Randhir Singh

Gurmeet Kaur, Randhir Singh, Mehakpreet Singh et al.

In the present work, a new approach is proposed for finding the analytical solution of population balances. This approach is relying on idea of Homotopy Perturbation Method (HPM). The HPM solves both linear and nonlinear initial and boundary value problems without nonphysical restrictive assumptions such as linearization and discretization. It gives the solution in the form of series with easily computable solution components. The outcome of this study reveals that the proposed method can avoid numerical stability problems which often characterize in general numerical techniques related to this area. Several examples including Austin's kernel available in literature are examined to demonstrate the accuracy and applicability of the proposed method.

Randhir Singh

In this work, we apply Adomian decomposition method for solving nonlinear derivative-dependent doubly singular boundary value problems: $(py')'= qf(x,y,y')$. This method is based on the modification of ADM and new two-fold integral operator. The approximate solution is obtained in the form of series with easily determinable components. The effectiveness of the proposed approach is examined by considering three examples and numerical results are compared with known results.

Randhir Singh

In this paper, we present the optimal homotopy analysis method (OHAM) with Green's function technique to acquire accurate numerical solutions for the nonlocal elliptic problems. We first transform the nonlocal boundary value problems into an equivalent integral equation, and then use an OHAM with convergence control parameter $c_0$. To demonstrate convergence and accuracy characteristics of the OHAM method, we compare the OHAM and Adomian decomposition method (ADM) with Green's function. The numerical experiments confirm the reliability of the approach as it handles such nonlocal elliptic differential equations without imposing limiting assumptions that could change the physical structure of the solution. We also discuss the convergence and error analysis of proposed method. In summary: $(i)$ the present approach does not require any additional computational work for unknown constants unlike ADM and VIM \cite{khuri2014variational} $(ii)$ guarantee of convergence $(iii)$ flexibility on choice of initial guess of solution and $(iv)$ useful analytic tool to investigate a class of nonlocal elliptic boundary value problems.

Randhir Singh, Himanshu Gargyand, Apoorv Garg

The Haar wavelet based quasilinearization technique for solving a general class of singular boundary value problems is proposed. Quasilinearization technique is used to linearize nonlinear singular problem. Second rate of convergence is obtained of a sequence of linear singular problems. Numerical solution of linear singular prob- lems is obtained by Haar-wavelet method. In each iteration of quasilinearization technique, the numerical solution is updated by the Haar wavelet method. Conver- gence analysis of Haar wavelet method is discussed. The results are compared with the results obtained by the other technique and with exact solution. Eight singular problems are solved to show the applicability of the Haar wavelet quasilinearization technique.

Randhir Singh

In this paper, we consider the nonlinear doubly singular boundary value problems $(p(x)y'(x))'+ q(x)f(x,y(x))=0,~0<x<1$ with Dirichlet/Neumann boundary conditions at $x=0$ and Robin type boundary conditions at $x=1$. Due to the presence of singularity at $x=0$ as well as discontinuity of $q(x)$ at $x=0$, these problems pose difficulties in obtaining their solutions. In this paper, a new formulation of the singular boundary value problems is presented. To overcome the singular behavior at the origin, with the help of Green's function theory the problem is transformed into an equivalent Fredholm integral equation. Then the optimal homotopy analysis method is applied to solve integral form of problem. The optimal control-convergence parameter involved in the components of the series solution is obtained by minimizing the squared residual error equation. For speed up the calculations, the discrete averaged residual error is used to obtain optimal value of the adjustable parameter $c_0$ to control the convergence of solution. The proposed method \textbf{(a)} avoids solving a sequence of transcendental equations for the undetermined coefficients \textbf{(b)} it is a general method \textbf{(c)} contains a parameter $c_0$ to control the convergence of solution. Convergence analysis and error estimate of the proposed method are discussed. Accuracy, applicability and generality of the present method is examined by solving five singular problems.

Arya Chakraborty, Randhir Singh

Conventional momentum strategies, despite their proven efficacy in generating alpha, frequently suffer from the "Winner's Curse", a structural vulnerability in which high performing assets exhibit clustered volatility and severe drawdowns during market reversals. To counteract this propensity for momentum crashes, this study presents the Adaptive Equity Generation and Immunisation System (AEGIS), a novel framework that fundamentally reengineers the trade-off between growth and stability. By leveraging a volatility-adjusted momentum filter to identify trend strength and employing a minimax correlation algorithm to enforce structural diversification, the model utilises sequential least squares programming (SLSQP) to optimise capital allocation for the sortino ratio. This architecture allows the portfolio to dynamically adapt to distinct market regimes: explicitly lowering the intensity of crashes during bear markets by decoupling correlated risks, while retaining asymmetric upside participation during bull runs. Empirical validation via a comprehensive 20-year walk-forward backtest (2006-2025), which covers significant stress events like the 2008 Global Financial Crisis, confirms that the framework produces substantial excess alpha relative to the standard S&P 500 benchmark. Notably, the strategy successfully matched the capital appreciation of the high-beta NASDAQ-100 index while achieving significantly reduced downside volatility and improved structural resilience. These results suggest that synthetic beta can be effectively engineered through mathematical regularisation, enabling investors to capture the high-growth characteristics of concentrated portfolios while preserving the defensive stability typically associated with broad-market diversification.