Ritu Kumari | Searcharxiv

arXiv2026

Fokker--Planck Dynamics on Star Graphs with Variable Drift: Well-Posedness, Adjoint Analysis, and Numerical Approximation

Stochastic transport processes on networked domains (modelled on metric graphs) arise in a variety of applications where diffusion and drift mechanisms interact with an underlying graph structure. The Fokker--Planck equation provides a natural framework for describing the evolution of probability densities associated with such dynamics. While Fokker--Planck equations on metric graphs have been studied from an analytical viewpoint, their optimal control remains largely unexplored, particularly in settings where the control acts through the drift term. In this paper, we investigate an optimal control problem governed by the Fokker--Planck equation on a star graph, with a bilinear control appearing in the drift. We establish the well-posedness of the state equation and prove the existence of at least one optimal control. The associated adjoint system is derived, and first-order necessary optimality conditions are formulated. A wavelet-based numerical scheme is proposed to approximate the optimal solution, and its performance is illustrated through representative numerical experiments. These results contribute to the analytical and computational understanding of controlled stochastic dynamics on network-like domains.

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arXiv2025

Fractional Quadrature rule and using its Exactness for the Müntz-Legendre Scaling Functions for Solving Fractional Differential Equations

Fractional operators (derivatives/integrals) are defined via the integration of the functions. When the function is produced by a spanning set of fractional power functions, traditional quadrature rules often need to be revised, failing to provide exact evaluations for fractional power functions and thus introducing approximation errors. In this paper, we have formulated a fractional quadrature rule that achieves exact integration for functions within this specific set to address this issue. Some properties of the fractional quadrature rule have been proved, and the absolute error bound in the proposed fractional quadrature rule has been derived. The behavior of roots of the orthogonal Müntz polynomial has also been observed for its application as nodes in the fractional quadrature rule. To illustrate the effectiveness of the newly proposed fractional quadrature rule, we focus on fractional differential equations that incorporate the left Caputo fractional derivative. In this context, Müntz-Legendre scaling functions are utilized to approximate the Caputo derivative of functions involved in these equations. Additionally, we have derived an operational matrix for Riemann-Liouville integration to approximate the respective functions with the help of the fractional quadrature rule. To demonstrate the practical utility of our method, we provide illustrative examples that compare the $L_2$-error estimates in the solutions of fractional differential equations using our approach against those obtained with the Block-pulse method. These comparisons underscore the superior accuracy of our proposed method.

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