This study develops and analyzes a nonlinear mathematical model to describe the dynamics of marriage, extramarital affairs, marital breakdown, and divorce. The population is divided into five compartments: single $S(t)$, married $M(t)$, individuals in extramarital affairs $A(t)$, broken marriages $B(t)$, and divorced $D(t)$. The model is formulated as a system of nonlinear ordinary differential equations capturing transitions between these states through parameters representing marriage formation, affair initiation, reconciliation, separation, and divorce. Basic qualitative properties, including positivity and boundedness of solutions, are established to ensure the model is well posed. Equil... More >

Rapid social transformation in metropolitan regions has significantly reshaped crime dynamics, particularly in large urban centers such as Delhi. Understanding whether crime escalation reflects structural social change or random fluctuation is crucial for policy formulation. This study investigates the relationship between social transformation and crime escalation in Delhi within the broader national context of India over the period 2010-2020. The analysis employs a quantitative time-series and econometric framework, including trend analysis, regression modeling, correlation, elasticity estimation, and structural diagnostics. Secondary data are obtained from the National Crime Records Burea... More >

This study explores the application of the Adomian Decomposition Method (ADM), combined with the Laplace transform, to obtain approximate solutions of the time--fractional Wazwaz--Benjamin--Bona--Mahony (WBBM) equations. These equations, which describe wave phenomena in fluid dynamics within a fractional--time framework, present significant analytical challenges. By integrating the Laplace transform with ADM, a modified technique---referred to as the Laplace Transform Adomian Decomposition Method (LTADM)---is introduced. The time--fractional WBBM equations are solved using LTADM, and three--dimensional solution plots are generated for six different values of the fractional order \( \delta \)... More >

A unified fractional simplest equation framework is developed for nonlinear power-law wave equations whose travelling-wave reductions lead to $U'' + \lambda U^p = 0$, $p>1$. In contrast to classical simplest equation methods restricted to integer polynomial expansions, we introduce a fractional power ansatz $U(\xi)=A\phi^\alpha(\xi)$ derived from a dominant balance principle. A general scaling theorem is established, proving that the admissible travelling-wave exponents are uniquely determined by $\alpha(p-1)=2$, yielding the explicit fractional law $\alpha=\frac{2}{p-1}$. Consequently, the equation admits algebraic travelling waves of the universal form $U(\xi)=A(\mu\xi+C_0)^{-2/(p-1)}$, wi... More >

A nonlinear compartmental model is developed to analyze crime dynamics in a structured society. The population is stratified into eight compartments: \(S(t)\) (susceptible), \(E(t)\) (exposed), \(C(t)\) (active criminals), \(C_v(t)\) (convicted criminals), \(P_h(t)\) (passive-honest), \(P_c(t)\) (committed-honest), \(J_h(t)\) (honest judges), and \(J_c(t)\) (corrupt judges). The model incorporates nonlinear mechanisms such as institutional corruption (\(\kappa_1\)), judicial correction (\(\kappa_2\)), recidivism feedback (\(\rho_1,\rho_2\)), exposure intensity (\(\eta_1\)), and rehabilitation (\(r_2\)), providing a realistic portrayal of crime--justice interactions. Solutions remain positive... More >