper is to develop a robust numerical scheme for a class of singularly perturbed delay Sobolev (pseudo-parabolic) problems that have wide application in various branches of mathematical physics and fluid mechanics.For the small perturbation parameter, the standard numerical schemes for the solution of these problems fail to resolve the boundary layer(s) and the oscillations occur near the boundary layer. Thus, in this paper to resolve the boundary layer(s), im-plicit Euler scheme for the time derivatives on uniform mesh and extended B-spline basis functions consisting of free parameter λ are presented for spatial variable on Bakhvalov type mesh. The stability and uniform convergence analyisis of the proposed method are established. The error estimation of the developed method is shown to be firts order accurate in time and second order accurate in space. Numerical exprementation is carried out to validate the applicability of the developednumerical method. The numerical results reveals that the computational result is in agreement with the theoretical estimations

Dynamic spatial–temporal conflict quantification of construction machinery for pouring blocks in arch dams

PurposeThe spatial–temporal conflicts in the construction process of concrete arch dams are related to the construction quality and duration, especially for pouring blocks with a continuous high-strength and high-density construction process. Furthermore, the complicated construction technology and limited space resources aggravate the spatial–temporal conflicts in the process of space resource allocation and utilization, directly affecting the pouring quality and progress of concrete. To promote the high-strength, quality-preserving and rapid construction of dams and to clarify the explosion moment and influence degree of the spatial–temporal conflicts of construction machinery during the pouring process, a quantification method and algorithm for a “Conflict Bubble” (CB) between construction machines is proposed based on the “Time–Space Microelement” (TSM).Design/methodology/approachFirst, the concept of a CB is proposed, which is defined as the spatial overlap of different entities in the movement process. The subsidiary space of the entity is divided into three layered spaces: the physical space, safe space and efficiency space from the inside to the outside. Second, the processes of “creation,” “transition” and “disappearance” of the CB at different levels with the movement of the entity are defined as the evolution of the spatial–temporal state of the entity. The mapping relationship between the spatial variation and the running time of the layered space during the movement process is defined as “Time–Space” (TS), which is intended to be processed by a microelement.FindingsThe quantification method and algorithm of the CB between construction machinery are proposed based on the TSM, which realizes the quantification of the physical collision accident rate, security risk rate and efficiency loss rate of the construction machinery at any time point or time period. The risk rate of spatial–temporal conflicts in the construction process was calculated, and the outbreak condition of spatial–temporal conflict in the pouring process was simulated and rehearsed. The quantitative calculation results show that the physical collision accident rate, security risk rate and efficiency loss rate of construction machinery at any time point or time period can be quantified.Originality/valueThis study provides theoretical support for the quantitative evaluation and analysis of the spatial–temporal conflict risk in the pouring construction process. It also serves as a reference for the rational organization and scientific decision-making for pouring blocks and provides new ideas and methods for the safe and efficient construction and the scientific and refined management of dams.

A joint time–space extrapolation approach within the Wiener path integral technique for efficient stochastic response determination of nonlinear systems

A joint time–space extrapolation approach within the Wiener path integral (WPI) technique is developed for determining, efficiently and accurately, the non-stationary stochastic response of diverse nonlinear dynamical systems. The approach can be construed as an extension of a recently developed space-domain extrapolation scheme to account also for the temporal dimension. Specifically, based on a variational principle, the WPI technique yields a boundary value problem (BVP) to be solved for determining a most probable path corresponding to specific final boundary conditions. Further, the most probable path is used for evaluating, approximately, a point of the system response joint probability density function (PDF) corresponding to a specific time instant. Remarkably, the BVP exhibits two unique features that are exploited in this paper for developing an efficient joint time–space extrapolation approach. First, the BVPs corresponding to two neighboring grid points in the spatial domain of the response PDF not only share the same equations, but also the boundary conditions differ only slightly. Second, information inherent in the time-history of an already determined most probable path can be used for evaluating points of the response PDF corresponding to arbitrary time instants, without the need for solving additional BVPs. In a nutshell, relying on the aforementioned unique and advantageous features of the WPI-based BVP, the complete non-stationary response joint PDF is determined, first, by calculating numerically a relatively small number of PDF points, and second, by extrapolating in the joint time–space domain at practically zero additional computational cost. Compared to a standard brute-force implementation of the WPI technique, the developed extrapolation approach reduces the associated computational cost by several orders of magnitude. Two numerical examples relating to an oscillator with asymmetric nonlinearities and fractional derivative elements, and to a nonlinear structure under combined stochastic and deterministic periodic loading are considered for demonstrating the reliability of the extrapolation approach. Juxtapositions with pertinent Monte Carlo simulation data are included as well.