IntroductionWe study a time-fractional diffusion equation with Caputo memory on the real line and develop a self-contained analytical-computational benchmark framework.MethodsThe solution is obtained using the Laplace transform in time and the Fourier transform in space, leading to Mittag–Leffler relaxation factors. The Green kernel is expressed through the M-Wright function and a Mellin–Barnes contour integral with explicit assumptions, contour conventions, and branch choices. A continuous L2-type stability estimate is derived, and a smooth manufactured solution is used to verify a semi-implicit fractional time-stepping method on a truncated computational interval.ResultsThe analytical validation recovers the established M-Wright kernel, the Gaussian limit, mass conservation, and the subdiffusive second moment. The numerical tests report final-time errors, residual diagnostics, separated spatial and temporal convergence behavior, and sensitivity with respect to the fractional order.DiscussionThe framework organizes classical transform and special-function representations together with explicit assumptions, stability bounds, and a reproducible manufactured-solution benchmark. It can serve as a reference problem for finite-difference, spectral, collocation, physics-informed neural-network, and hybrid solvers when those methods are implemented and assessed independently.
Understanding temporal changes in betting-related behavior is important for characterizing digital engagement with gambling in Ghana, yet direct monthly measures of population-level betting participation remain unavailable. This study therefore investigates betting-search salience using a preserved 72-month Google Trends composite spanning January 2020 to December 2025 and complements the empirical analysis with conditional SEIR-type behavioral scenarios. Six parsimonious autoregressive specifications were evaluated using an expanding validation period covering 2023–2024, with model selection fixed before an independent 2025 test period. The selected ARX2 specification achieved an in-sample (R2 = 0.690) and a validation RMSE of 0.1031 on the logarithmic scale. However, its 2025 RMSE of 0.1276 exceeded that of a no-change benchmark (0.1202), indicating limited out-of-sample predictive improvement and evidence of temporal instability. Model coefficients are therefore interpreted as conditional associations with search salience rather than estimates of betting prevalence or causal behavioral effects. The scenario analysis treats behavioral transition rates, initial state proportions, and the bounded mapping from fitted search salience to contact-rate trajectories as explicit assumptions rather than estimated Ghanaian quantities. Across most sensitivity configurations, enhanced-exit and combined scenarios reduced peak and cumulative active occupancy, although the magnitude of these effects varied substantially with assumed transition rates and initial conditions. The findings provide a cautious characterization of betting-search dynamics and their predictive limitations while avoiding unsupported inference about population-level betting prevalence or causal policy effects. More broadly, the study identifies the behavioral measurements required to move from digital search signals toward empirically calibrated latent-state models and quantitative evaluation of intervention effects in Ghana.
Islamic financing risk is frequently analyzed as a single undifferentiated category, despite substantial heterogeneity in how ownership, control, payment obligation, and loss are allocated across financing modes. This paper develops a contract-level risk-mitigation framework across ten Islamic financing instruments and empirically tests a hypothesis motivated by that framework: whether regulatory capital adequacy (CAR) dampens the relationship between profit-and-loss-sharing (PLS) financing share and non-performing financing (NPF) at the country-banking-system level, using a quarterly six-country panel drawn from the Islamic Financial Services Board’s Prudential and Structural Islamic Financial Indicators (PSIFI) database (256 country-quarter observations, 2014–2024). After confirming non-stationarity in the levels data via Augmented Dickey-Fuller tests and re-specifying the model in first differences with entity and time fixed effects and Driscoll-Kraay standard errors, we find no statistically significant CAR × PLS interaction effect on credit risk across three specifications and five robustness checks (including annual GDP-growth and inflation controls), the point estimate’s sign is inconsistent across specifications, and the models explain little of the within-country variation in credit risk (within R2 between 0.02 and 0.06). The confidence intervals are wide and span both zero and economically meaningful effects of either sign, so the results reflect limited estimation precision at the country-banking-system level: they cannot distinguish a true effect close to zero from a modest effect estimated imprecisely. The paper’s contribution is therefore twofold: a reusable conceptual framework for contract-specific Islamic finance risk analysis, and a methodologically transparent demonstration of the limits of country-level macro panels for this class of question.
Probability distribution modeling for bounded data provides a probabilistic representation of real-world processes. However, data bounded within the unit interval pose statistical challenges due to asymmetry and boundary point masses. This study investigates the suitability of probability distributions for modeling the proportion of visitors engaged in physical–recreational activities (PRAs) in parks of Pune and develops a framework for fitting and assessing these distributions. Classical Gaussian and non-Gaussian distributions were fitted to the proportion of PRA from 88 urban parks of Pune. These models were evaluated using information criteria (Akaike Information Criterion [AIC]/Corrected Akaike Information Criterion [AICC]/Bayesian Information Criterion [BIC]), Graphical diagnostics, and a parametric bootstrap Kolmogorov–Smirnov (K–S) goodness-of-fit test. The Zero–One Inflated Beta (ZOIB) distribution provided the most suitable representation of the PRA proportions among all the selected distributions, enabling a parametric characterization through its shape parameters and moments. ZOIB regression and likelihood ratio (LR) test results (LR = 12.1444, p = 0.734) indicated no statistically significant impact of zones on PRA proportions, demonstrating consistent park usage pattern across zones. The fitted ZOIB results showed that overall, 67% of visitors were engaged in PRA. It highlighted park visitors’ behavior toward PRA and hence the importance of incorporating demand for physical–recreational activities into urban park infrastructure planning and management. This evidence supports planning for inclusive and accessible urban green spaces (public parks), contributing to Sustainable Development Goal (SDG) 11, particularly Target 11.7 (inclusive and equitable access to green spaces).
